Differential Geometry through Supersymmetric Glasses 9811206775, 9789811206771

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11457_9789811206771_tp.indd 1

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b2530   International Strategic Relations and China’s National Security: World at the Crossroads

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Control Number: 2020023691 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

DIFFERENTIAL GEOMETRY THROUGH SUPERSYMMETRIC GLASSES Copyright © 2020 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-120-677-1 (hardcover) ISBN 978-981-120-678-8 (ebook for institutions) ISBN 978-981-120-679-5 (ebook for individuals)

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Contents

Introduction

1

GEOMETRY

9

1.

Real Manifolds 1.1 1.2 1.3 1.4

2.

Riemannian Geometry . . . . . . . Differential Forms . . . . . . . . . . Principal Fiber Bundles and Gauge Vielbeins and Tangent Space . . .

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Complex Manifolds 2.1 2.2 2.3

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11

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Complex Description . . . . 2.1.1 Complex differential Real Description . . . . . . Examples . . . . . . . . . .

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Hyper-K¨ ahler and HKT Manifolds 3.1 3.2

Hyper-K¨ ahler Manifolds 3.1.1 Examples . . . . HKT Manifolds . . . . . 3.2.1 Examples . . . .

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29 36 39 47 51

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PHYSICS 4.

11 18 22 24

51 57 61 67

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Dynamical Systems with and without Grassmann Variables

79

4.1

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Ordinary Classical Mechanics . . . . . . . . . . . . . . . . v

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7.4

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Basic Definitions . . . . . . . Supersymmetric Oscillator . . Electrons in a Magnetic Field Witten’s Model . . . . . . . . Extended Supersymmetry . . Classical Supersymmetry . . .

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134 138 144 145 147 150

159 161

Basic Structures . . . . . . . . . . . . . . . . . . . . . . . 161 Euler Characteristic . . . . . . . . . . . . . . . . . . . . . 168 Deformations  Morse Theory  Quasitorsions . . . . . . 171

Supersymmetric Description of the Dolbeault Complex 9.1

115 121 125 129 133

Supersymmetric Description of the de Rham Complex 8.1 8.2 8.3

97 99 101 105 108 109 115

Path Integrals in Quantum Mechanics Grassmann Evolution Kernel . . . . . Witten Index . . . . . . . . . . . . . . 6.3.1 Examples . . . . . . . . . . . .

N =1 . . . . . . . . . . . . . N =2 . . . . . . . . . . . . . N =4 . . . . . . . . . . . . . 7.3.1 (2, 4, 2) . . . . . . . 7.3.2 (3, 4, 1) and (4, 4, 0) Harmonic Superspace . . . . .

86 88 91 93 97

SYNTHESIS

9.

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Superspace and Superfields 7.1 7.2 7.3

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Path Integrals and the Witten Index 6.1 6.2 6.3

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4.1.1 Algebra of symmetries Ordinary Quantum Mechanics Grassmann Variables . . . . . Grassmann Dynamics . . . .

Supersymmetry 5.1 5.2 5.3 5.4 5.5 5.6

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4.2 4.3 4.4 5.

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N = 2 Superfield Description . . . . . . . . . . . . . . . . 177 9.1.1 Holomorphic torsions . . . . . . . . . . . . . . . . 183

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Contents

9.2 9.3

vii

N = 1 Superfield Description . . . . . . . . . . . . . . . . 184 Dolbeault and Dirac Complexes on S 4 \{·} . . . . . . . . . 192

10. Sigma Models with Extended Supersymmetries

197

10.1 K¨ ahler Manifolds and Around . . . . . . . . . . . 10.1.1 Quasicomplex K¨ahler models . . . . . . . 10.1.2 Bi-K¨ ahler models . . . . . . . . . . . . . 10.2 Symplectic Sigma Models . . . . . . . . . . . . . 10.2.1 Bi-symplectic models . . . . . . . . . . . 10.3 HKT and bi-HKT Models . . . . . . . . . . . . . 10.3.1 Bi-HKT models . . . . . . . . . . . . . . 10.4 N = 8 Models . . . . . . . . . . . . . . . . . . . . 10.4.1 Hyper-K¨ ahler models . . . . . . . . . . . 10.4.2 N = 8 supersymmetric bi-K¨ahler models 10.4.3 N = 8 bi-HKT models . . . . . . . . . . 10.4.4 Symplectic model of the second kind . . .

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11. Taming the Zoo of Models

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11.1 Similarity Transformations . . . . . . 11.2 Hamiltonian Reduction . . . . . . . . 11.2.1 A toy model . . . . . . . . . 11.2.2 (2, 2, 0) −→ (1, 2, 1) . . . 11.2.3 (4, 4, 0) −→ (2, 4, 2) . . . 11.2.4 Symplectic and other models

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12. HK and HKT through Harmonic Glasses 12.1 12.2 12.3 12.4

The Hyper-K¨ ahler Model . . . . . The HKT Model . . . . . . . . . . Metric  Obata Families  HKT → Examples . . . . . . . . . . . . . . 12.4.1 Taub-NUT metric . . . . . 12.4.2 Delduc-Valent metric . . . 12.5 Gauge Fields . . . . . . . . . . . .

13. Gauge Fields on the Manifolds

197 203 204 207 211 211 217 220 220 223 224 226

229 234 234 237 239 242 245

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246 252 259 267 267 270 272 275

13.1 Spinors  Dirac Operator . . . . . . . . . . . . . . . . . . 275 13.1.1 Dirac operator and the supercharges . . . . . . . . 278 13.2 Magnetic Field on S 2 . . . . . . . . . . . . . . . . . . . . . 283

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13.2.1 Fractional magnetic charge . . . . . . . . . 13.3 Line Bundles on CPn . . . . . . . . . . . . . . . . . 13.4 Non-Abelian Fields  Instantons on S 4 . . . . . . . 13.4.1 Non-Abelian bundles . . . . . . . . . . . . 13.4.2 Self-dual fields  Semi-dymanical variables 13.4.3 Explicit solutions . . . . . . . . . . . . . . 13.5 KLW Theorem and its HKT Generalization . . . .

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14. Atiyah-Singer Theorem 14.1 Generalities . . . . . . . . . . . . 14.2 Hirzebruch Signature . . . . . . . 14.3 Dirac Index . . . . . . . . . . . . 14.3.1 K¨ ahler manifolds . . . . . 14.3.2 Non-K¨ ahler manifolds . . 14.4 Functional Integral Calculation . 14.4.1 Non-Abelian Dirac index

289 292 294 294 296 301 302 305

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305 308 310 310 315 317 321

Bibliography

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Index

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Introduction

In 1982, Edward Witten published his seminal paper [1] where he showed that some classical problems of differential geometry and differential topology such as the de Rham complex and Morse theory can be described in a very simple and transparent way using the language of supersymmetric quantum mechanics. However, according to what I could judge from my conversations with mathematicians, the language of supersymmetry has not become yet a common working tool in this branch of mathematics. The experts working in this field prefer to use more traditional methods. This is a pity, because supersymmetry does not only allow to describe in a simple way the results known before, it allows to derive many new mathematical results. And this is the raison d’ˆetre for this book. I will try to explain the supersymmetry formalism in a way understandable both for physicists and mathematicians and will use it to derive both old and new mathematical results for differential geometry of manifolds. One can make here a general remark. The relationship of mathematics and theoretical physics is obvious. One can recall that geometry was first developed in ancient Babylon as an applied discipline to serve the needs of farmers and tax officials. Only later people realized that geometry is not only useful, but also beautiful and began to play with geometric problems for their own sake. The dialog between pure maths and theoretical physics is mutually beneficial—there is no need to list here a vast number of examples justifying this point. Unfortunately, this dialog is now often hampered by a difference in scientific languages used by the two communities. It is difficult for a physicist to understand a paper written by a pure mathematician, even if it is written on a subject that s/he knows fairly well, because it is expressed in a different form, to which s/he is not accustomed. On the other hand, it is

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difficult for a mathematician to understand papers written by physicists. This was not always so. Such a rift did not exist 200 years ago, while 100 years ago it was already there, but not so deep. It deepened in the middle of the last century after Nicolas Bourbaki published an influential series of monographs in different branches of mathematics written in an extremely austere formal manner. Like it or not, but this rift is now a fact of life that one cannot ignore. Going back to our book, we have written it in a slang representing a mixture of the two languages. We did so in a hope that this slang will be understandable to both communities. More precisely: (1) When studying geometry of a manifold, we are interested in its invariant properties that do not depend on the choice of coordinates. It is possible (and mathematicians do so) to formulate the statements of Riemannian geometry in an austere way without any use of the coordinates. We find it, however, convenient to use in our description some coordinate parametrization of a manifold (bearing in mind, of course, that many different such parametrizations are possible). In particular, a vector field on the manifold of dimension D will be understood simply as a set of D functions V N of the coordinates xM that transform under general coordinate transformations in an appropriate way. By affine connection we will understand the object GP M N carrying three indices, which enters the definition of covariant derivative. GP M N may be symmetric under the interchange M ↔ N if there is no torsion, or not symmetric if the torsion is present. (2) The author is a physicist and speaks maths with a heavy accent. Still, as the subject of the book is mathematical, I will try to organize the book in a mathematical way, including theorems, lemmas, etc. (But heuristic reasoning and clarifying examples will also play an essential role.) I will do so even when describing purely physical issues such as classical mechanics. All the notions (except the most elementary) will be defined in this book, so that both the physicists and mathematicians will be able to understand what am I talking about. A warning: these definitions will inevitably have a distinct physical flavour. And the proofs of the theorems that we give will not be incorrect, but they will lack an austere mathematical rigour. First of all, not being a mathematician, I am simply not able to provide it and, second, our goal is to be understandable to both communities.

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The basic idea of Witten was the following. Consider a smooth manifold of dimension D parametrized by the coordinates xM . Consider the classical de Rham complex, i.e. the set of all p-forms, α(p) = αM1 ...Mp (xN ) dxM1 ∧ · · · ∧ dxMp ,

(0.1)

p = 0, . . . , D. One defines the operator of exterior derivative d: dα = ∂N αM1 ...Mp dxN ∧ dxM1 ∧ · · · ∧ dxMp

(0.2)

and the conjugate operator d† that acts on p-forms as (−1)pD+D+1 ? d?, where ? is the duality operator (see Chap. 1 for further details). The operators d and d† are nilpotent, while the anticommutator {d, d† } is a certain second-order differential operator. In fact, it is the Laplace-Beltrami operator 4, generalized to operate on differential forms. One can now observe that the algebra d2 = (d† )2 = 0,

{d, d† } = −4

(0.3)

is isomorphic to the simplest supersymmetry algebra ˆ 2 = (Q ˆ † )2 = 0, Q

ˆ Q ˆ†} = H ˆ, {Q,

(0.4)

ˆ and Q ˆ † are Hermitially conjugate supercharges and H ˆ is the Hamilwhere Q † ˆ ˆ ˆ ˆ tonian. It follows that [Q, H] = [Q , H] = 0, so that a system involves one complex or two real integrals of motion. In this language, a p-form (0.1) is interpreted as a wave function belonging to the Hilbert space where the supercharges and the Hamiltonian act. This wave function depends on the ordinary real dynamical variables xM and also on the Grassmann anticommuting variables ψ M . Any wave function Ψ(xM , ψ M )

(0.5) M

can be expanded into a Taylor series in ψ . It is noteworthy that this QD+1 series is finite, as the product of D + 1 anticommuting factors i=1 ψ Mi vanishes. Thus, a p-form (0.1) maps into a wave function (0.5) involving a product of p Grassmanian factors. The usefulness of this mapping stems from two remarks: (1) It allows one to use the whole bunch of physical methods that were developed since Newton and Schr¨odinger to study dynamical systems. (2) All dynamical systems relevant for geometric applications are supersymmetric. And one can use powerful methods (in particular, the superspace and superfield formalism) developed by physicists during half a century that passed since 1971 when supersymmetry was born [2].

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One should note the following. Supersymmetry was discovered and mostly studied then by physicists as a new interesting symmetry of field theory systems. However, for geometrical applications we will need to deal only with supersymmetric quantum mechanical (SQM) systems where the dynamical variables depend only on time, but not on spatial coordinates, as they also do in field theories.1 Equation (0.4) describes the simplest SQM algebra with one complex or two real supercharges. We will see later that this algebra is relevant for the description of arbitrary real manifolds equipped with the de Rham differential and its conjugate and also of arbitrary complex manifolds equipped with the Dolbeault differential and its conjugate. But the supersymmetry algebra can be richer and involve more than two real supercharges. We will see that such extended supersymmetries are relevant for the description of certain special classes of the manifolds, in particular, of the K¨ ahler and hyper-K¨ ahler manifolds and also the so-called HKT manifolds.2 All these and some other manifolds will be discussed at length later. The plan of the book is the following. The part Geometry includes mathematical preliminaries— some standard facts of the theory of smooth manifolds and is mainly addressed to physicists. In particular, we will briefly describe Riemannian geometry (though a physicist who studied general relativity should already be familiar with this subject) and the de Rham complex, then complex geometry and the Dolbeault complex, and finally the hyper-K¨ahler and HKT manifolds (the latter subject is less known and the corresponding section may also represent an interest to mathematicians). Physics provides the reader with a necessary information concerning ordinary and supersymmetric dynamical systems. We start with brief review of the ordinary analytical mechanics and of quantum mechanics. In principle, this is studied not only at physical, but also in mathematical departments at good universities, but my experience tells me that many of my mathematical colleagues are not so well familiar with these issues. What is definitely not studied at the universities are dynamical systems with Grassmann variables. We explain what they are in Chap. 4. And then we go over to supersymmetric systems. In Chap. 5, a supersymmetric quantum system is defined as a system admitting the al1 For this reason, such independent spatial coordinates will not, or almost will not appear in our book. Still, to prevent a possible confusion, we note that they have nothing to do with the dynamical variables xM (t). 2 HKT stands for “hyper-K¨ ahler with torsion”.

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gebra (0.4) (a mathematical definition) or as a system where all excited eigenstates in the spectrum of the Hamiltonian are double degenerate (a physical definition). We discuss the simplest physical example of such a system (it is the motion of an electron in a homogeneous magnetic field — the problem solved by Landau almost 90 years ago), some other simple examples, and unravel in Chap. 7 their mathematical structure introducing the notion of superspace (for quantum-mechanical system it is rather “supertime”) and 1-dimensional superfields (or “supervariables”). Finally, we introduce 1-dimensional harmonic superspace. This notion is indispensable for the description of generic HKT manifolds. In Chap. 6, we review the path integral approach to ordinary and supersymmetric systems. The important notion of the Witten index is introduced. The part Synthesis is central in the book. We go back to the geometric structures discussed in the first part, but describe now the known and not so well known new results in supersymmetric language. Chapter 8 is devoted to the de Rham complex (we also touch upon Morse theory, which can be explained in very simple terms using supersymmetry). Chapter 9 is devoted to the Dolbeaux complex. Among other things, we give there the supersymmetric interpretation of the Newlander-Nirenberg theorem and show that the Dolbeaux complex can well be defined not only for complex manifolds, but also for certain non-complex even-dimensional manifolds such as S 4 . In Chap. 10, we discuss SQM systems enjoying extended supersymmetry ˆj , Q ˆ † ). These with several pairs of Hermitially conjugated supercharges (Q j models describe K¨ ahler, hyper-K¨ahler and HKT manifolds as well as some other types of manifolds (quasicomplex, spinorial, bi-spinorial, bi-K¨ahler and bi-HKT manifolds) that have not yet attracted much attention of mathematicians. In Chap. 11, we establish a genetic relationship between the different kinds of models (and hence—between the different kinds of manifolds). We show in particular that any supersymmetric model described in Chaps. 8– 10 can be derived from a trivial noninteracting flat space model by applying two operations: (i) similarity transformation of the supercharges and (ii) Hamiltonian reduction. Chapter 12 is devoted to generic hyper-K¨ahler and HKT manifolds described using the harmonic superspace language. We show that, by the same token as any K¨ ahler metric is derived from the K¨ahler potential, hj k¯ = ∂j ∂k¯ K(z n , z¯n ), any hyper-K¨ahler metric can be derived from the harmonic prepotential L+4 . Similarly, the data including the pair of pre-

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potentials (L+3 , L) provide us at the end of a long day3 with a particular HKT metric. In Chap. 13, we give a detailed discussion of the gauge fields living on manifolds and the associated supersymmetric models. In mathematical description, gauge fields represent connections of principal fiber bundles. Such a bundle is characterized by an integer topological charge (called the Chern class). On the physical side, it is possible to consider systems with a fractional Chern class, but these systems are not supersymmetric. The last chapter is devoted to the Atiyah-Singer theorem that relates the index of certain elliptic operators to topological characteristics of the manifold where they are defined. We show that this index can be interpreted as the Witten index of an appropriate SQM system and calculate it in some nontrivial cases.

Notes on notation • To distinguish the ordinary functions from the operators, the latter will be in most cases equipped with hats: H is a classical Hamiltonian and ˆ is a quantum Hamiltonian. But sometimes we will not put a hat: d H and d† in Eq. (0.3) are operators, even though they do not carry hats. • We mostly use the bar for the complex conjugation of ordinary functions ¯ In some cases, the star A∗ will also be or Grassmann functions, A → A. used, but only for the ordinary functions. ˆ¯ but some• The Hermitially conjugated operators will be denoted as A, † ˆ times also by A . • The real world tensor indices will be denoted by capital latin letters: M, N , etc. For the corresponding tangent space indices, we take the letters from the beginning of the alphabet: A, B, etc. The world indices are written at two levels: this distingushes covariant and contravariant vectors and tensors. The position of the tangent space indices does not have a mathematical significance and will be chosen exclusively out of esthetic considerations. • In complex spaces, the holomorphic world tensor indices will be denoted by small letters: m, n, . . . The antiholomorphic indices will be denoted as m, ¯ n ¯ , . . . The corresponding tangent space indices are a, b, . . . ; a ¯, ¯b, . . . 3 The procedure here is essentially more complicated than for the K¨ ahler manifolds where the metric is obtained simply by differentiating the K¨ ahler potential K.

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The position of the latter also carries some significance (see the footnote on p. 33). • In some cases (but not always), one of the bars in the expressions like V¯ n¯ will be skipped and we will write V n¯ or V¯ n instead. Acknowledgments I am deeply indebted to my Dubna collaborators: Evgeny Ivanov and Sergei Fedoruk, without whom I would not be able to derive many results reported in this book. Sergei Fedoruk, Zohar Komargodski, George Papadopoulos, and Alexei Rosly took the pain to read the manuscript at the preliminary stage and made a lot of valuable comments. It is also a pleasure for me to appreciate the help of Boris Smilga in drawing the figures. Finally, I would like to thank the whole World Scientific team, and especially Lakshmi Narayanan, for encouragement and support.

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PART 1

GEOMETRY

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Chapter 1

Real Manifolds

This and two next chapters of this book are addressed to a reader-physicist. We can also recommend to him two excellent reviews by T. Eguchi, P.B. Jilkey and A.J. Hanson [3] and by P. Candelas [4]. 1.1

Riemannian Geometry

Consider a D-dimensional manifold M without boundaries. A neighbourhood1 of each point on M is homeomorphic to RD . Any such manifold can be represented as a union [ M = Uα (1.1) α

of a finite number of open subsets Uα ⊂ M, each of these subsets being homeomorphic to RD . For example, the surface of the Earth can be represented as a union of two open disks: one of them has a center at the North Pole and extends down to the Tropic of Capricorn and the second is centered at the South Pole and extends up to the Tropic of Cancer. Introduce the natural Cartesian coordinates on RD . Then an invertible map Uα ↔ RD or Uα ↔ QD , where QD is a subset of RD with the topology of the open ball, provides us with a certain coordinate chart on Uα —each point of Uα is characterized by a set of coordinates {xM }. There are many different such maps and, correspondingly, many different charts {xM }. The transitions from one chart into another are called reparameterizations or general coordinate transfromations. The sets Uα may overlap. The intersection Uα ∩ Uβ belongs thus to two different charts (a map of Europe and a map of Russia both contain Yalta) 1 We said “neighbourhood” and this means that our manifold is metric, i.e. a distance between any two of its points is defined. Mathematicians like to play with more general not necessarily metric manifolds, but we will not do so in our book.

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M and is described by two different sets of coordinates {xM (α) } and {x(β) }. One set can be expressed into another, N M xN (α) = fαβ (x(β) ) .

(1.2)

N fαβ

The functions are called the transition functions. If three charts Uα , Uβ , Uγ have a nonempty intersection, the transition functions should satisfy there an obvious condition fαβ ◦ fβγ = fαγ . Definition 1.1. A smooth manifold is a manifold where the transition functions have an infinite number of derivatives. Definition 1.2. An orientable manifold is a manifold where one can choose N the charts such that all the transition Jacobians |∂M fαβ | are positive. A well-known example of a nonorientable manifold is the so-called Klein bottle. But we will be interested only in the smooth orientable manifolds in our book. One can define scalar, vector and tensor fields living on M. A scalar field depends only on the point and is invariant under reparameterizations. There are two types of vectors. Definition 1.3. A set of D functions of coordinates that transform under reparameterizations in the same way as dxM , V 0M =

∂x0M N V , ∂xN

(1.3)

is called a contravariant vector. Definition 1.4. A set of D functions of coordinates that transform under reparameterizations in the same way as the gradient operator ∂M , 0 VM =

∂xN VN , ∂x0M

(1.4)

is called a covariant vector. Following the usual convention, we wrote the index for the contravariant vector upstairs and for the covariant vector downstairs. The summation over the repeated indices is assumed. If M is flat Euclidean space RD and we limit our consideration to the transformations representing rigid rotations and translations, there is no distinction between these two types of vectors. We can further consider tensors with an arbitrary number of covariant or contravariant indices. For example, a tensor TNM of rank (1, 1) is

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defined as a set of D2 functions of the coordinates that transform under reparameterizations as ∂x0M ∂xQ P T . (1.5) ∂xP ∂x0N Q The generalization to an arbitrary case is obvious. The product of two tensors also represents a tensor. This product can keep all the indices of the factors or some of them may be contracted. We leave to the reader the M proof of the fact that, if TNM and SN are tensors [i.e. transform as in (1.5)], M M P N the object RN = TP SN is also a tensor, while TNM SM is a scalar. Metric smooth manifolds are also called Riemannian. All the manifolds to be discussed in this book are Riemannian. If the distance between two points X and Y is small enough, one can assume that these points belong to the same chart Uα and their coordinates are close to each other. Let ds be the distance between the points X and X + dX characterized by the coordinates xM and xM + dxM . Generically, ds2 represents a quadratic form (TNM )0 =

ds2 = gM N (xP ) dxM dxN .

(1.6)

In our book, we will consider only the manifolds with ds2 > 0 for nonvanishing dxM . But the reader understands that the latter condition is not fulfilled for flat Minkowski space or curved space with Minkowskian signature—the space we live in. Bearing in mind that ds2 is a scalar, we conclude that gM N is a symmetric covariant tensor of the second rank. It is called the metric tensor. One P . can define also the inverse metric tensor g M N satisfying gM N g N P = δM M N For any contravariant vector V , the vector VM = gM N V is covariant. For any covariant vector VM , the vector V M = g M N VN is contravariant. Theorem 1.1. The object EM1 ...MD =

√

g εM1 ...MD ,

(1.7)

where g = det(gM N ) and εM1 ...MD is totally antisymmetric with normalization ε1...D = 1, is an invariant tensor (i.e. it preserves its form under reparameterizations). Proof. Any antisymmetric tensor of rank D has the form TM1 ...MD = f (x) εM1 ...MD . The tensor nature of TM1 ...MD dictates its transformation law: ∂xN1 ∂xND 0 TM = . . . TN ...N , (1.8) 1 ...MD ∂x0M1 ∂x0MD 1 D

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which means that f (x0 ) = Jf (x) ,

(1.9)

where J =

∂(x1 , . . . , xD ) ∂(x01 , . . . , x0D )

is the Jacobian of pthe transformation. It is not difficult to check that (1.9) holds iff f (x) = g(x), so that TM1 ...MD coincides with (1.7). Our next task is to learn how to differentiate tensor fields. The gradient of a scalar ∂M φ is a covariant vector, but the object ∂N V M is not a tensor, it does not transform according to (1.5). Definition 1.5. The covariant derivative of the contravariant vector field V M (x) is defined as P ∇N V M = ∂N V M + GM NP V

with the condition that ∇N V

M

(1.10)

transforms as a tensor.

2 The object GM While N P will be called in this book the connection. M ∇N V is a tensor, the connection is not a tensor. Its transformation law is   ∂ 2 xS ∂x0M ∂xR ∂xQ S 0 G + (1.11) (GM ) = NP ∂xS ∂x0N ∂x0P RQ ∂x0N ∂x0P

Definition 1.6. The covariant derivative of the covariant vector field VM (x) is defined as ∇N VM = ∂N VM − GP N M VP .

(1.12)

with the same GP N M as in (1.10). It is also a tensor (prove it). The generalization of the above definitions for an arbitrary tensor is quite straightforward. For example, Q Q M ∇P TNM = ∂P TNM + GM P Q TN − GP N TQ .

(1.13)

The “physical meaning” of the covariant derivatives (1.10) and (1.12) is the following. The ordinary partial derivatives give the difference of the values of the function in neighbouring points: f (x + dx) − f (x) ≈ 2 This definition is convenient for us. Mathematicians usually define it somewhat differently—for them the whole covariant derivative (1.10) is the affine connection.

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(∂M f )dxM . The presence of the second term in (1.10) indicates that before being subtracted from V M (x + dx), the vector V M (x) should be modified a little bit: M P N V M (x) → V˜ M (x) − GM ⇒x+dx (x) = V N P V dx + o(dx) .

(1.14)

This modification can be interpreted as a parallel transport of the vector V M from the point x to the point x + dx. Similarly, the presence of the second term in (1.12) indicates a parallel transport of the covariant vector VM : VM (x) → V˜M,

⇒x+dx (x)

N = VM (x) + GP N M VP dx + o(dx) . (1.15)

The origin of the term “connection” is now clarified. The connection GM NP specifies a particular way that the different points of the manifold are connected by the operator of the parallel transport. A lot of different choices for GM N P are possible. A distinguished role is played by the Levi-Civita connection3 ΓM NP . It satisfies two conditions: (1) The covariant derivative of the metric tensor vanishes. M (2) The connection is symmetric, ΓM N P = ΓP N . The first condition (the metricity condition) actually means that, after a parallel transport to the point x + dx, the vectors V˜ M and V˜M are still related to each other by the metric tensor at the point x + dx: V˜M (x + dx) = gM N (x + dx)V˜ N (x + dx) . Or speaking differently, the contravariant vector V M is rotated a little bit after an infinitesimal parallel transport, but its length is not changed: gM N (x + dx)V˜ M (x + dx)V˜ N (x + dx) = gM N (x)V M (x)V N (x). (1.16) When also the second condition is imposed, the connection is determined uniquely and reads 1 MQ g (∂N gQP + ∂P gN Q − ∂Q gN P ) . 2 People say that this connection is torsionless. ΓM NP =

(1.17)

Definition 1.7. The torsion is the antisymmetric part of a generic connection GM NP , M M M CN P = GN P − GP N 3 Another

name for

ΓM NP

is Christoffel symbols.

(1.18)

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Remark. In contrast to GM N P , the torsion is a tensor. Indeed, the unhandy term with second derivatives in the transformation law (1.11) is symmetric in the indices N, P and disappears after antisymmetrization. An important local characteristics of a manifold is its curvature. We will be interested not with the extrinsic curvature that depends on embedding, but with the invariant intrinsic curvature (for example, the intrinsic curvature vanishes for a cylinder, but not for a sphere).

N

π 2

Q L

Fig. 1.1 Rotation of a tangent vector after going around a closed circuit on the Earth surface. N is the North Pole and Q and L represent Quito in Ecuador and Libreville in Gabon—these cities lie almost at the equator and their longitudes differ by ∼ 90o .

The ordinary Riemannian curvature is defined by inspecting the rotation M of a vector after its parallel transport (defined with GM N P = ΓN P ) along a small closed contour. To understand that the vector can be rotated after this, the reader is welcome to meditate over Fig. 1.1. For 2-dimensional surfaces it does not matter what was the original vector direction and how the contour in the manifold is oriented. But for higher dimensions it does. Consider a parallel transport of V M along a small square lying in the plane (P Q) (see Fig. 1.2). Bearing in mind (1.14) and its relationship to (1.10), it is easy to see that the vector is shifted by ∆V M = −a2 [∇P , ∇Q ]V M ,

(1.19)

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where a is the size of the square. Now, ∇P V M is a tensor, hence [∇P , ∇Q ]V M is also a tensor. The latter can be presented as [∇P , ∇Q ]V M = RM N P Q V N , (1.20)

Q

a P a Fig. 1.2

Parallel transport along a small square.

The object RM N QP is called the Riemann curvature tensor. It is convenient to lower its first index and consider the totally covariant tensor of the 4th rank RM N QP . It has the following properties: RM N QP = −RN M QP = −RM N P Q = RQP M N , RM N QP + RM QP N + RM P N Q = 0 . (1.21) 2 2 For a D-dimensional manifold, the Riemann tensor has D (D − 1)/12 independent components. It follows from (1.20) and (1.10) that M M R M R R M N P Q = ∂ P ΓM (1.22) QN − ∂Q ΓP N + ΓP R ΓQN − ΓQR ΓP N . Besides the ordinary Riemann tensor, one can define a curvature tensor for any connection GM N P . It is given by the same formula (1.22), but with M substituted for Γ GM NP N P . A situation when the ordinary Riemann tensor (1.22) vanishes (so that the manifold is flat in the standard meaning of this word), but some other curvature tensor does not vanish or vice versa, is quite possible and not uncommon. The contractions of the Riemann tensor give the Ricci tensor RM Q = RM N Q N and the Riemann scalar curvature R = RM M . For 2-dimensional surfaces, R coincides with the Gaussian curvature R (the inverse product of the principal radii of the tangent ellipsoid), multiplied by 2.

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1.2

Differential Forms

Consider a special class of tensor fields, the totally antisymmetric covariant tensors T[M1 ···Mp ] . These fields play a distinguished very important role in many mathematical and physical applications. A nice fact is that one can describe these fields using a convenient compact notation not explicitly displaying the indices.4 For example, instead of the covariant vector VM , one can consider the linear form V = VM dxM , where dxM is an infinitesimal displacement.5 For an antisymmetric covariant tensor of the second rank T[M N ] or higher ranks, we meet some problem, however. The contraction T[M N ] dxM dxN is simply zero. But one can define the so-called wedge product dxM ∧dxN , which is antisymmetric under the permutation of the differentials, dxM ∧ dxN = −dxN ∧ dxM . Likewise, dxM1 ∧ · · · ∧ dxMp is totally antisymmetric in M1 , . . . , Mp . With this definition, T = T[M N ] dxM ∧ dxN , etc. do not vanish. On a D-dimensional manifold, one can define such forms of degree p varying from 0 (for the scalars) up to D. The forms of degrees p and D − p are interrelated by the Hodge duality transformation ?. Definition 1.8. If α = αM1 ···Mp dxM1 ∧ · · · ∧ dxMp ,

(1.23)

then ?α =

1 E M1 ...Mp Mp+1 ...MD αM1 ···Mp dxMp+1 ∧ · · · ∧ dxMD , (1.24) (D − p)!

where EM1 ...MD is the covariant totally antisymmetric tensor defined in (1.7).6 With this definition, the property ? ? αp = (−1)p(D−p) αp 4 The

(1.25)

plethora of indices is an inherent feature of the Riemann formalism described above. This makes explicit calculations (for example, the calculations in general relativity) difficult. Sometimes, one has to live with it, but for antisymmetric tensors the life can be considerably simplified. 5 Let F M be the field of forces that act on a particle moving along the manifold. Then FM dxM is the work done by the force FM as the particle moves from xM to xM + dxM . 6 In most mathematical textbooks, the coefficients α M1 ···Mp of the differential forms on the real manifolds are assumed to be real. But they may also be assumed to be complex and that is what we will do, anticipating the mapping of the forms to the wave functions of an appropriate supersymmetric quantum mechanical model to be discussed in Chap. 8.

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holds. Thus, in ordinary 3-dimensional flat space, the magnetic field can be described either by the vector Bi or by the dual antisymmetric tensor Fij = 1 2 εijk Bk . The dual of the invariant volume form7 √ (1.26) αV = g dx1 ∧ · · · ∧ dxD is the constant scalar α0 = 1. The set of all forms is obviously a vector space. But besides adding, one can also multiply forms using the wedge product. Clearly, αp ∧ βq is a (p + q)-form. If p + q > D, it vanishes. In addition, with the definition (1.24) in hand, we introduce the inner product. It is defined only for the forms of the same degree p and reads Z Z D √ Y N dx . (1.27) (αp , βp ) = αp ∧ ?βp = p! αM1 ···Mp β M1 ···Mp g M

M

N =1

Note that (?αp , ?αp ) = (αp , αp ). The forms can also be differentiated. We define the exterior derivative operator d as follows [it was already written in (0.2), but we repeat it here]: Definition 1.9. If α = αM1 ...Mp dxM1 ∧ · · · ∧ dxMp , then dα = ∂N αM1 ...Mp dxN ∧ dxM1 ∧ · · · ∧ dxMp .

(1.28)

The exterior derivative of a p-form is a (p + 1)-form. The operation (1.28) is a generalization of the ordinary curl. In flat 3-dimensional space, d

?

Ai dxi → ∂i Aj dxi ∧ dxj −→ (εijk ∂i Aj ) dxk .

(1.29)

The operator d is nilpotent, d2 α = 0. Remark. Note that one could replace in the definition (1.28) the ordinary derivative ∂N by the covariant Levi-Civita derivative ∇N . The extra terms in ∇N αM1 ...Mp vanish after antisymmetrization. This means that 7 If

the manifold is compact, the integral Z

√

Z αV = M

gives its volume.

M

g

D Y N =1

dxN

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the operator d may be viewed of as an operator acting on the Hilbert space including all antisymmetric tensor fields. It makes a tensor out of a tensor. Finally, we define the adjoint of d with respect to the norm (1.27): d† = (−1)Dp+D+1 ? d ? .

(1.30)

For any even D and for all p, d† = − ? d?. For an odd D, d† = (−1)p ? d?. The result of the action of d† on a p-form is a (p − 1)-form. For any couple of forms αp and βp−1 , the property (αp , dβp−1 ) = (d† αp , βp−1 )

(1.31)

holds. d† is a generalization of the divergence, as can be easily seen by inspecting the action of d† on a 1-form in flat space: d† (Ai dxi ) = −(∂i Ai ). The operator d† is nilpotent, as immediately follows from the definition (1.30), from the property (1.25) and from the nilpotency of d. Consider the operator 4 = −(dd† + d† d) .

(1.32)

It is a differential operator of the second order which makes p-forms out of p-forms. It is called the Laplace-de Rham operator. When one restricts the action of 4 on ordinary functions (p = 0), we are dealing with the Laplace-Beltrami operator 1 √ 4f = √ ∂M ( gg M N ∂N f ) , g

(1.33)

a generalization of the ordinary Laplacian for curved manifolds. The operator −4 is positive definite: (αp , −4αp ) = (αp , d† d αp ) + (αp , dd† αp ) = (dαp , dαp ) + (d† αp , d† αp ) ≥ 0 .

(1.34)

The properties [4, d] = [4, d† ] = 0

(1.35)

hold. The Hilbert space of the differential forms with the operators d, d† and 4 acting on it is called the de Rham complex. At the end of this section, we give four more definitions. Definition 1.10. The form αp is closed if dαp = 0. The form αp is co-closed if d† αp = 0.

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x x C1 C1 C2

C2

0

0

b) An obstacle

a) No obstacles Fig. 1.3

Exact and not exact.

Definition 1.11. The form αp is exact if it can be represented as αp = dβp−1 . The form αp is coexact if it can be represented as αp = d† γp+1 . Definition 1.12. The form αp is harmonic if 4αp = 0. An exact form is always closed (this follows from the nilpotency of d), but a closed form is not always exact. The following theorem, however, holds:8 Theorem 1.2. In a topologically trivial space homeomorphic to RD , a closed form is also exact. Proof. The proof is especially simple for 1-forms, in which case the theorem can be reformulated as follows: In RD , a vector field AM (x) with zero curl, ∂[M AN ] = 0, can be presented as a gradient of a scalar function φ(x). The proof is simple. Given the value φ(0) at the origin, we determine φ(x) at any other point as Z x φ(x) = φ(0) + AM (x) dxM , (1.36) 0

where the intergal is done along any line connecting 0 and x (see Fig. 1.3a). The condition ∂[M AN ] = 0 and the Stokes theorem ensures that the integral (1.36) does not depend on the path. 8 We will later use it on several occasions to prove the existence of a solution to an equation or equation system.

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A similar proof can be given for higher p. Remark. Note that the requirement that our space has a trivial topology is essential. If the space has a “hole” (has a nontrivial fundamental group), not all paths can be continuously transformed to one another, and the velocity field of liquid that flows around a hole cannot be presented as a gradient of some potential (see Fig. 1.3b). For a nontrivial manifold M and a given p, the coset closed/exact is a linear space known as the de p Rham cohomology group HdR (M). We will discuss at length cohomology of manifolds in Chap. 8. 1.3

Principal Fiber Bundles and Gauge Fields

A fiber bundle E over the base M with fiber F is a manifold which looks locally as a direct product of M and F. To be more exact: if we represent M as a union (1.1) of the open subsets Uj , the bundle represents a union of the direct products Uj × F. Note that the bundle may be glued from the pieces Uj ×F in a nontrivial “twisted” way, so that the bundle does not necessarily coincide globally with the product M × F. The simplest nontrivial bundle is the M¨obius strip. In this case, the base is the circle S 1 and the fiber is the interval I. But the M¨ obius strip does not coincide with S 1 × I. Definition 1.13. A principal fiber bundle is a bundle with the fiber representing a Lie group G. To avoid unnecessary (for us) complications, we will assume in the following that this group is compact. Definition 1.14. A vector bundle associated with a given principal bundle is a bundle with the fiber being a representation space Φ of the group G. It will be more convenient for us to work with this latter object. Note here that what mathematicians call “representation space” physicists simply call “representation”, and we will stick in the following to the physical terminology. Definition 1.15. Take in each fiber that grows over the point x ∈ M a particular element φ(x) ∈ Φ. We obtain what is called a section of our vector bundle.

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Given a section φ(x), we can act at each point on φ(x) by a group element g(x) to obtain another section: φ0 (x) = g(x)φ(x)

(1.37)

The transition (1.37) will be called a gauge transformation. The word “gauge” means that the group transformation (1.37) is local: its parameters depend on x ∈ M so that we are dealing here not just with the group G, Q but with an infinite-dimensional group x G(x). We are interested only in the smooth functions φ(x), g(x) that can be differentiated. However, an ordinary derivative ∂M φ(x) is an inconvenient object, because it does not transform as in (1.37): ∂M φ0 (x) involves an extra term ∼ ∂M g(x). Proceeding in the same way as we did in (1.10) and (1.12), we define a covariant derivative ∇M φ(x) = [∂M + iAM (x)] φ(x) ,

(1.38)

where the components of AM (x) belong to the Lie algebra9 g. AM (x) is section-dependent. We require that it transforms when going from φ(x) to φ0 (x) in such a way that ∇M φ0 (x) = g(x)∇M φ(x) .

(1.39)

It is easy to derive the law A0M (x) = gAM (x)g −1 + i(∂M g)g −1 .

(1.40)

A kinship of AM (x) to the affine connection GM N P discussed in the previous section is obvious. Mathematicians call AM (x) the principal G-connection, but the physicists call it a gauge field living on the manifold. It is the latter name that we are going to use in our book. We know that the manifold represents generically a union of several different charts. Up to now, we only defined AM (x) in a particular chart. (α) (β) The fields AM (x(α) ) and AM (x(β) ) in two different charts may coincide when x(β) is expressed via x(α) , but it is not a necessary condition. In (α)

(β)

many interesting cases,10 AM and AM do not coincide, but are related by a nontrivial gauge transformation (β)

AM

−1 −1 = gαβ AM (x)gαβ + i(∂M gαβ )gαβ .

(1.41)

The only condition is that, if by any chance three different charts Uα , Uβ and Uγ overlap, the functions gαβ , gβγ and gαγ satisfy in the overlap 9 We

use the physical convention: AM (x) are represented by Hermitian matrices. will be discussed in Chap. 13.

10 They

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region the consistency condition (the cocycle condition in the mathematical language) gαβ gβγ = gαγ . Gauge fields play a tremendously important role in physics. An example is the electromagnetic Maxwell field. In that case, G = U (1), g(x) = exp{iχ(x)} and the law (1.40) is reduced to A0µ (x) = Aµ (x) − ∂µ χ(x), where µ = 0, 1, 2, 3 is the Minkowski space index. Gauge fields associated with the non-Abelian groups SU (3) and SU (2) play the fundamental role in the physics of strong and weak interactions. In the Riemann geometry section, we discussed the affine connection related to the infinitesimal parallel transport of vectors and tensors and also the curvature tensor describing how the vectors and tensors transform after going around a small closed contour. The notion of curvature can also be introduced for principal fiber bundles. To this end, one should consider, as in Eqs. (1.19), (1.20) the commutator of two covariant derivatives (1.38), def

−i[∇M , ∇N ] = ∂M AN − ∂N AM + i[AM , AN ] = FM N (1.42) Clearly, FM N ∈ g. It is an antisymmetric tensor.11 Mathematicians call the form F = (1/2)FM N dxM ∧ dxN “the curvature form of a principal G-connection”. We will use the physical language and call the tensor FM N the field density tensor (and AM will be called the gauge potential or just the gauge field). In the Maxwell case, the components of the tensor Fµν = ∂µ Aν − ∂ν Aµ are the electric and magnetic fields. The Abelian field density is invariant under the gauge transformation (1.40). In the non-Abelian case, it transforms as 0 −1 FM . (1.44) N = g FM N g The notation of differential forms allows one to represent (1.42) in a somewhat more compact way. Introduce a matrix-valued 1-form A = AM dxM . Then the definition (1.42) is equivalent to 1 (1.45) F = dA + iA ∧ A = FM N dxM ∧ dxN . 2 1.4

Vielbeins and Tangent Space

Suppose we define at each point x of the manifold M a set of D orthonormal vectors eA ; eA ·eB = δAB . In coordinates, the latter condition reads N gM N eM (1.46) A eB = δAB . 11 To

prove the tensor nature of FM N , note that P ∂M AN − ∂N AM = ∇M AN − ∇N AM + CM N AP ,

(1.43)

P where ∇M is the Levi-Civita covariant derivative and CM N is the torsion tensor defined in (1.18).

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12 Definition 1.16. The object eM A is called the vielbein.

One can imagine (though it is not necessary) that M is embedded in a higher-dimensional Euclidean space and a tangent space13 (denoted as Tx M) is traced at each point P of the manifold (Fig. 1.4). The points in Tx M are related to the points in M at the vicinity of P by the orthogonal projection. The set {eA } represents then an orthonormal frame in Tx M.

P

M

Fig. 1.4

A zweibein.

N Consider the tensor QM N = eM A eA . Multiplying it by eM B and using MN N (1.46), we derive Q eBM = eB for all B. Bearing in mind the completeness of the basis {eA } in Tx M, this implies that N MN QM N = eM . A eA = g

(1.47)

eM A

is in some sense a “square root” of the metric. The Thus, the vielbein indices M, N , etc. will be called world indices and A, B, etc.— tangent space indices. Any tensor can be projected onto the tangent space by contracting it with an appropriate number of the vielbein factors eM A and eAM . In particular, the tensor (1.7) can be alternatively represented as EM1 ...MD = eA1 M1 · · · eAD MD εA1 ...AD .

(1.48)

It is clear that there is a considerable freedom in the choice of the vielbein. One can multiply eM A at each point by an orthogonal matrix OAB (x), and 12 A

“manylegger”.

13 Mathematicians

distinguish the notions of tangent space and cotangent space, and in this case it is rather cotangent than tangent. This distinction will not, however, be relevant for us in this book and we will use a sloppy terminology and omit the prefix co.

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this does not affect the metric. This is nothing but gauge freedom discussed in the previous section. For a mathematician, the manifold M with the tangent space attached at its each point defines the tangent bundle of M, but one can also define the so-called orthonormal fiber bundle with the fiber representing the set of all orthonormal frames in Tx M, which is homeomorphic to O(D). This is a particular case of the principal fiber bundle. Define now the covariant derivative of the vielbein. We want it to transform as a tensor under general coordinate transformations and as a vector in the tangent space. The latter condition means that the rotation M e0M A = OAB eB implies 0 M (∇N eM A ) = OAB ∇N eB .

(1.49)

The only way to ensure it is to add to the covariant derivative defined in (1.10) an extra term and write M M P M ∇N e M A = ∂N eA + GN P eA + ωAB, N eB .

(1.50)

14

The object ωAB, M is called the spin connection. ωAB, M is anti-Hermitian and belongs to the Lie algebra so(D). It is nothing but the gauge field for the group SO(D) (multiplied by i in our convention): −iω transforms under rotations in the same way as in (1.40), giving 0 ωAB, M = OAC (x) ωCD, M OBD (x) − [∂M OAC (x)] OBC (x) . (1.51)

Many different choices for ωAB, M are in principle possible. But the cleverest (and the standardly used) choice consists in imposing the requirement ∇N e M A = 0.

(1.52)

Its meaning is the following. Define the vielbein field at all points of the manifold. Take the vielbein at point x and perform its parallel transport to x + dx. It is rotated due to nonzero affine connections, as dictated in (1.14). But then we use the gauge freedom of local frame rotations associated with nonzero ωAB, M to “neutralize” the effect due to nonzero GM N P . The condition (1.52) says that, after a parallel transport from x 14 This term is due to the fact that ω AB, M is indispensible for describing spinor fields on the manifolds. It is used in general relativity, where M represents a physical curved spacetime and the spinor fields are physical fermion fields. In our book, M will never be interpreted as physical space (the latter will involve only time), but rather as an external target space whose coordinates are time-dependent dynamical variables. Still, spin structures may also be considered for many such target spaces, and we will do so in Chaps. 13,14 where we will discuss the Dirac operator and the index theorem.

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to x + dx and a subsequent frame rotation, the vielbein coincides with eM A (x + dx)! In this case, the spin connection ωAB, M is expressed via the vielbeins N P and affine connections: ωAB, M = −eBN (∂M eN A + GM P eA ) One can prove N (we leave it to the reader) that, if GM P are chosen so that the covariant derivative of the metric tensor vanishes, the spin connection as given in (1.53) is antisymmetric under A ↔ B and hence belong to so(D), indeed! We arrive then at the standard textbook expression,
N P ωAB, M = eAN ∂M eN (1.53) B + GM P eB . This result is quite natural. As was mentioned, the metricity condition ∇P gM N = 0 means that the vector lengths do not change under parallel transport. Thus, a parallelly transported vielbein is still a vielbein— an orthonormal frame in the tangent space. And then the local rotations generated by the antisymmetric spin connection can bring, indeed, the vielbein to eM A (x + dx). There exist affine connections not respecting the metricity (one of them, the so-called Obata connection will be discussed in Chap. 3 and Chap. 12). Then ωAB, M as defined in (1.53) is not antisymmetric and does not belong to so(D). But in this case, the very notions of vielbein and spin connection are probably not so useful. The relationship (1.53) can be rewritten in a nice compact way using the language of differential forms. Introduce the 1-forms eA = eAM dxM ,

ωAB = ωAB, M dxM .

(1.54)

Eq. (1.53) is equivalent to a Cartan structure equation deA + ωAB ∧ eB = TA ,

(1.55)

where 1 1 M N P eAM CN = CA,BC eB ∧ eC (1.56) P dx ∧ dx 2 2 is the torsion 2-form. When there is no torsion, we simply obtain deA + ωAB ∧ eB = 0. Even nicer is the second Cartan’s structure equation,15 1 def dωAB + ωAC ∧ ωCB = RABCD eC ∧ eD = RAB , (1.57) 2 which relates the curvature tensor to the spin connection. More explicitly, TA =

RM N,AB = ∂M ωAB,N − ∂N ωAB,M + ωAC,M ωCB,N − ωAC,N ωCB,M . 15 Do

not mix up RAB thus defined with the Ricci tensor.

(1.58)

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Equation (1.57) has exactly the same meaning as (1.45). It is remarkable that the same curvature tensor may be expressed via the affine connections as in (1.22) or via the spin connections as in (1.57). But this is stipulated by the requirement (1.52) that relates parallel transports to local frame rotations. Taking the exterior derivative of Eq. (1.55) and using (1.57), we derive the so-called consistency condition dTA + ωAB ∧ TB = RAB ∧ eB .

(1.59)

When the torsion is absent, we derive the constraint RAB ∧ eB = 0

(1.60)

for the ordinary Riemann curvature. This condition is none other than the second line in (1.21).

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Chapter 2

Complex Manifolds

2.1

Complex Description

Definition 2.1. An even-dimensional manifold representing a union (1.1) of open subsets Uα homeomorphic to RD=2d is called a complex manifold if n=1,...,d each subset can be parameterized by the complex coordinates {z(α) } in such a way that the transition functions are analytic, n n m z(α) = fαβ (z(β) ).

(2.1)

The manifold may or may not be equipped with a metric, but we will always assume that it is. Theorem 2.1. For a complex manifold, one can always choose the metric in the Hermitian form: n ¯ ds2 = 2hnm zm ¯ dz d¯

(2.2)

with hm¯n = hnm ¯. The factor 2 is introduced for further conveniences. Proof. For a particular chart Uα , it is obvious. The property hm¯n = hnm ¯ follows from the reality of ds2 . Then the existence of a globally defined Hermitian metric can be proven using the so-called partition of unity technique. The proof is the same as for real manifolds, which we skipped in the previous chapter; the crucial observation is that, in the intersection Uα ∩Uβ of two overlapping charts, a Hermitian metric in Uα stays Hermitian in Uβ after a holomorphic change of variables. Let a Hermitian metric ds2(α) be defined on each Uα . We introduce a set of smooth functions fα (z(α) , z¯(α) ) such that 29

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• fα = 0 at the border of Uα . P • α fα = 1 at every point of the manifold. Then the metric ds2 =

X

fα ds2(α)

(2.3)

α

is the metric on the union ∪α Uα that we are looking for. To understand it, imagine a manifold consisting of only two overlapping charts, like in the example that follows. Then the metric (2.3) coincides with ds2(1) at all points of U(1) not belonging to U(2) , coincides with ds2(2) at all points of U(2) not belonging to U(1) and represents a smooth interpolation between the two in the overlap region. The simplest nontrivial example of a complex manifold is S 2 (Fig. 2.1). It can be represented as a union of two charts. One of them includes all the points of the sphere except its north pole. It can be parameterized by the complex coordinate θ (2.4) z− = cot eiφ 2 given by the stereographic projection on the plane tangent to the south pole. Another chart can be parameterized by the complex coordinate θ z+ = tan e−iφ (2.5) 2 given by the stereographic projection on the plane tangent to the north pole. In the overlap region, which includes all the points of the sphere except its poles, the transition function z+ = 1/z− is analytic. The metric is convenient to choose in the form 2 dzd¯ z (2.6) ds2 = 2h dzd¯ z = (1 + z z¯)2 for both charts. In the overlap region, ds2+ = ds2− , and we do not need to bother√in this case about the partition of the unity. We also pose z = (x + iy)/ 2 so that the area of our sphere is equal to1 Z Z 1 (2.7) A = h dzd¯ z = dxdy  2 = 2π , 2 2 1 + x +y 2 √ and its radius is equal to 1/ 2. Theorem 2.2. Any orientable 2-dimensional manifold is complex. 1 See

the footnote on p. 36.

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z+

31

N

S Fig. 2.1

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z−

Complex coordinates on S 2 .

Proof. Consider two overlapping charts on M. In the overlap region, the manifold can be described by the coordinates {x, y} or {u, v}. It is convenient to choose the coordinates such that ds2 = λ(x, y)(dx2 + dy 2 ) = µ(u, v)(du2 + dv 2 ) . (2.8) Choose the complex coordinates in each chart: . u + iv x + iy w = √ . (2.9) z = √ , 2 2 Then λ dzd¯ z = µ dwdw ¯. (2.10) Generically, ∂w ∂w dw = dz + d¯ z. (2.11) ∂z ∂ z¯ Substituting it into (2.10), we obtain, however, ∂w ∂ w ¯ = 0, ∂z ∂z which means that w is either holomorphic or antiholomorphic function of z. If it were antiholomorphic, the Cauchy-Riemann equations would read ∂u ∂v ∂u ∂v = − , = ∂x ∂y ∂y ∂x and the Jacobian  2  2 ∂(u, v) ∂u ∂v = − − (2.12) ∂(x, y) ∂x ∂y would be negative. But that would contradict the assumption that the manifold is orientable.

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Not all 4-dimensional manifolds are complex, however. For example, S is not. The reader can understand why, if s/he tries to repeat the construction in Fig. 2.1 for S 4 . It is easy to find a couple of complex n=1,2 coordinates z− parameterizing the “southern” hemisphere on S 4 . It is n=1,2 also easy to find a couple of complex coordinates z+ parameterizing the “northern” hemisphere. But it is impossible to perform it in a way that n n the transition functions are holomorphic. If z− and z+ are obtained by stereographic projections, like in Fig. 2.1, they are related as zn z¯n n n or z+ (2.13) = m−m . z+ = m−m z¯− z− z¯− z− In neither case the transition functions are holomorphic. Of course, the fact that the simplest attempt to build up a complex description of S 4 does not work is not yet a proof that it is impossible. In the case of S 4 it can be proven, however. Surprisingly, no such proof exists for S 6 . We are not able to construct complex charts with analytic transition functions, but we are not quite sure that it is impossible. The simplest example of a complex 4-dimensional manifold is2 CP2 . We will write its metric at the end of this chapter. The metric (2.10) is Hermitian: the components gzz and gz¯z¯ are absent and λ(z, z¯) is real. For D = 2, any metric can be brought to this form by a proper choice of coordinates. In a generic multidimensional case, it is not so, but, as we mentioned above, the metric can always be chosen to be Hermitian, and we will always assume that it is. mp ¯ We will express the inverse metric tensor as hn¯ m , so that hnm = δnp ¯h p¯ pn ¯ and h hnm ¯ = δm ¯. A manifold of complex dimension d can, of course, be also treated as a real manifold of dimension D = 2d. The translation from the real to the complex language and inverse is performed by trading the set of coordinates ¯ xM to the set {z m , z¯m }. Also 4

∂M = (∂m , ∂m ¯),

gm¯n = gn¯ m = hm¯n ,

gmn = gm¯ ¯n = 0.

(2.14)

With this choice, the convention (2.2) for the Hermitian metric exactly matches the definition (1.6). For d = 1 these conventions mean that z is expressed via x, y as in (2.9). Also   ∂ ∂ 1 ∂ = √ −i . (2.15) ∂z ∂y 2 ∂x 2 A complex projective space CPn is defined as a set of all complex (n + 1)tuples {z 0 , z 1 , . . . , z n } with at least one nonzero z j , factorized over the identification {z 0 , . . . , z n } ≡ {λz 0 , . . . , λz n } with any nonzero complex λ.

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In a multidimensional case, x3 + ix4 x1 + ix2 √ √ , z2 = ... (2.16) z1 = 2 2 The tensor fields living on the manifold may now have the indices of four types: holomorphic or antiholomorphic and covariant or contravariant. When an index is raised, it changes its holomorphicity: V n¯ = hn¯ m Vm , etc. Holomorphicity is also changed, of course, after a complex conjugation: Vn = V¯n¯ . In these terms, the only nonvanishing components of the Levi-Civita connection (1.17) are 1 1 Γm,np = (∂n hpm Γm,¯np¯ = (∂n¯ hmp¯ + ∂p¯hm¯n ) , ¯ ¯ + ∂p hnm ¯ ), 2 2 1 1 Γm,np¯ = Γm,pn = Cmnp¯, Γm,p¯ Cm¯ (2.17) ¯ ¯ n = Γm,¯ ¯ np = ¯ np 2 2 with Cmnp¯ = ∂n hmp¯ − ∂m hnp¯ , Cm¯ ¯ hpm ¯ − ∂m ¯ hp¯ n . (2.18) ¯ np = Cmnp¯ = ∂n We now introduce a very important notion of the Bismut connection [5, 6]. Definition 2.2. The Bismut connection is a torsionful connection 1 (B) (2.19) GM,N P = ΓM,N P + CM N P , 2 with the torsion tensor CM N P ≡ CM,N P completely antisymmetric under exchange of any pair of indices, whose only nonzero components are those given in (2.18) and those restored from them by antisymmetry. (B)

The nonzero components of Gm,np¯ are (B)

Gm,np = ∂p hnm ¯ , ¯

(B)

Gm,¯np¯ = ∂p¯hm¯n ;

(B)

(B)

Gm,np¯ = Cmnp¯ ,

Gm,¯ ¯ np , ¯ np = Cm¯

(2.20)

while (B)

¯ eam ¯

(B)

Gm,¯np = Gm,n (2.21) ¯ p¯ = 0. a We define next the complex vielbeins em and the complex conjugated with the properties3 n ¯m eam ean¯¯ = hm¯n , ena¯¯ em , a =h n eam ena = δm ,

¯ n ¯ n ¯ eam ¯ ea ¯ = δm ¯ ,

b b em a em = δa ,

¯

¯

¯ b b em a ¯ em ¯ = δa ¯,

(2.23)

3 In

contrast to the real case, we write here the tangent space indices at two levels, bearing in mind that they are complex conjugated when lowered or lifted. The scalar product XA YA of two tangent space vectors acquires in the complex notation the form XA YA = Xa Y a + Xa¯ Y a¯ = Xa Ya¯ + Xa¯ Ya .

(2.22)

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The metric is invariant under unitary tangent space rotations eam → U ab ebm ,

¯

¯ b † a ¯ eam ¯ → em ¯ (U )¯ b .

(2.24)

The spin connections (1.53) can also be expressed in this complex notation. The Levi-Civita spin connections read ω¯ba,m = −ωa¯b,m = ebp (∂m epa + Γpmk eka ) , 1 ¯ ωab,m = 0 , ωa¯¯b,m = epa¯¯e¯kb Cp¯km (2.25) ¯ . 2 and other components are related to (2.25) by complex conjugation: ω¯ba,m a,m , ¯ = ωb¯

ωab,m ¯ = ωa ¯¯ b,m .

(2.26)

For the spin connections associated with the Bismut connection (2.19), the (B) (B) components ωab,M and ωa¯¯b,M vanish and only the components of mixed (B)

(B)

tangent space holomorphicity ωa¯b,M and ωa¯b,M are left. That means that the form (B)

(B)

(B)

¯ ωAB,M dxM = ωa¯b,m dz m + ωa¯b,m d¯ zm ¯

(2.27)

represents an anti-Hermitian matrix d × d. It belongs to the algebra u(d). (B) Bearing in mind (1.57), the same concerns the curvature form Ra¯b , while (B)

(B)

the components Rab and Ra¯¯b vanish. This implies that the curvature (B)

tensor of the Bismut connection has the only nonzero components Rm¯n,p¯q and those obtained by permutations m ↔ n ¯ , p ↔ q¯, which is not true for the ordinary Riemann tensor. Now consider the covariant derivative (1.50). As was discussed, the third term describes an infinitesimal frame rotation. Generically, the latter is an element of SO(D), but we have seen that in this case it belongs to a smaller group U (d) ⊂ SO(2d)—a holomorphic vielbein stays holomorphic and an antiholomorphic vielbein stays antiholomorphic. As we discussed in Chap. 1, the spin connection and the affine connection are related by the condition that (1.50) vanishes, which means that local frame rotations and rotations due to parallel transport compensate each other. Thus, the latter also belong to U (d) ⊂ SO(2d), i.e. any holomorphic vector stays holomorphic and any antiholomorphic vector stays antiholomorphic after a parallel transport.4 This group is called the holonomy group. Note that all this concerns only the frame rotations and parallel transports induced by the Bismut spin and affine connections. For a generic 4 This can also be checked directly by substituting explicit expressions of the Bismut connections (2.20) to the definitions (1.14) and (1.15).

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complex manifold, the holonomy group corresponding to the ordinary LeviCivita connection is still SO(D), like it is for a generic D-dimensional real manifold. A very important class of complex manifolds are K¨ ahler manifolds. Definition 2.3. A complex manifold endowed with a Hermitian metric (2.2) is K¨ ahler if Cmnp¯ = ∂n hmp¯ − ∂m hnp¯ = 0 .

(2.28)

One can make the following remarks: (1) We have seen that all 2-dimensional orientable manifolds are complex. They carry complex dimension d = 1 and it is clear from (2.28) that they are all K¨ ahler. (2) A metric satisfying the condition (2.28) can be presented as hm¯n = ∂m ∂n¯ K(z, z¯) .

(2.29)

The function K(z, z¯) is called the K¨ ahler potential. (3) If Cmnp¯ = 0, the Bismut connection coincides with the torsionless LeviCivita connection. Hence the ordinary holonomy group of a K¨ahler manifold is U (d) (and if the holonomy group is U (d), the manifold is K¨ ahler). (4) In this case, the only nonzero components of the Christoffel symbols (1.17) are holomorphic and antiholomorphic: Γpmn = hq¯p ∂m hn¯q ,

¯ pq ¯ Γpm¯ ¯ hq n ¯. ¯ n = h ∂m

(2.30)

Inversely, if the Christoffel components of mixed holomorphicity like Γpm¯n vanish, the metric is K¨ahler. (5) The only nonzero components of the K¨ahler spin connections are ω¯ba,m = −ωa¯b,m = e¯nb¯ ∂m ean¯¯ ,

n a ωb¯a,m ¯ = −ωa ¯b,m ¯ = eb ∂m ¯ en . (2.31)

The only nonzero components of the K¨ahler Riemann tensor are Rmpn¯ ¯ q = −Rpmn¯ ¯ q = −Rmp¯ ¯q n = Rpm¯ ¯ qn = ∂m ∂p¯hn¯q − hs¯t (∂m hn¯s )(∂p¯ht¯q ) .

(2.32)

The relation ¯

a b Rmpn¯ ¯ q = en eq¯(∂m ωa¯ c,m ωc¯ c,p¯ ωc¯ b,p¯ − ∂p¯ωa¯ b,m + ωa¯ b,p¯ − ωa¯ b,m ) (2.33)

[it follows from (1.58)] also holds.

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Complex differential forms

In complex terms, a generic differential form (1.23) is expressed as αp,q = αm1 ···mp n¯ 1 ···¯nq dz m1 ∧ · · · ∧ dz mp ∧ d¯ z n¯ 1 ∧ · · · ∧ d¯ z n¯ q . (2.34) The expression (2.34) is called a (p, q)-form. p and q do not exceed the complex dimension of the manifold d, otherwise the wedge product in the right-hand side vanishes. We can now adjust all the constructions and the definitions of Chap. 1.2 to the complex case. The Hodge duality operator is convenient to define as 1 ¯ 1 ···m ¯p n1 ···nq Em mp+1 ···md n ¯ q+1 ···¯ nd (d − p)!(d − q)! αm1 ···mp n¯ 1 ···¯nq dz mp+1 ∧ · · · ∧ dz md ∧ d¯ z n¯ q+1 ∧ · · · ∧ d¯ z n¯ d ,

(2.35)

Em1 ···md n¯ 1 ···¯nd = det(hm¯n ) εm1 ···md εn¯ 1 ···¯nd

(2.36)

∗αp,q =

where

is the invariant tensor (1.7) in the complex notation and the indices are ¯ raised with hmn . The operator (2.35) transforms a (p, q)-form to a (d − p, d − q)-form. The property ∗ ∗ αp,q = (−1)p(d−p)+q(d−q) αp,q

(2.37)

holds. The inner product is defined for the forms of the same type p, q in analogy with (1.27):5 Z Z (αp,q , βp,q ) = αp,q ∧ ∗βp,q = p!q! αr1 ···rp s¯1 ···¯sq βm1 ···mp n¯ 1 ···¯nq M M Y ¯ 1 r1 ¯ p rp s¯1 n1 × hm · · · hm h · · · hs¯q nq det(hnm dz t d¯ z t . (2.39) ¯) t

The dual form ∗αp,q has the same norm as αp,q . Consider now a subset of (2.34) involving only the (p, 0)-forms (or else holomorphic p-forms): αp,0 = αm1 ...mp dz m1 ∧ · · · ∧ dz mp . 5 The

(2.40)

convention Y t

dz t d¯ zt =

Y

dxM

(2.38)

M

(without the factors i that are sometimes inserted) is consistently chosen throughout the book.

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Define the operator of holomorphic exterior derivative ∂ in analogy with (1.28): ∂α = ∂n αm1 ...mp dz n ∧ dz m1 ∧ · · · ∧ dz mp .

(2.41)

We introduce also the conjugate [with respect to the inner product (2.39)] operator ∂ † . In the full analogy with (1.30), it is expressed as ∂ † = (−1)dp+d+1 ∗ ∂ ∗ . †

(2.42) †

The operators ∂ and ∂ are nilpotent. The anticommutator −{∂, ∂ } is a Hermitian operator called the Dolbeault laplacian. Definition 2.4. The set of all holomorphic forms (2.40) equiped by the operators ∂ and ∂ † is called the Dolbeault complex. One can also define the anti-Dolbeault complex involving the antiholomorphic forms ¯1 ¯p α ¯ = αm zm ∧ · · · ∧ d¯ zm (2.43) ¯ 1 ...m ¯ p d¯ † and the nilpotent operators ∂¯ and ∂¯ . Generically, the anti-Dolbeault lapla¯ ∂¯† } does not coincide with the Dolbeault one. cian equal to −{∂,

There is, however, a very important class of the manifolds where it does—these are K¨ ahler manifolds. Before discussing this, note that the action of the operators ∂, ∂ † , ∂¯ ¯ and ∂¯† can also be defined on the space of all (p, q)-forms. For ∂ and ∂, † this action represents a rather obvious generalization of (2.41), and ∂ and ∂¯† are their Hermitian conjugates. It is rather obvious that ¯ p,q = αp,q+1 , ∂αp,q = αp+1,q , ∂α ∂ † αp,q = αp−1,q , ∂¯† αp,q = αp,q−1 . (2.44) † † ¯ ¯ The sums ∂ + ∂ and ∂ + ∂ are our old friends, the operators d and d† of the de Rham complex. Definition 2.5. For a K¨ ahler manifold, the set of all forms (2.34) and the † ¯ † ¯ operators ∂, ∂ , ∂ and ∂ is called the K¨ ahler–de Rham complex. One may ask why we impose the condition for the manifold to be K¨ahler. Have we not just seen that the action of the holomorphic exterior derivatives and their conjugates can be defined for any Hermitian complex manifold? The matter is that K¨ ahler manifolds have a following special property: Theorem 2.3. For a K¨ ahler manifold, the Dolbeault and anti-Dolbeault ¯ ∂¯† }. In addition, the anticommutators laplacians coincide, {∂, ∂ † } = {∂, † † ¯ ¯ {∂, ∂ } and {∂ , ∂} vanish.

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Proof. We will not prove this theorem here. We will do so in Chap. 10 using supersymmetric methods. It follows that for K¨ ahler manifolds the Dolbeault laplacian coincides up to the factor 1/2 with the Laplace–de Rham operator 4 = −{d, d+ }. The latter commutes with both exterior derivatives and their conjugates. ¯ ∂ † and ∂¯† form a closed superalgebra This means that the operators ∂, ∂, (while for a generic complex manifold it is not the case). This is the first time that the word “superalgebra” appeared in the main body of our book. Superalgebra is a generalization of Lie algebra including not only commutators, but also anticommutators. The simplest superalgebra (0.3) describes the de Rham complex. And for the K¨ahler–de Rham complex, we are dealing with an extended superalgebra. Rich mathematics associated with the notion of superalgebra will be displayed and discussed later in the book. For the time being, we simply illustrate the construction that we just described by the picture in Fig. 2.2—an adaptation of what is called the Hodge diamond.

p

q

†

†

Fig. 2.2 K¨ ahler–de Rham complex. Each dot corresponds to a set of forms with definite values of p and q.

For any complex manifold, one can define a real (1, 1)-form K = n ¯ ihnm ¯m . It follows from the definition (2.28) that, if the mani¯ dw ∧ dw fold is K¨ ahler, this form is closed: ¯ = 0. dK = (∂ + ∂)K

(2.45)

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In this case, the form K is called the K¨ ahler form. The closedness of K is also a sufficient condition for the manifold to be K¨ahler. 2.2

Real Description

An interesting and important fact is that one can describe complex manifolds without explicitly introducing complex charts, but working exclusively in the real terms. Definition 2.6. An almost complex manifold is a manifold equipped with a tensor field IM N satisfying the properties (i) IM N = −IN M and (ii) P IM N IN P = −δM . The tensor IM N is called the almost complex structure. It is clear that the tangent space projection IAB belongs to SO(D). It is easy to prove that an almost complex manifold is necessarily even-dimensional. Indeed, the determinant of an odd-dimensional skewsymmetric matrix IAB vanishes, which contradicts the condition I 2 = −1. But not any even-dimensional manifold is almost complex. One can prove (though it is not so easy) that S 4 is not. On the other hand, S 6 is almost complex, as we will see soon. A reader who meets for the first time this definition might be somewhat perplexed. It might not be clear for him why an antisymmetric real tensor is called complex structure and the word almost in the definition looks enigmatic. To understand this, consider first the simplest possible example—flat 2dimensional Euclidean space. It can be parametrized by the real Cartesian √ coordinates x1 , x2 or by the complex coordinate z = (x1 + ix2 )/ 2. An obvious relation ∂z/∂x2 = i∂z/∂x1 holds, which can also be presented in the form6 ∂z ∂z + iεM N = 0 ∂xM ∂xN

(2.46)

(with ε12 = 1). The tensor εM N satisfies both conditions in the definition above and is the complex structure in this case. (Up to the sign. It will be convenient for us to choose IM N = −εM N .) For a curved 2-dimensional surface, the complex structure looks a little bit more complicated. For example, for S 2 with the metric ds2 = dθ2 + 6 The property (2.46) holds not only for z, but for any holomorphic function f (z). In the latter case, the real and imaginary parts of (2.46) are none other than the CauchyRiemann conditions.

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sin2 θ dφ2 , it may be chosen as

⇒ Iθ θ

Iθθ = Iφφ = 0 , Iθφ = −Iφθ = sin θ , 1 = Iφ φ = 0 , Iφ θ = − sin θ , Iθ φ = . sin θ

(2.47)

However, when we project it on a tangent plane, we obtain N IAB = IM N eM A eB = ±εAB .

(2.48)

depending on the choice of zweibeins. IAB may be interpreted as the orthogonal matrix rotating the vielbein by ±π/2. Note also that one can describe IAB as a tensor in R3 , Iij = εijk nk

(2.49)

(nk being a unit vector orthogonal to the surface of S 2 ), projected on the tangent plane. Now consider a general multidimensional case. Theorem 2.4. Suppose that a tensor IM N satisfying the conditions above exists. With a proper vielbein choice, its tangent space projection can be brought to the canonical form7   0 1 IAB = diag (−ε, . . . , −ε) with ε = . (2.50) −1 0 Proof. To construct an orthonormal base in the tangent space E where the complex structure acquires the form (2.50), we start with choosing in E an arbitrary unit vector e0 . It follows from I = −I T and I 2 = −1 that the vector e1 = Ie0 has also unit length and is orthogonal to e0 . Obviously, Ie1 = I 2 e0 = −e0 . Consider the subspace E ∗ ⊂ E that is orthogonal to e0 and e1 . If it is not empty, choose there an arbitrary unit vector f0 and consider f1 = If0 . One can easily see that f1 also belongs to E ∗ , is orthogonal to f0 and has unit length. Now consider the subspace E ∗∗ ⊂ E ∗ ⊂ E that is orthogonal to e0 , e1 , f0 , f1 and, if E ∗∗ is not empty, repeat the procedure. We arrive at the matrix (2.50). We further suppose that the equation system ∂z n ∂z n − iIM N = 0 M ∂x ∂xN

(2.51)

7 Again, I AB can be interpreted as an orthogonal matrix rotating the vielbeins and satisfying the condition I 2 = −1.

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has d independent solutions. We may trade now xM for {z n , z¯n¯ } and express IM N in this frame. The expression [it follows from (2.51)] is very simple:   n n ∂xM ∂xM N ∂z N P ∂z n IM = IM iIN Im = ∂z m ∂xN ∂z m ∂xP = −i n ¯ n ¯ Im = iδm ¯ ¯ ,

∂z n ∂xM n = −iδm , ∂xM ∂z m n ¯ n I m = Im = 0. ¯

(2.52)

We now express the metric tensor gM N in the frame with the coordinates {z n , z¯n¯ }. One can derive, using (2.51), that the metric (1.6) acquires a Hermitian form (2.2).8 One can formulate and prove the following theorem: Theorem 2.5. If the equation system (2.51) has d independent solutions, the manifold is complex. Proof. We have to show that the transition functions in the overlap region of two charts parameterized by the coordinates {z n , z¯n¯ } and {wp , w ¯ p¯} are n n p n p¯ holomorphic, z = f (w ), so that ∂z /∂ w ¯ = 0. As we have just shown, if the condition (2.51) is fulfilled, the tensor IM N acquires the form9 (2.52), and this is true for both charts. Assume a generic dependence z n = f n (wp , w ¯ p¯) and consider the transformations of N the component of IM when going from z to w. We derive ∂wq ∂ z¯n¯ ∂wq ∂z n − = δpq , ∂z n ∂wp ∂ z¯n¯ ∂wp ∂wq ∂z n ∂wq ∂ z¯n¯ − = 0. (2.54) n p ¯ ∂z ∂ w ¯ ∂ z¯n¯ ∂ w ¯ p¯ On the other hand, we have the obvious relations ∂wq ∂z n ∂wq ∂ z¯n¯ + = δpq , ∂z n ∂wp ∂ z¯n¯ ∂wp ∂wq ∂z n ∂wq ∂ z¯n¯ + =0 (2.55) ∂z n ∂ w ¯ p¯ ∂ z¯n¯ ∂ w ¯ p¯ It follows that ∂wq ∂z n ∂wq ∂z n q = δ and = 0. (2.56) p ∂z n ∂wp ∂z n ∂ w ¯ p¯ 8 Incidentally,

the conditions (i) and (ii) in the Definition 2.6 above, which imply g M N = IM P IN Q g P Q ,

(2.53)

were crucial in deriving this property. Sometimes, people consider the pairs (g, I) that do not satisfy (2.53), but we will not do so. 9 It is called in this case the complex structure; not “almost” any more.

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The first identity says that the matrix ∂wq /∂z n is not degenerate. But then the second identity dictates that ∂z n /∂ w ¯ p¯ = 0. It is useful for some applications to introduce the tensors QM N =

∂xN ∂z m , ∂z m ∂xM

PM N =

∂xN ∂ z¯m . ∂ z¯m ∂xM

(2.57)

Using (2.51) and its corollary (2.52), we derive QM N =


1 N δ + iIM N , 2 M

PM N =


1 N δ − iIM N . 2 M

(2.58)

The tensor PM N also enters the condition (2.51). It can be interpreted as the projector from the space of all vectors to the space of antiholomorphic vectors. Likewise, QM N is the projector onto the space of holomorphic vectors. The properties QM N QN S = QM S , PM N PN S = PM S , QM N PN S = 0 (2.59) hold. It is now natural to ask in what case the system (2.51) has a solution and in what case it has not. The answer to this question is provided by the Newlander-Nirenberg theorem [7]: Theorem 2.6. The complex coordinates satisfying the condition (2.51) can be introduced and the manifold is complex iff the condition NM N K = ∂[M IN ] K − IM P IN Q ∂[P IQ] K = 0

(2.60)

holds.10 Proof. Necessity. Represent the system (2.51) as DM z n = 0 with DM = ∂M − iIM N ∂N .

(2.61)

10 This

combination is a tensor, in spite of the presence of the ordinary rather than covariant derivatives. This is so because the terms in the covariant derivatives involving the Christoffel symbols cancel out in this case. Using a sloppy language, we will call the L.H.S. of Eq. (2.60) the Nijenhuis tensor. A conventional definition of the Nijenhuis tensor is a little bit different: NM N K(conventional) = IM P NP N K(this

book)

= IM P ∂[P IN ] K + IN P ∂[M IP ] K .

We will do so because the object (2.60) has a more transparent structure, and it is this combination that directly appears in (2.62) below.

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For self-consistency, the conditions [DM , DN ]z n = 0 should also hold. We derive   [DM , DN ]z n = −i∂[M IN ] Q − I[M P (∂P IN ] Q ) ∂Q z n   = −i∂[M IN ] K − iI[M P (∂P IN ] Q )IQ K ∂K z n − I[M P (∂P IN ] Q )DQ z n = −iNM N K ∂K z n

(2.62)

(we used DQ z n = 0 and IQ K ∂P IN Q = −IN Q ∂P IQ K that follows from I 2 = −1). For this to vanish, the tensor NM N K should also vanish (to see that, choose the real coordinates xM as the real and imaginary parts of z n ). Sufficiency. This part of the theorem [the proof of existence of the solution to the system (2.51) under the condition (2.60)] is more diffucult. Well, it might be not so difficult for mathematicians in the case when the complex structures IM N represent analytic functions of the coordinates (it is more complicated when only a limited number of derivatives of IM N over the coordinates exist, but we are definitely not interested in such fine mathematical issues). Then the sufficiency of the conditions [DM , DN ] = KM N Q DQ for the equation system DM z n = 0 to have a solution is a corollary of the classical Frobenius theorem [8]. We will give here another “physical” proof of this fact [9] which will elucidate the meaning of the constraint (2.60). Its linearized version is similar in spirit to multidimensional Cauchy-Riemann conditions. • Let the complex structure IM N have a canonic form (2.50).11 Then the solutions to (2.51) obviously exist. One of the solutions is 1 z(0) =

x1 + ix2 √ , 2

2 z(0) =

x3 + ix4 √ ,... , 2

(2.63)

n but any analytic function of z(0) is also a solution. Suppose now that the complex structure does not coincide with (I0 )M N = diag(−ε, . . . , −ε), but is close to it: I = I0 + ∆, ∆  1. As a first step in the proof, we show that, after such an infinitesimal deformation, solutions to (2.51) still exist. • Let us first do it in the simplest case D = 2. Then the condition (2.60) is fulfilled identically. The condition I 2 = −1 means that {∆, I0 } = 0. This implies that ∆ = ασ 1 + βσ 3 or

∆11 = −∆22 , 11 We

∆21 = ∆12 .

(2.64)

know that, for any almost complex structure, its tangent space projection IAB can always be chosen in this form. But the requirement that the same is true for IM N is not trivial.

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Look now at the system (2.51). We set z = z(0) + δz. The equations acquire the form ∂ i∆1 1 − ∆1 2 ∂ √ (δz) + i (δz) = , ∂x1 ∂x2 2 ∂ i∆2 1 − ∆2 2 ∂ √ (δz) − i (δz) = . (2.65) ∂x2 ∂x1 2 Bearing in mind (2.64), these two equations coincide. They can be expressed as i ∂(δz) = ∆1 1+i2 , (2.66) ∂ z¯(0) 2 which can be easily integrated on a disk. • The simplest nontrivial case is D = 4. The condition {∆, I0 } = 0 implies ∆1 1 = −∆2 2 , ∆1 2 = ∆ 2 1 , ∆1 3 = −∆2 4 ,

∆1 4 = ∆ 2 3 ,

∆3 1 = −∆4 2 ,

∆3 2 = ∆ 4 1 ,

∆3 3 = −∆4 4 , ∆3 4 = ∆ 4 3 . (2.67) 1 2 We pose z → z, z → w. A short calculation shows that, bearing the relations (2.67) in mind, the equations (2.51) are reduced to i ∂(δz) = ∆1 1+i2 , ∂ z¯(0) 2 i ∂(δz) = ∆3 1+i2 , ∂w ¯(0) 2 i ∂(δw) = ∆1 3+i4 , ∂ z¯(0) 2 ∂(δw) i = ∆3 3+i4 . (2.68) ∂w ¯(0) 2 If D > 2, the conditions (2.60) provide nontrivial constraints. Their linearized version is   ∂P ∆N M − ∂N ∆P M = (I0 )P Q (I0 )N S ∂Q ∆S M − ∂S ∆Q M . (2.69) Again, bearing in mind (2.67), one can show that, for D = 4, out of 24 real conditions in (2.69), only 4 independent real or 2 independent complex constraints are left. The latter have a simple form ∂ ∂ ∆3 1+i2 − ∆1 1+i2 = 0 , ∂ z¯(0) ∂w ¯(0) ∂ ∂ ∆3 3+i4 − ∆1 3+i4 = 0 . (2.70) ∂ z¯(0) ∂w ¯(0)

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The first equation in (2.70) is the integrability condition for the system of the first two equations in (2.68). It is sufficient for the solution of this system to exist. Indeed, it implies that the (0,1)-form ω = ∆1 1+i2 d¯ z(0) + ∆3 1+i2 dw ¯(0) is closed, ∂¯0 ω = 0. Bearing in mind the trivial topology of a chart of our complex manifold that we are discussing, ω is also exact (see Theorem 1.2), which is tantamount to saying that the solution exists. The second relation in (2.70) is the necessary and sufficient integrability condition for the system of the third and fourth equations in (2.68). • This reasoning can be translated to the case of higher dimensions. For an arbitrary D = 2d, the equations (2.51) are reduced, bearing in mind I 2 = −1, to d2 conditions similar to (2.68) but with differentiation n ¯ over each antiholomorphic variable z¯(0) for each complex function δz n . The conditions (2.60) lead to d2 (d − 1)/2 complex constraints which represent integrability conditions of the type (2.70). They imply that the forms 1 2 ω1 = ∆1 1+i2 d¯ z(0) + ∆3 1+i2 d¯ z(0) + ..., 1 2 ω2 = ∆1 3+i4 d¯ z(0) + ∆3 3+i4 d¯ z(0) + ...,

(2.71)

etc. are all closed. Due to the trivial topology of the chart, it also means that they are exact. • Once the complex coordinates z = z(0) + δz satisfying the equations (2.51) are found, the complex structure acquires in these new coordinates the canonical form (2.52) and (2.50). Thus we have actually proven that, after a small deformation, IM N can be brought to the form (2.50) by an infitesimal diffeomorphism, provided the condition (2.60) is satisfied. • Let now the complex structure be arbitrary, not necessarily close to I0 of Eq. (2.50). If the tensor IM N (x) is analytic in a given chart (and we assume that), we can expand it into a formal series in a small parameter α: I(x) = I0 + αI1 (x) + α2 I2 (x) + . . .

(2.72)

Do the same for the solutions z n (x) we are looking for: n n n z n (x) = z(0) + αz(1) (x) + α2 z(2) (x) + . . . n αz(1) (x)

(2.73) n

n The correction was determined before. Let z˜ (x) = z(0) + n αz(1) (x). As was just mentioned, the complex structure in these new

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coordinates has the canonical form (2.52) up to the terms ∝ α2 . Introducing the real and imaginary parts of z˜n (x) and calling them x ˜M , we may bring it in the form (2.50). • Taking also into account the term α2 I2 (x) in (2.72), we may express the complex structure in the new coordinates x ˜ as 2˜ I(˜ x) = I0 + α I2 (˜ x) + higher-order terms. (2.74) n Repeating the same procedure that we used to determine z(1) (x), we n n can now determine z˜(2) (˜ x), from that z(2) (x), and likewise all the terms in the series (2.73). • With the only reservation that we did not address a difficult question of the convergence of the series (2.73), the theorem is proven.

In Chap. 9 we will give a supersymmetric interpretation of this theorem. The next important observation is Theorem 2.7. The complex manifold is K¨ ahler iff the ordinary covariant derivatives of its complex structure vanish. Proof. Choose the frame where the complex structure has the form (2.52). One can then derive ∇P Im n = ΓnP q Im q − ΓqP m Iq n = 0 for both P = p and n ¯ P = p¯. The covariant derivative ∇P Im also vanishes. On the other hand, ¯ ∇P Im n¯ = ΓnP¯ q Im q − ΓqP¯ m Iq¯n¯ = −2iΓnP¯ m , q n n q¯ n ∇P Im = 2iΓnP m ¯ = ΓP q¯Im ¯ − ΓP m ¯ . ¯ Iq

(2.75)

These derivatives vanish iff the Christoffels of mixed holomorphicity vanish, i.e. iff the manifold is K¨ ahler. The conditions ∇P IM N = 0 or the condition (2.45) with K = IM N dxM ∧ dxN can serve as alternative definitions of the K¨ahler manifolds. For a generic complex manifold, the Levi-Civita covariant derivatives (2.75) do not vanish. On the other hand, the complex structure is covariantly constant with respect to a class of affine connections with vanishing components GnP¯ q¯ and GnP q , as a glance at (2.75) reveals. Another glance at Eq. (2.20) reveals that the Bismut connection belongs to this class. We arrive at Theorem 2.8. The complex structure of a generic complex manifold is covariantly constant with respect to its Bismut connection.

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There is a beautiful formula that expresses the torsion tensor of the Bismut connection in the real terms. It reads [10, 11] CM N P = IM Q IN S IP R (∇Q ISR + ∇S IRQ + ∇R IQS ) ,

(2.76)

where ∇M is the ordinary Levi-Civita covariant derivative.12 Its validity is easy to check going into the frame M ≡ {m, m}, ¯ using (2.52) and comparing (2.76) with (2.18). 2.3

Examples

Let us now see how all these general constructions work in some particular cases. (1) S 2 ≡ CP1 . We already discussed it earlier. Within the standard latitude-longitude parametrization, its metric is ds2 = dθ2 + sin2 θ dφ2 . The complex structure was written in (2.47). The reader can be easily convinced that the Nijenhuis tensor (2.60) vanishes, as it should. The complex coordinates for the south and north hemispheres can be chosen as in (2.4) and (2.5) and the metric as in (2.6). This metric is derived from the K¨ ahler potential: h =

1 = ∂ ∂¯ ln(1 + z z¯) (1 + z z¯)2

(2.77)

The K¨ ahler form K = ih dz ∧ d¯ z is closed. (2) Not all 4-dimensional manifolds are complex (S 4 is not) and not all complex D = 4 manifolds are K¨ahler. As an example of a complex but not K¨ ahler manifold, consider the Hopf manifold H2 [12]. Topologically, it is S 3 × S 1 . It can be described by a couple of complex coordinates z m=1,2 with the points z m and 2z m being identified (we can thus choose z m to lie in the spherical band 1 ≤ |z| < 2). We choose the conformally invariant metric ds2 =

dz m d¯ zm , z k z¯k

(2.78)

which is consistent with this identification. Obviously, the metric (2.78) does not satisfy (2.28) and is not K¨ahler. 12 Actually, one can replace here the covariant derivatives by the ordinary ones because the contribution of the symmetric Christoffel symbols cancels out in the sum. But we have written ∇I to emphasize the tensor nature of CM N P .

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(3) An example of a K¨ ahler manifold of complex dimension 2 is the complex projective space CP2 . It is a set of complex triples (w0 , w1 , w2 ) identified under the multiplication by a nonzero complex number λ: (w0 , w1 , w2 ) ≡ (λw0 , λw1 , λw2 ) . A natural symmetrically-looking “round” metric on this space reads ds2 = 2

w ¯ α w α dw ¯ β dwβ − w ¯ α w β dw ¯ β dwα (w ¯ α wα )2

(2.79)

(α = 0, 1, 2). Now, CP2 is topologically nontrivial, but it can be covered by three disks with the topology C2 . One of these charts excludes the points with w0 = 0 (the set of all such points has the topology CP1 ). The complex coordinates uniquely describing the points on this chart can be chosen as z 1,2 = w1,2 /w0 . Then the metric (2.79) reduces to the Fubini-Study form:   ¯ z n z¯m 1 δm¯n − , (2.80) hm¯n = ∂m ∂n¯ ln(1 + z z¯) = 1 + z z¯ 1 + z z¯ ¯

z z¯ ≡ z k z¯k , k = 1, 2. Evidently, this metric is K¨ahler. The second chart excludes the points with w1 = 0 and is conveniently described by the coordinates u1 = w0 /w1 , u2 = w2 /w1 . The third chart excludes the points with w2 = 0 and is conveniently described by the coordinates v 1 = w1 /w2 , v 2 = w0 /w2 . The metrics in the second and the third charts expressed in these coordinates have exactly the same functional form as the metric (2.80). A generalization for CPn>2 is rather obvious. This manifold represents a set of complex (n + 1)-tuples, (w0 , . . . , wn ), identified under multiplication by λ 6= 0. It has complex dimension n. The minimal number of the charts that cover CPn is n + 1. In each chart, the metric can be chosen in the form (2.80). (4) S 6 . As was mentioned, we do not know today whether this manifold admits a complex structure or not. But an almost complex structure exists. It represents a generalization of the complex structure (2.47) for S 2 represented as the 3-dimensional tensor IM N (S 2 ) = εM N P xP

(2.81)

projected on the surface |x| = 1. Similarly, an almost complex structure on S 6 can be chosen as IM N (S 6 ) = fM N P xP ,

(2.82)

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where M, N, P = 1, . . . , 7, the condtion |x| = 1 is imposed, and fM N P is a totally antisymmetric tensor describing the algebra of octonions. One of the choices for fM N P is f123 = f176 = f257 = f365 = f246 = f145 = f347 = 1, (2.83) with all other nonzero components being restored by antisymmetry. N Obviously, IN M = −IM N . The validity of IM P IP N = −δM is best seen if projecting (2.82) on the tangent plane. For example, for the “north pole” xP = δ P 7 , we obtain   0 0 0 0 0 −1 0 0 0 0 1 0      0 0 0 1 0 0  (2.84) IAB =  ,  0 0 −1 0 0 0     0 −1 0 0 0 0  1 0 0 00 0 and the property I 2 = −1 is obvious. Thus, it is an almost complex structure, indeed. But it is not a complex structure: the Nijenhuis tensor (2.60) calculated with (2.82) does not vanish.

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Chapter 3

Hyper-K¨ ahler and HKT Manifolds

3.1

Hyper-K¨ ahler Manifolds

This notion was introduced in mathematics relatively late [13]. We begin with Definition 3.1. A hypercomplex manifold is a manifold admitting three different integrable complex structures I p that satisfy the quaternion algebra N (I p )M R (I q )R N = −δ pq δM + εpqr (I r )M N .

(3.1)

N p The complex structures (I p )M N as well as (I p )AB = eM A eB (I )M N , are skew-symmetric matrices. The following theorem holds:

Theorem 3.1. The dimension of a hypercomplex manifold is a multiple integer of 4. Any triple (I p )AB of skew-symmetric matrices satisfying the algebra (3.1) can be brought by an appropriate choice of basis to the following canonical form I 1 ≡ I = diag(I, . . . , I),

I 2 ≡ J = diag(J, . . . , J),

I 3 ≡ K = diag(K, . . . , K) ,

(3.2)

where  0 −1 0 0 1 0 0 0   I=  0 0 0 −1  , 0 0 1 0   0 0 0 −1  0 0 −1 0   K= 0 1 0 0 

0 0 −1 0 0 0 J= 1 0 0 0 −1 0



10 0



 0 1 , 0 0

(3.3)

0 51

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Proof. The proof follows the same pattern as the proof of Theorem 2.4. Choose a unit vector e0 in the tangent space E. The algebra (3.1) and the p skew symmetry of IAB dictate that the vectors e0 and ep = Iˆp e0 have also unit length and are mutually orthogonal. Thus, the dimension of the tangent space is at least 4. In the subspace spanned by the vectors e0 , e1 , e2 , e3 , the matrices I p have the form (3.3). Now consider the space E ∗ ⊂ E orthogonal to e0 and ep . If it is not empty, choose there a unit vector f0 and add this vector, as well as the vectors fp = Iˆp f0 ∈ E ∗ , to the basis. Thus, the dimension of E ∗ is either 0 or at least 4. Consider now the subspace E ∗∗ ⊂ E ∗ ⊂ E orthogonal to (e0 , ep , f0 , fp ). If it is not empty, repeat the procedure. We arrive at (3.2).

For sure, the vector e0 is not special, and one could start with e.g. e1 and restore the vectors e0,2,3 as e0 = −I 1 e1 , e2 = I 3 e1 , e3 = I 2 e1 . The triple (−I 1 , I 3 , I 2 ) is also quaternionic. In the following, we restore the “democracy” between the basis vectors and use the notation (e0 , ep ) → (e1 , e2 , e3 , e4 ), (f0 , fp ) → (e5 , e6 , e7 , e8 ), etc. Then I12 = −I21 = −1, I34 = −I43 = −1 and similarly for J, K. Remark. For the complex manifolds with a single complex structure, one can always choose the real coordinates (the real and imaginary parts of the complex coordinates z n ) such that the tensor IM N acquires the same canonical skew-symmetric form as IAB [cf. Eq. (2.52)]. For a hypercomplex manifold, this is always possible to do for one of the complex structures, but not for all of them. Generically, JM N and KM N would have in this case a nontrivial complicated form.1 The matrices (3.3) are skew-symmetric and belong to the algebra so(4). But an important observation is that they are self-dual,

(I, J, K)AB =

1 εABCD (I, J, K)CD 2

(3.4)

That means that I, J, K belong to one of the su(2)-components of so(4) = su(2) ⊕ su(2). A reader-physicist may recognize in these matrices the so-called ’t Hooft 1 They

obey, however, certain restrictions spelled out in Theorems 3.4, 3.7 below.

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53

p symbols ηAB [14]. The latter are defined as follows:2 p η44 = 0,

p p ηa4 = −η4a = δap ,

p ηab = εabp ,

(a, b = 1, 2, 3) .

(3.6)

In these terms, 3 IAB = −ηAB ,

2 JAB = ηAB ,

1 KAB = −ηAB .

(3.7)

We now introduce the convenient symplectic notation which plays the same role for hypercomplex manifolds as the complex notation for complex manifolds. Represent a vector tangent-space index A by the pair of indices (ia), where i = 1, 2 and a = 1, . . . 2n. We will raise and lower the indices according to X i = εij Xj , Y a = Ωab Yb , where εjk = −εjk with the convention ε12 = 1 and Ωab is the symplectic matrix, which we choose in the form Ωab = diag(−ε, . . . , −ε) [like in Eq. (2.50), though that formula had, of course, a completely different meaning]; Ωab = −Ωab = diag(ε, . . . , ε). In this notation, only the subgroup SU (2) × Sp(n) of the tangent space rotation group is manifest. These restricted rotations leave the matrix εij Ωab invariant. To establish the precise relation between the vector and spinor notations, we introduce 4n rectangular matrices ΣA :
ja
ja (Σ1,2,3,4 )ja = σµ† , 0, . . . , (Σ5,6,7,8 )ja = 0, σµ† , 0, . . . , etc. (3.8) with3 σµ† = {σ, −i1} , where σ are the Hermitian Pauli matrices:     01 0 −i 1 2 σ = , σ = , 10 i 0

(3.11)

σ

3

 =

1 0 0 −1

 .

(3.12)

p 2 In the following, we will also need the dual ’t Hooft symbols η ¯AB , which differ from p ηAB by the signs in the first line: p p η¯a4 = −¯ η4a = −δap ,

The symbols 3 Also

p η¯AB

p η¯ab = εabp .

(3.5)

are anti-self-dual. σµ = {σ, i1} .

(3.9)

Note the identities † σµ σν† + σν σµ = 2δµν .

(3.10)

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Then for any tensor (in the flat tangent space) we have the correspondence i i T ...ja... = √ (ΣA )ja T ...A... , T ...A... = √ (ΣA )ja T ...ja... , (3.13) 2 2 where (ΣA )ja = εjk Ωab (ΣA )kb , and the dots stand for all other indices. In these terms, the flat metric is expressed as 1 g ja, kb = − (ΣA )ja (ΣA )kb = εjk Ωab , 2 gja, kb = εjk Ωab = g ja, kb . (3.14) Note that, for a real vector V A , the components V ja obey the pseudoreality condition V ja = εjk Ωab V kb ≡ Vja .

(3.15)

In the symplectic notation, the canonical flat complex structures (3.2), (3.3) can be expressed as 1 (I p )ja,kb = − (ΣA )ja (ΣB )kb (I p )AB = i(−σ 3 , σ 2 , −σ 1 )jk Ωab , (3.16) 2 where (σ p )jk = (σ p )kj = εkl (σ p )j l ,

(σ p )jk = εjl , (σ p )l k

(3.17)

and (σ p )l k are the Pauli matrices (3.12). The matrices (3.17) are symmetric.4 In the following, when we need to express the complex structures in the symplectic notation, we will use another, more convenient representation: (I p )ja,kb = −i(σ p )jk Ωab

⇒

(I p )ja,kb = −i(σ p )jk Ωab . (3.18)

This corresponds to another choice of the complex structures: (I, J, K) → (K, −J, I). The triple (K, −J, I) is also quaternionic. Definition 3.2. A hyper-K¨ ahler manifold is a hypercomplex manifold with all three complex structures covariantly constant with respect to the LeviCivita connection. Remark. In fact, it is not necessary to stipulate the hypercomplex nature of the manifold in the definition. It is sufficient to require the presence of two different anticommuting covariantly constant complex structures I, J. Indeed, it is not difficult to prove that the matrix K = IJ is also in 4 Note that the matrices (σ p )ja in (3.11) (with both upper or both lower indices) and (σ p )l k in (3.17) (with the indices placed at the different levels) coincide. The difference in conventions is justified by the fact that the indices in (3.17) refer to one and the same subgroup SU (2) ⊂ SO(4n) whereas the indices j, a in (3.11) have a different nature.

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this case a covariantly constant (and hence integrable, see Theorem 2.7) complex structure. A hyper-K¨ ahler manifold is K¨ahler. As was discussed in the previous chapter, the holonomy group of a generic K¨ahler manifold of complex dimension 2n is U (2n) ⊂ SO(4n). But the presence of two extra complex structures brings about additional constraints on the holonomy group. We will now prove an important theorem: Theorem 3.2. A manifold of real dimension 4n is hyper-K¨ ahler iff its holonomy group is Sp(n) ⊂ U (2n) ⊂ SO(4n). Proof. This statement is very well known to mathematicians. We give here its detailed proof in explicit “physical” terms. Direct theorem We prove first the direct theorem: that the holonomy group of a hyperK¨ ahler manifold is Sp(n). After passing to the tangent space, the covariant p constancy condition ∇M IN S = 0 for the triplet of the complex structures p IN S takes the form p p p ∂M IAB + (ωAC,M ICB − IAC ωCB,M ) = 0 ,

(3.19)

N where ωAB,P is the spin connection (1.53) with GN M P = ΓM P . Substitutp ing the constant expression (3.18) for IAB in (3.19), we observe that this condition is reduced to p p ωAC,M ICB − IAC ωCB,M = 0 ,

(3.20)

which tells us that the spin connection, understood as a matrix ωAB in tangent space whose entries are 1-forms, commutes with all complex structures. The 4-dimensional case (n = 1) is especially simple. As was noticed, the complex structures (3.3) are the elements of so(4) forming a basis in one of its su(2) components. A skew-symmetric matrix ωAB that commutes with them belongs to another su(2) component. And5 su(2) ≡ sp(1). If n > 1, we use the symplectic notation introduced above. Bearing in mind (3.18), the condition (3.20) takes the form ωia jb (σ p )jk − (σ p )i j ωja kb = 0 . 5 It

(3.21)

is instructive to rediscuss from this angle the holonomy group of a K¨ ahler 4dimensional manifold equipped with only one complex structure I. A spin connection that commutes with I belongs to the subalgebra su(2) × u(1) = u(2).

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A generic antisymmetric connection ωia jb parametrized as

=

−ωjb ia can be

q ωia jb = εij T(ab) + B[ab] (ij) = εij T(ab) + B[ab] (σ q )ij

(3.22)

q with arbitrary T(ab) and B[ab] . When one substitutes this into (3.21), the first term ∝ εij does not contribute and we are led to

q [σ q , σ p ]ik = 0 . B[ab]

(3.23)

This holds for any p, which implies q = 0. B[ab]

(3.24)

ωAB = ωia jb = εij T(ab) .

(3.25)

We thus derived

But any symmetric matrix of dimension 2n can be presented as T(ab) = Ta c Ωbc ,

(3.26)

where6 Ta c ∈ sp(n). Thus, ωAB (or rather ωA B ) belongs to sp(n). But then RA B = dωA B + ωA C ∧ ωC B also belongs to sp(n), and the theorem is proven. Inverse theorem The proof of the inverse theorem—that, if the holonomy group is Sp(n), the existence of three quaternionic covariantly constant complex structures follows—is based on the following Lemma: Lemma. Let g be a Lie algebra and h be its subalgebra. Let AM and FM N be the gauge potential and the field density for the algebra g. Let FM N ∈ h. Then one can always choose the gauge where also AM ∈ h. Proof. Such a gauge is well-known, it is the Fock-Schwinger gauge xM AM = 0 [15]. In this gauge the potential is expressed via the field density as Z 1 AM = dα αxN FN M (αx) . (3.27) 0 6 Indeed,

Ωh = 0.

an element h of sp(n) is a Hermitian 2n-dimensional matrix satisfying hT Ω +

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Indeed, in the gauge xM AM = 0, we may write AM (y) =

∂AP (y) ∂ [y P AP (y)] − y P ∂y M ∂y M   ∂AM (y) = y P FP M (y) − . ∂y P

(3.28)

Substitute αx for y and rewrite it in the form ∂ ∂ AM (αx) = [αAM (αx)]. (3.29) P ∂x ∂α Integrating this over α between 0 and 1, we arrive at (3.27). αxP FP M (αx) = AM (αx) + xP

In the case of interest, g = so(4n), h = sp(n), A ≡ ω and F ≡ R. If R ∈ sp(n), one can choose the coordinates and vielbeins with ω ∈ sp(n). And once ω ∈ sp(n), it commutes with the quaternionic complex structures (3.18). Bearing in mind (3.19), it follows that the convolutions of these flat structures with the vielbeins are covariantly constant. The proven theorem means that one can define a hyper-K¨ahler manifold as a manifold where the Riemann curvature form RAB lies in the sp(n) algebra. This definition and Definition 4.2 are equivalent. 3.1.1

Examples

Evidently, flat space R4n is a hyper-K¨ahler manifold. The flat complex structures are the constant matrices (3.2). R4n is not compact, but one can easily impose periodic boundary conditions for all Cartesian coordinates to obtain a compact flat torus. There are many nontrivial compact hyper-K¨ahler manifolds, but in four dimensions there is only one such example—the so-called K3 manifold. It is, however, rather complicated and will not be discussed here. We will describe instead two most popular and most discussed in the literature examples of the noncompact hyper-K¨ahler 4-dimensional manifolds: the Taub-NUT manifold [16] and the Eguchi-Hanson manifold [17]. According to the theorem just proven, the curvature form RAB of a 4-dimensional hyper-K¨ ahler manifold belongs to sp(1) ≡ su(2). That means that RAB belongs to one of the su(2) components of so(4) and is either self-dual or anti-self-dual. If the complex structures are chosen to be anti-self-dual, as in (3.3), the form RAB must be self-dual, and vice versa. Another (physical) name for 4-dimensional hyper-K¨ahler manifolds is gravitational instantons. The point is that the Ricci tensor (Ricci)M N for

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these manifolds vanish and they thus represent solutions to the Einstein equations in empty Euclidean space, (Ricci)M N = 0. Indeed, the (anti)-self-duality of the form RAB implies that the Riemann tensor satisfies the condition 1 (3.30) RABCD = ± εABEF REF CD . 2 Multiplying this by δAC , we derive 1 (Ricci)BD = RABAD = ± εABEF REF AD = 0 (3.31) 2 [the last relation is a corollary of the second line in (1.21)]. This explains the word “gravitational”. And “instantons” is a common name by which the physicists call Euclidean solutions of the classical equations of motion. In Chap. 14 we will write such solutions for non-Abelian gauge fields, representing a particular interest for physics. We say few more words about the physical interpretation of the gravitational instantons at the end of this section. (1) Taub-NUT manifold. Consider R4 parametrized by the ordinary Cartesian coordinates xM and impose the constraint r2 = (x1 )2 + (x2 )2 + (x3 )2 + (x4 )2 > a2 (the topology is thus S 3 × R). Choose the metric r−a 2 1r+a 2 dr + (r2 − a2 )(σ12 + σ22 ) + 4a2 σ , (3.32) ds2 = 4r−a r+a 3 where σ1,2,3 are the Maurer-Cartan forms η p xM dxN σp = M N 2 (3.33) r p with ηM N defined in (3.6). The property dσp = 2εpqr σq ∧ σr

(3.34)

holds. A natural choice of the vierbein r 1-forms is r p r−a 1 r+a 2 2 σ3 , e4 = dr. (3.35) e1,2 = r − a σ1,2 , e3 = 2a r+a 2 r−a The connections ωAB can be found as a solution to the Cartan equation deA + ωAB ∧ eB = 0. They are:7   4a2 2a 2a ω12 = 2 − σ3 , ω23 = σ1 , ω31 = σ2 , (r + a)2 r+a r+a 2r 2r 4a2 ω14 = σ1 , ω24 = σ2 , ω34 = σ3 . (3.36) r+a r+a (r + a)2 7 Attention! In the review [3], this and the next formula are given with wrong coefficients.

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These connections are neither self-dual nor anti-self-dual, but the curvatures (1.57) are anti-self-dual as they should be: 4a (e4 ∧ e1 + e2 ∧ e3 ) , R14 = −R23 = (r + a)3 4a R24 = R13 = (e4 ∧ e2 − e1 ∧ e3 ) , (r + a)3 8a (e4 ∧ e3 + e1 ∧ e2 ) . (3.37) R34 = −R12 = − (r + a)3 The fact that the connections (3.36) do not have a definite duality is due to our choice of vierbeins in (3.35). However, as was mentioned above, one can always bring ωAB into the subalgebra su(2) where RAB belongs by using the gauge freedom O(4) to modify the vielbein choice. The form (3.32) of the metric is convenient to demonstrate that it is hyper-K¨ ahler, but for field-theory applications another form [related to (3.32) by a variable change] is more convenient and, probably, more familiar for a reader-physicist. It reads 2 ds2 = V −1 (r) (dΨ + Adx) + V (r) dxdx , (3.38) where x is a 3-vector, Ψ ∈ (0, 2π), 1 V (r) = +λ (3.39) r (r = |x|) is a kind of “Coulomb potential”, and y x Ax = , Ay = − , Az = 0 (3.40) r (r + z) r (r + z) is the vector potential of a magnetic monopole. The parameter λ in (3.39) is positive and can be rescaled to 1. But one can as well consider the metric (3.38), (3.39) with negative λ. The latter metric has several interesting physical interpretations. In particular: the Lagrangian describing the motion of a particle along such manifold also describes the dynamics of two interacting magnetic monopoles of a special type (the so-called BP S monopoles) at large distances r  λ−1 [18]. (The dynamics of the monopoles at arbitrary distances is also described by a somewhat more complicated hyper-K¨ahler metric—the Atiyah-Hitchin metric [19].) The dynamics of the system of an arbitrary number N of monopoles is described by a hyper-K¨ahler manifold of dimension D = 4(N − 1) [20, 21]. The latter enjoys SU (N ) isometry. By Hamiltonian reduction (see Chap. 11.2), one can derive from these generalised Taub-NUT metrics many other hyper-K¨ahler metrics associated with other simple Lie groups [20, 22, 23].

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(2) Eguchi-Hanson manifold. Topologically, this manifold represents a tangent bundle (i.e. a bundle with the fiber representing a tangent space at each point of the manifold)8 over S 2 . Its metric can be chosen in the form    a 4   dr2 2 2 2 + r (3.41) σ + σ + 1 − σ32 ds2 = 1 2 1 − (a/r)4 r r ≥ a. (We included the value r = a in the allowed range, because the apparent singularity of the metric at r = a is in this case removable.) The vierbeins may be chosen as p dr . (3.42) e1,2 = r σ1,2 , e3 = r 1 − (a/r)4 σ3 , e4 = p 1 − (a/r)4 The connections are self-dual: ω14 = ω23 =

1 − (a/r)4 σ1 ,

p

1 − (a/r)4 σ2 ,   = 1 + (a/r)4 σ3 .

ω24 = ω31 = ω34 = ω12

p

(3.43)

The curvature forms are also, of course, self-dual: 2a4 4 (e ∧ e1 + e3 ∧ e2 ) , r6 2a4 R24 = R31 = 6 (e4 ∧ e2 + e1 ∧ e3 ) , r 4a4 R34 = R12 = − 6 (e4 ∧ e3 + e2 ∧ e1 ) . (3.44) r A nice feature of the Eguchi-Hanson metric is its asymptotic flatness. Indeed, it is clear from (3.41) that the metric becomes flat in the limit r → ∞. However, the boundary of the Eguchi-Hanson manifold at infinity is not S 3 (as for the ordinary 4-dimensional space), but rather RP 3 = S 3 /Z2 . Asymptotic flatness allows one to suggest a nice physical interpretation of the Eguchi-Hanson solution [24]. As was mentioned, it represents a “gravitational instanton”. In physics, instantons (Euclidean solutions) can often be interpreted as the quasiclassical trajectories for the tunneling transitions from one vacuum to another. And this is also the case for the Eguchi-Hanson solution. It can be interpreted as a tunneling trajectory between two flat R3 spaces of opposite orientations. The 3-space turns inside out during such a transition. R14 = R23 =

8 Actually, “cotangent bundle” and “cotangent space”, but, as you remember, we have chosen to use sloppy language at this point.

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61

HKT Manifolds

Up to now we discussed the geometrical issues worked out first by mathematicians and then described by physicists in the language of supersymmetry. The reader will learn about the supersymmetric description of generic Riemannian, complex, K¨ ahler and hyper-K¨ahler geometries in a proper place in this book. But HKT manifolds is a different story: they were discovered first by physicists doing supersymmetry [25, 26] and then these geometries were grasped and assimilated by mathematicians [27,28]. In this geometrical chapter we will not talk about supersymmetry, however, and present the HKT geometry using the same language as in the preceding discussions. Definition 3.3. An HKT manifold is a hypercomplex manifold where the complex structures are covariantly constant with respect to one and the same Bismut connection. The following remarks are in order: (1) As was the case for the hyper-K¨ahler manifolds, it is not necessary to stipulate the hypercomplex nature of the manifold in the definition. It is sufficient to require the presence of two anticommuting complex structures I, J that are covariantly constant with respect to one and the same Bismut connection. It follows then that K = IJ is also a complex structure that is covariantly constant with respect to the same connection. (2) “HKT” stands for “hyper-K¨ahler with torsion”. This term is in fact misleading, because, if the torsion is present, HKT manifolds are not hyper-K¨ ahler and not even K¨ahler. But it is now firmly established in the literature, nothing better has been suggested, and we will follow the crowd. (3) The words “one and the same” in the definition are crucial. Indeed, according to Theorem 2.8, each complex structure of a hypercomplex manifold is covariantly constant with respect to its Bismut connection. Generically, these three Bismut connections are different. But they all coincide for an HKT manifold, and this is not trivial. (4) Not any hypercomplex manifold is HKT. But any 4-dimensional hypercomplex manifold is (see Theorem 3.6 below). Examples of hypercomplex non-HKT manifolds of dimension 8 and higher were found in [29].

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The bi-HKT models to be described in Chap. 10.3.1 are also not HKT, but hypercomplex. The following important theorem holds. Theorem 3.3. The holonomy group of an HKT manifold with respect to its universal Bismut connection is Sp(n). Inversely: if there exists a torsionful metric-preserving connection whose curvature form lies in sp(n), one can find three quaternionic complex structures that are covariantly constant with respect to this connection, and we are dealing with an HKT manifold. Proof. We can use literally the same reasoning as in the proof of Theorem 3.2. The presence or absence of torsion in the universal connection with respect to which three quaternionic complex structures are covariantly constant is irrelevant. There is a 2-form associated with each complex structure: ωI = IM N dxM ∧ dxN ,

ωJ = JM N dxM ∧ dxN ,

ωK = KM N dxM ∧ dxN .

(3.45)

For a hyper-K¨ ahler manifold, all these forms are closed [see Eq. (2.45)], but for a generic hypercomplex or HKT manifold, they are not. These forms possess, however, some special properties. We will prove two theorems. One of them is rather simple and another one is a little bit more complicated. Theorem 3.4. Consider a hypercomplex manifold with the quaternionic complex structures I, J, K = IJ and the corresponding forms (3.45). Then the form ωJ + iωK is purely holomorphic and the form ωJ − iωK is purely antiholomorphic with respect to the complex structure I. Proof. Consider the tensor I + = (J + iK)/2. Bearing in mind the quaternion algebra for I, J, K , we obtain II + = −iI + and I + I = iI + . In the frame where I has the form (2.52), we derive that the only nonvanishing components of (I + )M N are (I + )m n¯ . And this means that (I + )M N has only the holomorphic components and the form ωJ + iωK is holomorphic. The antiholomorphicity of the form ωJ − iωK is proven in exactly the same way. Corollary. The only nonzero components of the tensor JM N are Jm n¯ n N and Jm ¯ , and the same is true for KM . It follows in addition that Kmn¯ = −iJmn¯ ,

n n Km ¯ = iJm ¯ .

(3.46)

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Remark. The tensor (I + )mn plays for a hypercomplex manifold roughly the same role as the complex structure IM N for a complex manifold. It is thus natural to call (I + )mn a hypercomplex structure.9 In the flat case, it has a simple form (I + )m n¯ = diag(−ε, . . . , −ε) , (3.47) which looks like the canonical complex structure (2.50) but, of course, the dimension of the latter is two times larger. One can also define the tensor I − = (J − iK)/2. Its only nonzero comn ponents are (I − )m ¯ . In the flat case, it has exactly the same form (3.47) as I + modulo the interchange n ↔ n ¯ . The products (J +iK)M P (J −iK)P N /4 P N and (J − iK)M (J + iK)P /4 give the holomorphic and antiholomorphic projectors (2.58). Theorem 3.5. A hypercomplex manifold is HKT iff the (2, 0)-form ωJ + iωK is annihilated by the action of the operator of the holomorphic (with respect to the complex structure I) exterior derivative ∂I . Proof. We follow the paper [27]. Direct theorem We will prove first that ∂I (ωJ + iωK ) = 0 for an HKT manifold. Let d be the ordinary exterior derivative operator. It can be represented as the sum d = ∂I + ∂¯I . Introduce the operator dI representing (with the factor 2) the “imaginary part” of ∂I : dI = i(∂¯I − ∂I ) . (3.48) Working in the frame where I has the form (2.52), one derives dI α = IM S ∂S αN1 ...NP dxM ∧ dxN1 ∧ · · · ∧ dxNP . (3.49) ˆ Another operator α → Iα is defined as ˆ = IM N1 . . . IM NP ωN ...N dxM1 ∧ · · · ∧ dxMP . Iα (3.50) 1 1 P P When acting on the form of the type (p, q), it multiplies α by the factor ˆ J, ˆK ˆ satisfy the same quaternionic algebra as the (−i)p−q . The operators I, complex structures. We prove now some simple lemmas. Lemma 1. ˆ Iα) ˆ , dI α = (−1)P Id( where P = p + q is the real degree of the form.

(3.51)

9 People usually understand by hypercomplex structure simply the presence of a quaternionic triple of complex structures. But in our book, this term will always refer to the tensors I ± .

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Proof. Choose the complex coordinates. Consider the R.H.S. of (3.51) and use the complex expression (2.52) for I. The components IM N are thus constant and the partial derivatives do not act upon them. The form ˆ has the order P + 1 and, according to (3.50), the expression Id( ˆ Iα) ˆ d(Iα) has altogether (P + 1) + P = 2P + 1 factors of I. This involves P pairs giving I 2 = −1 [this compensates the factor (−1)P ] and we are left with just one unpaired factor. We obtain the expression (3.49). Lemma 2. For any complex manifold, dI ωI =

1 CM RS dxM ∧ dxR ∧ dxS . 3

(3.52)

Proof. Choosing complex coordinates so that ωI = −2ihm¯n dz m ∧d¯ z n¯ , and bearing in mind (2.52), (3.49) and (2.18), we derive ¯ dI ωI = Cmr¯s dz m ∧ dz r ∧ d¯ z s¯ + Cm zm ∧ d¯ z r¯ ∧ dz s , ¯ r¯s d¯

(3.53)

which coincides with (3.52). Corollary: For the HKT manifolds where the Bismut torsions for I, J, K coincide, dI ωI = dJ ωJ = dK ωK .

(3.54)

Lemma 3. Let I, J, K be quaternion complex structures. Then ˆ I = ωI , Iω ˆ I = Kω ˆ I = −ωI , Jω

ˆ J = ωJ , Jω

ˆ K = ωK , Kω

ˆ J = Kω ˆ J = −ωJ , Iω

(3.55) ˆ K = Jω ˆ K = −ωK . Iω

ˆ I = −ωI . By definition, Proof. Let us prove the relation Jω ˆ I = JM R JN S IRS dxM ∧ dxN . Jω On the other hand, JM R JN S IRS = −KM S JN S = −IM N . Other relations are proved similarly. Remark. The condition (3.54) can be rewritten, bearing in mind (3.51) and the first line in (3.55), as ˆ I = Jdω ˆ J = Kdω ˆ K. Idω

(3.56)

We are ready now to prove the main theorem and show that ∂I (ωJ + iωK ) = 0 .

(3.57)

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The real and imaginary parts of (3.57) give a kind of Cauchy-Riemann conditions10 dωJ − dI ωK = 0,

dωK + dI ωJ = 0 .

(3.58)

Consider the first relation. We obtain 1 ˆ ˆ 3 ˆ K = −JˆKdω ˆ K dI ωK = Id( IωK ) = −Idω

remark

=

−Jˆ2 dωJ = dωJ .(3.59)

The number “1” above the equality sign means in virtue of Lemma 1, etc. The relation dωK + dI ωJ = 0 is proved in a similar way. Inverse theorem All the steps of this reasoning can be retraced. Indeed, (3.57) implies (3.58). Take e.g. the relation dωJ = dI ωK and act by Jˆ upon it. We see that 1 ˆˆ ˆ 3 ˆ J = Jd ˆ I ωK = ˆ K. Jdω J Id(IωK ) = Kdω

(3.60)

From this and Lemma 1, one derives that dJ ωJ = dK ωK . Bearing in mind (3.52), this means that the Bismut connections for the complex structures J and K coincide. But then I = JK is also covariantly constant with respect to the same connection and the manifold is HKT.

It follows that the condition (3.57) together with the quaternionic algebra of the three complex structures can be taken as an alternative definition of HKT manifolds. The conditions ∂J (ωK + iωI ) = 0 and ∂K (ωI + iωJ ) = 0 follow in virtue of the direct theorem just proven. With this definition in hand, we can prove the theorem: Theorem 3.6. Any 4-dimensional hypercomplex manifold is HKT. Proof. The form ωJ + iωK has type (2, 0) with respect to the complex structure I. But then, for a manifold of complex dimension 2, ∂I (ωJ +iωK ) vanishes identically. One can now notice that HKT manifolds relate to generic hypercomplex manifolds in roughly the same way as K¨ahler manifolds relate to generic complex manifolds. Indeed, (i) K¨ahler manifolds are complex manifolds with the closed K¨ ahler form and (ii) any complex manifold of real dimension 2 is K¨ ahler. On the other hand, (i) HKT manifolds are hypercomplex 10 Cf.

Eq.(14) in Ref. [25].

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manifolds for which the form ωJ + iωK is closed with respect to ∂I and (ii) any hypercomplex manifold of complex dimension 2 is HKT. We have discussed up to now the ordinary Levi-Civita connection and the torsionful Bismut connection. Another very useful notion that is tailormade for studying the hypercomplex and HKT manifolds is the Obata connection [30]. Definition 3.4. The Obata connection is a torsionless connection with respect to which all three quaternionic complex structures of a hypercomplex manifold are covariantly constant. For a hyper-K¨ ahler manifold, the Obata connection coincides with the Levi-Civita connection, but it is not so in a generic HKT or still more generic hypercomplex case. The essential difference is that the Obata covariant derivative of the metric tensor does not vanish if the manifold is not hyper-K¨ ahler! This means in particular that vectors may not only rotate under parallel transports, but also change their length; the holonomy group is not compact and complicated (see e.g. [31]). Theorem 3.7. [32] Let I, J, K be three integrable quaternionic complex structures. Choose the complex coordinates associated with I. Then the Obata connection is given by the formula ¯

¯

(GO )kmn = Jnl ∂m J¯l k = Knl ∂m K¯l k ,

¯

O k (GO )km¯ ¯ n = (G )mn ,

(3.61)

and all other components of (GO )K M N vanish. Proof. It consists of three steps Lemma 1. The expression (3.61) is symmetric under permutation m ↔ n. Proof. This follows from integrability. Indeed, the condition (2.60) for the structure J implies JS M (∂M JN K − ∂N JM K ) = JNQ (∂Q JS K − ∂S JQ K ) . Choose S = s, N = n, K = k. Then, bearing in mind Theorem 3.4 and its Corollary, we obtain ¯ k ¯ k Js m ∂n Jm = Jn m ∂s Jm ¯ ¯ .

Lemma 2. The covariant derivatives with the connection (3.61) of all complex structures vanish.

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Proof. It can be checked rather directly using Eqs. (3.61), (2.52), and Theorem 3.4. We leave it to the reader. Lemma 3. The Obata connection is unique. Proof. Suppose there are two different Obata connections. Let their difference be ∆N P M . The identities Q N N ∆Q P M I Q − ∆P Q I M = 0 ,

∆Q P M (J

±

iK)QN

−

∆N P Q (J

±

iK)MQ

= 0

(3.62) (3.63)

should hold. N It follows from (3.62), (2.52) and from the symmetry ∆N P M = ∆M P that all the components except ∆npm and ∆np¯¯m ¯ vanish. Consider Eq. (3.63) for the structure J + iK. According to Theorem 3.4, the only nonzero components of the latter are (J + iK)qn¯ . Choose ¯ = 0, as we just proved, N =n ¯ , P = p, M = m. Bearing in mind that ∆npm we obtain ∆qpm (J + iK)qn¯ = 0 . Multiplying that by (J − iK)n¯s and using JK = I and Iq s = −iδqs , we derive ∆qpm = 0. Considering (3.63) for J −iK and choosing there N = n, P = p¯, M = m, ¯ q¯ we prove that ∆p¯m ¯ = 0.

3.2.1

Examples Four dimensions

Theorem 3.8. Let M be a 4-dimensional manifold with the hyper-K¨ ahler ˜ with the metric g˜M N (x) = metric gM N (x). Then the manifold M λ(x) gM N (x) is HKT. Proof. M is K¨ ahler with respect to any of the complex structures I, J, K which we choose in the form (3.3). In particular, it is K¨ahler with respect to I. Choose the corresponding complex coordinates and write the metric m ˜ reads on M as ds2 = 2hnm z n¯ . Then the metric on M ¯ dz d¯ d˜ s2 = 2λ hm¯n dz m d¯ z n¯ .

(3.64)

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The corresponding Bismut connection with respect to which the complex structure I is covariantly constant has the torsion (2.18). Bearing in mind that hnm ahler, we derive ¯ is K¨ Cmnp¯ = (∂n λ) hmp¯ − (∂m λ) hnp¯ .

(3.65)

˜ It is Consider now the torsion tensor CM N P in a generic frame in M. totally antisymmetric and is linear with respect to the derivatives of the conformal factor ∂Q λ(x). With these restrictions, the only tensor structure that one can write is ˜M N P Q g˜QR ∂R (ln λ) CM N P = A E p = A det(g) εM N P Q g QR ∂R λ ,

(3.66)

˜M N P Q is the invariant antisymmetric tensor where A is a constant and E ˜ (1.7) on M. Comparing (3.66) with (3.65) for flat M with gM N = δM N , we can fix A = 1. Repeat this reasoning for the complex structures J and K. We arrive at the same result (3.66) for the corresponding Bismut torsions. Thus, the Bismut connections for the three complex structures coincide, and the ˜ is HKT. manifold M As a simplest example, consider a conformally flat 4-dimensional metric ds2 = λ(x) dxM dxM .

(3.67) 4

This manifold can represent a deformation of R or be topologically nontrivial. The choice λ(x) = 1/(xM xM ) together with the identification xM ≡ 2xM gives the Hopf manifold H2 = S 3 × S 1 [see Eq. (2.78)]. Note that the complex structures (I p )M N for all the manifolds (3.67) coincide with their constant flat expressions (3.3). This means that the Obata connection (3.61) vanishes in this case. We will see in Chap. 12 that the theorem just proven represents a very particular case of a much more general theorem. Using powerful supersymmetric methods, we will see there that all HKT manifolds are grouped in multi-parametric families stemming from a hyper-K¨ahler manifold (we will call such HKT manifolds reducible) or irreducible families not involving a HK manifold as a member. For all the members of one family, the Obata connections (or, better to say, the Obata curvatures—connections depend on the frame choice) coincide. In a certain sense, the Hopf manifold is irreducible, because its topology differs from that of R4 . However, as we have seen, its metric can be locally

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reduced to the flat hyper-K¨ ahler metric by a conformal transformation. An example of a 4-dimensional HKT metric which cannot be reduced by a conformal transformation to any HK metric even locally was constructed in [33]. We will discuss this Delduc-Valent metric in Chap. 12. Higher dimensions: group manifolds. The Hopf manifold H2 is a group manifold, H2 = S 3 × S 1 ≡ SU (2) × U (1). Its metric can be alternatively presented as ds2 = 2dφM dφN Tr{∂M ω ∂N ω −1 } + dχ2 , M

(3.68)

M

where φ parametrize SU (2), ω ∈ SU (2) = exp{iσM φ /2}, and χ is the angle on U (1). Consider now the next in complexity example, which is SU (3). It is a manifold of dimension 8 = 4·2. It was shown in [34,35] that this manifold (as well as many other group manifolds whose dimension represents a multiple integer of 4) is HKT. We start with proving the Samelson theorem [36]. Theorem 3.9. Any group manifold G of even dimension is complex. Proof. We write the metric on G as gM N = 2Tr{∂M ω ∂N ω −1 )} ,

(3.69)

M

where ω = exp{itM x } is a matrix describing an element of a Lie group G with the Hermitian generators tM normalized as Tr{tM tN } = 12 δM N . Obviously, this metric is invariant under the left and right group multiplications, ω → U ωV (U, V ∈ G). Consider a neighbourhood of any point ω on G. Bearing in mind the invariance above, this point can be brought to unity by group rotations, and we can assume ω ≈ 1. This means that xM  1. Expanding (3.69) in xM , we derive11 1 (3.71) gM N = δM N − fAM R fAN Q xR xQ + o(x2 ) , 12 11 We

used the property fAM R dAN Q xR xQ + {M ↔ N } = 0 .

It follows from the relation [83] fADE dBCE + fBDE dCAE + fCDE dABE = 0 , which is a corollary of the graded Jacobi identity [tA , {tB , tC }] + [tB , {tC , tA }] + [tC , {tA , tB }] = 0 .

(3.70)

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where fAM R are the structure constants of G. To prove the theorem, we should define an almost complex structure IM N and show that the Nijenhuis tensor for this structure vanishes. We take care in this definition that the components of the tensor IM N in the different points of the manifold are related to each other by the coordinate transformations generated by, say, a right group multiplication ω → ωV . For the close points, this gives 1 1 IM N (x) = IM N (0) + IM Q (0)fN QP xP − IN Q (0)fM QP xP 2 2 + o(x) . (3.72) The relation (3.72) can be alternatively written as IM N = eM A eN B IAB ,

(3.73)

1 1 eM A = δM A + fM AP xP − fAM R fAN Q xR xQ + o(x2 ) 2 6

(3.74)

where

are the vielbeins (the term ∝ x2 being restored from the condition eM A eN A = gM N ) and IAB ≡ IM N (0) is the tangent space projection of the complex structure, the same at all the points. The complex structure (3.72) is covariantly constant with respect to the Bismut connection with the torsion tensor C M N P = fM N P .

(3.75)

Indeed, it is straightforward to see that at the origin x = 0, 1 1 (B) ∇P IM N = ∂P IM N − fQP M I Q N − fQP N IM Q = 0 , 2 2

(3.76)

where we neglected the contribution of the ordinary Christoffel symbols ΓQ P M , which are of order O(x). Note that the torsion tensor (3.75) is invariant under group rotations, like the metric is, and does not depend on x. Let us define now the matrix IAB acting on the Lie algebra g. Subdivide the set of all generators tA into the set of all positive root vectors Eαj , the set of all negative root vectors E−αj = (Eαj )† and the elements of the Cartan subalgebra H. For a semi-simple group, the latter are spanned by the coroots12 αj∨ = [Eαj , E−αj ]. If the group includes a certain number of 12 This condition holds under a particularly convenient normalisation for the root vectors, the so-called Chevalleu normalisation.

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U (1) factors, their generators also belong to H. Define the action of I on the root vectors as ˆ α = −iEα , IE j j

ˆ −α = iE−α IE j j

(3.77)

ˆ = H. For example, for SU (3) with the standard and assume that IH nomenclature of the generators,     010 001 t1+i2 =  0 0 0  = Eα , t4+i5 =  0 0 0  = Eα+β , 000 000   000 t6+i7 =  0 0 1  = Eβ , 000 1 1 t8 = √ diag(1, 1, −2) (3.78) t3 = diag(1, −1, 0) , 2 2 3 [α and β being the simple roots of SU (3)], this means that I21 = I54 = I76 = −I12 = −I45 = −I67 = 1 , I83 = −I38 = 1 ,

(3.79)

where the first line is equivalent to (3.77) and the second line defines one of the possible actions of Iˆ on H compatible with the properties IAB = −IBA and I 2 = −1. Generically, we subdivide the generators of the Cartan subalgebra into pairs in an arbitrary way, and proceed analogously. Naturally, the dimension of the Cartan subalgebra and hence of the whole manifold should be even. We now substitute the complex structure (3.72) in the Nijenhuis tensor (2.60) and require it to vanish. We arrive at the algebraic condition fABC − IAD IBE fDEC − IBD ICE fADE − ICD IAE fEBD = 0 . (3.80) For SU (3), one can explicitly check that this identity is satisfied, but we will now show how to prove it for any group. Represent each positive root vector as Eαj = tAj + itA∗j .

(3.81) A∗j

with Hermitian tAj and tA∗j . Then for SU (3) Aj = 1, 4, 6 and = 2, 5, 7. Note that the commutator [tA , tB ] can only give a “starred” generator tC ∗ and the commutator [tA , tB ∗ ] can only give tC . In other words, fABC = fAB ∗ C ∗ = fA∗ BC ∗ = fA∗ B ∗ C = 0 , Now consider several cases.

(3.82)

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(1) Consider the L.H.S. of (3.80) with A, B associated with the same root vector Eα : A = A, B = A∗ . Bearing in mind that IAA∗ IA∗ A = −1, it is easy to see that the first term in (3.80) cancels the second one, while the third and the fourth terms vanish. (2) Let now A and B be associated with different root vectors Eα and Eβ , with tC ≡ h belonging to the Cartan subalgebra. The commutator [h, Eα ] is proportional to Eα and hence fABC = 0. In this case, all the terms in (3.80) vanish. (3) A somewhat less trivial case is when A, B, C are associated with three different root vectors. Note that, for any triple α, β, γ, one can find a couple (α, β) such that the commutator [Eα , Eβ ] has no projection on Eγ . Indeed, if by any chance [Eα , Eβ ] is proportional to Eγ , then γ = α+β. But then for the couple (α, γ), the commutator [Eα , Eγ ] may only be proportional to E2α+β (if such a root exists), but not to Eβ or Eα . Thus, we choose a couple α, β for which [tA + itA∗ , tB + itB ∗ ] ∼ 0

(3.83)

as far as fABC is concerned. It follows that fABC ∗ − fA∗ B ∗ C ∗ = fA∗ BC + fAB ∗ C = 0 .

(3.84)

Bearing this and (3.82) in mind, it is easy to see that the relation (3.80) holds for all star attributions. For example, for {ABC} → {ABC ∗ }, we deduce fABC ∗ − fA∗ B ∗ C ∗ − fCB ∗ A − fA∗ CB = 0 .

(3.85)

We concentrate now on SU (3). Theorem 3.10. The SU (3) manifold is HKT. Proof. We first prove the Lemma: Lemma 1. Let Ω be an automorphism of the algebra g so that Ω is an orthogonal matrix satisfying ΩAD ΩBE ΩCF fDEF = fABC .

(3.86)

Let I be an integrable complex structure. Then JAB = (ΩIΩT )AB

(3.87)

also gives after multiplying by eM A eN B an integrable complex structure, which is covariantly constant with the same Bismut connection.

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Proof. It is straightforward to see that J is antisymmetric and squares to −1. Bearing in mind the invariance of fABC , the matrix JAB satisfies the same condition (3.80) as IAB and hence JM N is integrable. Also the covariant derivative (3.76) with the torsion tensor (3.75) vanishes: when deriving this fact for the matrix I, we used the expression (3.74) for the vielbeins, but not a specific form of IAB . To prove the theorem, we have to find a quaternion triple of the complex structures with coinciding Bismut connections. We take I from Eq. (3.79) and search for J in the form (3.87) such that IJ + JI = 0. The third structure is then K = IJ. The automorphism Ω can be found by adjusting first its action on a subalgebra su(2) ⊕ u(1) ⊂ su(3). This subalgebra can be chosen in many different ways, and the theorem can be proven with any such choice [though a modification of the definition (3.79) for I might be necessary in the generic case], but the most convenient and universal choice, which works well also for other groups, involves the su(2) algebra associated with the highest root α + β. It has three Hermitian generators: t4 , t5 and √ t3 + 3 t8 1 (3.88) t˜3 = diag(1, 0, −1) = 2 2 ∨ ∨ (a half of the coroot α + β ). The generator of u(1) should then be chosen to be orthogonal to t˜3 : √ − 3 t3 + t8 1 . (3.89) t˜8 = √ diag(−1, 2, −1) = 2 2 3 With this choice, the elements of the matrix I in the rotated basis (1, 2, ˜ 3, 4, 5, 6, 7, ˜ 8) are the same as in (3.79). We see that the matrix I splits in two blocks: (1) The block in the subspace (4, 5, ˜3, ˜8) that corresponds to the subalgebra su(2) ⊕ u(1) of su(3) described above. (2) The block in the subspace (1,2,6,7) acting on the root vectors E±α , E±β . We note that each block has the form I defined in (3.3). To find the second complex structure, we calculate ΩIΩT with the automorphism Ω : ta → U † ta U , where     iπ iπ U = exp (Eα+β + E−α−β ) = exp t4 4 2   1 0 i 1  √  = √ (3.90) 0 20 . 2 i 0 1

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The transformations of the generators t4,5,˜3,˜8 of the first block and of the generators t1,2,6,7 in the second block can be found either directly by matrix multiplication or using the Hadamard formula: 1 1 eR Xe−R = X + [R, X] + [R, [R, X]] + [R, [R, [R, X]]] + . . . (3.91) 2 6 with R = −i π4 (Eα+β + E−α−β ). We observe that the sectors (45˜3˜8) and (1267) are not mixed. The generators in the first sector transform as Ω

t4,˜8 → t4,˜8 ,

Ω

Ω

t5 → t˜3 → −t5 .

(3.92)

The nonzero matrix elements of J = ΩIΩT in this sector are J58 = −J85 = J˜34 = −J4˜3 = 1, which gives the matrix J in (3.3). In the sector (1267), the automorphism acts in a little bit more complicated way: Ω

t1 →

t1 − t7 √ , 2

Ω

t2 →

t2 − t6 √ , 2

Ω

t6 →

t2 + t6 √ , 2

Ω

t7 →

t1 + t7 √ . (3.93) 2

Still, the action of this automorphism on I gives in this block the structure −J, the same up to a sign as in the first one! The two matrices I = diag(I, I) and J = diag(J, −J) anticommute. The matrix K = IJ gives the third member of the triple. The matrix K can ˜ † ta U ˜ also be obtained directly from I by applying the automorphism ta → U with o n ˜ = exp π (Eα+β − E−α−β ) . (3.94) U 4

As was mentioned, there are many other group manifolds admitting the HKT structure. The manifolds based on the group with only one nonAbelian factor are presented in the list below [34]: SU (2l + 1), SU (2l) × U (1), Sp(l) × [U (1)]l , SO(2l + 1) × [U (1)]l , SO(4l) × [U (1)]2l , SO(4l + 2) × [U (1)]2l−1 , G2 × [U (1)]2 , F4 × [U (1)]4 , E6 × [U (1)]2 , E7 × [U (1)]7 , E8 × [U (1)]8 .

(3.95)

We say here only few words concerning a general proof valid for all the manifolds from the list above, addressing the reader to the papers [34,35,38] for details.

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Consider the manifold SU (4) × U (1). SU (4) has three simple roots. The positive roots of SU (4) are schematically shown below:   ∗ α α+β α+β+γ ∗ ∗ β β+γ   . (3.96) ∗ ∗ ∗  γ ∗∗

∗

∗

We pick up the highest root θ = α + β + γ there and define the action of the matrix I in the Cartan subalgebra so that the coroot α∨ + β ∨ + γ ∨ is mixed with the generator t0 of u(1) and two remaining basic elements of the Cartan subalgebra are mixed with one another. The action of Iˆ on the root vectors is still defined by (3.77). The matrix I thus defined has a blockdiagonal form. There are four distinct blocks: (I) the outer block including the generators E±θ , θ∨ and t0 ; (II) the inner block including the generators E±β and two remaining generators of the Cartan subalgebra; (III) the block including E±α and E±(β+γ) ; (IV) the block including E±(α+β) and E±γ . All these blocks have the form I. Next, we apply to I the automorphism with   iπ (Eθ + E−θ ) . (3.97) U0 = exp 4 ˜ but in contrast to what we had This automorphism gives some matrix J, for su(3), it is not yet what we want: the conversion I → ±J has been effectuated only in the three out of the four blocks. The inner block associated with the nontrivial centralizer of the root vectors E±θ in su(4) [which is g(1) = su(2) ⊕ u(1)] remains unconverted and J˜ does not anticommute with I. To convert the fourth block, we have to pick up the highest root θ1 in g(1) (in the considered case, g(1) has only one root β ≡ θ1 ) and act on J˜ by the additional automorphism with   iπ U1 = exp (Eθ1 + E−θ1 ) . (3.98) 4 The consequent application of these two automorphisms give the matrix J with the structure J or −J in all the blocks. This procedure can be generalized to any group. One picks up the highest root in g, converts the “outer layer” in I with the automorphism (3.97), finds the centralizer g(1) of the highest root, finds the highest root (or highest roots if g(1) is not simple, which is quite often the case) in this centralizer, and then repeats this procedure until it is necessary. Note that not all group manifolds of dimension 4n are HKT. For example, SO(8) is not. The reason is that the Cartan subalgebra of so(8)

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is too small to serve the four “basic roots”—the highest root in so(8) and three highest roots in g(1) = su(2) ⊕ su(2) ⊕ su(2). One should bring about four U (1) factors to make the construction possible. The manifold SO(8) × [U (1)]4 is HKT. The fact that SU (3) and other group manifolds in the list (3.95) are HKT was proven by traditional mathematical methods. For SU (3), an alternative supersymmetric “physical” proof based on the formalism of harmonic superspace (see Chaps. 7,12) also exists [39]. There are also many examples of the HKT manifolds which are not group manifolds, but homogeneous spaces G/H. This was discovered in [10, 35]. The simplest way to see that is to use the method based on automorphisms, which was outlined above [38]. One of the simplest examples is the manifold SU (4)/SU (2). Its tangent space does not include the generators of the “internal” SU (2) group, and the complex structures are the matrices 12 × 12 including only three blocks of the “outer layer” of the complex structures of SU (4) × U (1). The structure J is obtained from the structure I by the action of the outer automorphism (3.97). The automorphism (3.98) is pointless.

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PART 2

PHYSICS

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b2530   International Strategic Relations and China’s National Security: World at the Crossroads

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Chapter 4

Dynamical Systems with and without Grassmann Variables

The main idea of this book is that, to study the geometric properties of manifolds, it is very useful to consider the associated dynamics. We will be interested in particular with the dynamics of particles moving along the geodesic trajectories in the manifold. These trajectories realize the shortest way between two points. A geodesic satisfies the equation P N d2 xM M dx dx . (4.1) = −Γ NP ds2 ds ds In geometry, s is just a parameter distinguishing different points on the line. But one can attribute to s the meaning of time. Then the equation (4.1) is a dynamical equation describing the motion of a particle along the geodesic. Various dynamical systems have been intensely studied in physics since Galileo and Newton, a lot of experience and different know-hows have been accumulated, and we can capitalize on that. Geodesics are, however, not enough. It turned out that, to obtain really strong and interesting results, one has to consider the motion along a supermanifold associated with a given manifold. This supermanifold is characterized by the ordinary coordinates, but on top of that also Grassmann anticommuting coordinates. Certain general remarks concerning classical and quantum dynamics of the ordinary systems and of the systems involving Grassmann variables is the subject of this chapter.

4.1

Ordinary Classical Mechanics

In this and in the following section, we briefly recall some facts very well known from the university course in classical and quantum mechanics. We do so for the benefit of those readers who, being pure mathematicians, probably know the geometry of manifolds that we talked about in the first 79

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part much better than the author, but might need to refresh their memory of certain basic physics notions. By dynamical system we understand a system described by a set of ordinary differential equations dxi = fi (xj ) . (4.2) dt The independent variable t is interpreted as time.1 The functions xi (t) are called dynamical variables. For the equation system (4.2), a Cauchy problem can be posed: if we know the initial conditions at t = 0, we can deduce the values of xi (t) at later times. We restrict our consideration to a special class of dynamical systems, the Hamiltonian systems. Definition 4.1. A Hamiltonian system is a dynamical system involving an even number 2n of dynamical variables which can be separated in two classes—the canonical coordinates qi=1,...,n and the canonical momenta pi=1,...,n —in such a way that the dynamical equations (4.2) acquire the form ∂H(pj , qj ) dqi ≡ q˙i = , dt ∂pi ∂H(pj , qj ) dpi ≡ p˙i = − , (4.3) dt ∂qi where the function H(qj , pj ) is called the Hamilton function or the Hamiltonian. The set of all variables {qi , pi } is called the phase space. It is obvious that2 ∂H dH ∂H ∂H ∂H ∂H ∂H = q˙i + p˙i = − = 0. (4.4) dt ∂qi ∂pi ∂qi ∂pi ∂pi ∂qi In other words, H(qj , pj ) is an integral of motion—a quantity preserving its value during the time evolution described by the equations (4.3). It has the physical meaning of energy. 1 Mathematicians often discuss nowadays another kind of dynamical systems where time is discrete. In this case, one writes instead of (4.2) (n+1)

xi

(n)

= fi [xj ] .

Cellular automata including the famous Conway’s Game of Life belong to this class. But discrete systems are beyond the scope of our book. 2 We still assume the summation over the repeated indices, although neither q , nor p i i represent vectors in the sense of Eqs. (1.3) and (1.4).

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Besides the energy, there might be other integrals of motion. Definition 4.2. The structure3 {A, B}P =

∂A ∂B ∂A ∂B − ∂pi ∂qi ∂qi ∂pi

(4.5)

is called the Poisson bracket of two observables A(qi , pi ) and B(qi , pi ). Theorem 4.1. A quantity f (qi , pi ) is an integral of motion iff the Poisson bracket {H, f }P vanishes. Proof. Just substitute the Hamilton equations of motion (4.3) in ∂f ∂f df = q˙i + p˙i dt ∂qi ∂pi to derive df = {H, f }P . dt

(4.6)

The equation (4.6) describes the time evolution of an arbitrary phase-space function. The Poisson bracket (4.5) has the following nice algebraic properties: {A, B}P = −{B, A}P ,

{AB, C}P = A{B, C}P + B{A, C}P ,

{A, {B, C}P }P + {B, {C, A}P }P + {C, {A, B}P }P = 0 .

(4.7)

Substituting the Hamiltonian H for C in the last (Jacobi) identity, we derive an important theorem: Theorem 4.2. If A(pi , qi ) and B(pi , qi ) are two integrals of motion, their Poisson bracket either vanishes identically or, if not, it is also an integral of motion. Now consider the function L(qi , q˙i ) = pi q˙i − H(pi , qi )

(4.8)

with the momenta pi in the R.H.S. to be replaced by their solution of the first of the Hamilton equations,4 q˙i = ∂H/∂pi . The function L is called the Lagrangian. And the transformation (4.8) is called the Legendre 3 In different textbooks, one can find different sign conventions for the Poisson bracket. We stick to the conventions of [40]. 4 We assume that the equation system q˙ = ∂H/∂p is not degenerate and has a unique i i solution for pi (qj , q˙j ).

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transformation. The latter obviously works both ways: one can start from the Lagrangian and derive the Hamiltonian. For the simplest one-dimensional Hamiltonian H = p2 /(2m) + U (q) , (4.9) we obtain L = mq˙2 /2 − U (q) . (4.10) Theorem 4.3. The set (4.3) of the Hamilton equations is equivalent to the following set of the equations  of Lagrange:  ∂L d ∂L − = 0. (4.11) dt ∂ q˙i ∂qi Proof. Consider the differential dL = q˙i dpi + pi dq˙i − dH . (4.12) Bearing in mind (4.3), we derive ∂H dqi . (4.13) dL = pi dq˙i − ∂qi 5 It follows that ∂L ∂L ∂H = pi , = − . (4.14) ∂ q˙i ∂qi ∂qi Differentiating over time the first relation in (4.14) that expresses the canonical momenta via the Lagrangian, substituting for p˙i the second Hamilton equation and taking also into account the second relation in (4.14), we derive (4.11). Introduce now the action functional Z t1 S[q(t), t1 − t0 ] = L[qi (t), q˙i (t)] dt .

(4.15)

t0

The fact that it depends, besides qi (t), only on the difference t1 − t0 , but not on t0 and t1 separately, follows from the absence of an explicit time dependence in the Lagrangian. Let us fix the boundary conditions: (0) (1) qi (t0 ) = qi , qi (t1 ) = qi , (4.16) (0,1)

where qi is a set of constants. The following very important theorem can be proven: Theorem 4.4. A set of functions qi (t) realizing an extremum of the functional (4.15) with the boundary conditions (4.16) satisfies also the Lagrange equations (4.11) and represents the classical trajectory of the system. 5 The partial derivatives in the L.H.S. of Eq. (4.14) are done, of course, in the assumption of fixed qi or q˙i , whereas the partial derivatives ∂H/∂qi in the R.H.S. are done in the assumption of fixed pi .

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Proof. The variation of the action functional is  Z t1  ∂L ∂L δ q˙i (t) + δqi (t) dt . δS = ∂ q˙i (t) ∂qi (t) t0

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(4.17)

Integrate the first term by parts, bearing in mind that δqi (t0 ) = δqi (t1 ) = 0. We obtain   Z t1   d ∂L ∂L − δqi (t) dt . (4.18) δS = − dt ∂ q˙i (t) ∂qi (t) t0 At an extremum, this variation should vanish, and it should be so for any variation δqi (t). And that is only possible if the expression in the square brackets vanishes. b)

z

a) z1 z0

c)

t0

t1

t

Fig. 4.1 Different trajectories of the stone: (a) the trajectory with minimal action; (b) too high kinetic energy; (c) too low potential energy.

Remark. For not too wild non-degenerate function L, the equation system (4.11) with the boundary conditions (4.16) has a unique solution. That means that the action functional has only one extremum. It is habitual to choose the sign of L in such a way that the action takes a minimal value. The least action principle is one of the major guiding principles in physics. Let us illustrate this general result with a simple example (Fig. 4.1). Take the Lagrangian mz˙ 2 L = − mgz ≡ T − U . 2 It describes the vertical motion of a stone of mass m under the action of gravity. Let us ask: what our stone should do to minimize the action

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R functional (4.15)? Well, to minimize the integral (T − U )dt, the stone would like to increase its potential energy U (z) = mgz and to climb higher. But, bearing in mind that it has an appointment at z1 at fixed time t1 , going too high would entail too high a velocity and a large positive contribution to the integral (4.15) coming from the first kinetic energy term, T = mz˙ 2 /2. The real trajectory is a negotiated compromise between these two effects. The last issue to be discussed in this section is the famous Noether’s theorem [41]. It asserts that any continuous symmetry of the Lagrangian entails the existence of an integral of motion. It is convenient to prove first its restricted version and then generalize. Theorem 4.5. Suppose that the Lagrangian L(qi , q˙i ) is invariant under an infinitesimal transformation of variables: qi → qi + δqi = qi + fi (qj , q˙j ) ,

  1.

(4.19)

Then the Noether charge Q = (∂L/∂ q˙i ) fi is an integral of motion. Proof. The variation of the Lagrangian δL vanishes and so does the variation of the action δS. The latter is expressed as in Eq. (4.17). Integrate it by parts, keeping now the boundary term. We obtain t   Z t1  d ∂L ∂L 1 ∂L (4.20) − fi dt +  fi . δS =  ∂qi dt ∂ q˙i ∂ q˙i t0 t0 This should vanish for any qi (t). In particular, it is true for the classical trajectories satisfying the classical equations of motion. For such trajectories, the vanishing of δS implies the vanishing of the boundary term in (4.20). This gives Q(t1 ) = Q(t0 ). Thus, Q(t) is an integral of motion, indeed. Theorem 4.6. Now suppose that the Lagrangian is not quite invariant under the transformation (4.19), but is shifted by a total time derivative, δL = (d/dt)Λ(q˙i , qi ). In this case, the integral of motion still exists, given by the expression Q(t) =

∂L fi − Λ . ∂ q˙i

(4.21)

Proof. The variation of the action (4.20) does not vanish any more. For a classical trajectory, we derive t ∂L 1 Λ(t1 ) − Λ(t0 ) = fi , ∂ q˙i t0 from which the conservation of (4.21) follows.

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Examples. (1) Consider the variation δqi = q˙i generated by a infinitesimal time shift, ˙ which is a total derivative t → t + . Under this variation, δL = L, so that the conditions of Theorem 4.6 are fulfilled. The corresponding conserved Noether charge is simply the energy, E =

∂L q˙i − L . ∂ q˙i

(2) Consider the Lagrangian x˙ 2 + y˙ 2 − U (x2 + y 2 ) . 2 It is invariant under rotations. Infinitesimally, L=m

δx = y ,

δy = −x .

(4.22)

(4.23)

The conserved Noether charge, Q = m(xy ˙ − yx) ˙ = px y − py x ,

(4.24)

is none other than the anglular momentum Jz (taken with the opposite sign). To avoid a possible confusion of a reader-physicist, one should make the following remark. In most textbooks on classical mechanics (see e.g. [42]) one understands by Noether’s theorem a related, but somewhat different statement. One requires an exact invariance of the Lagrangian under the transformations δqi = fi (qj , t) (no dependence on the general velocities q˙i but a time dependence is allowed) supplemented by a possible transformation of time, δt = f0 (qi , t). This is enough for conventional mechanical applications (for example, the energy conservation follows in that case from the invariance of the Lagrangian under constant time shifts not associated with shifts of qi ). But to apply Noether’s theorem for supersymmetric systems, as we will do in further chapters, it is Theorem 4.6 that we need: the supersymmetry transformations do involve velocity dependence and the supersymmetric Lagrangian is not exactly invariant under these transformations, but is shifted by a total time derivative. In supersymmetric field theories, supersymmetry transformations shift the Lagrangian density by a gradient ∂µ Λ of some function. People often say that such a transformation leaves the action invariant, but this assumes

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that the action integral is done over the whole space-time and that the fields vanish at t = ±∞ R t and at spatial infinity. The action of a mechanical system, calculated as t01 L dt with finite limits, may be modified. Note, however, that, though the action might be modified if a total derivative is added to the Lagrangian, the dynamical equations of motion are not. This is heuristically clear: a possible modification is due to the boundary terms while the equations of motion determine the behavior of the system inside the time interval (t0 , t1 ). One may prove a simple theorem: Theorem 4.7. Let ˙ i) = L + L0 = L + Λ(q

∂Λ q˙i . ∂qi

Then the Lagrange equations of motion following from L0 and L coincide. Proof. We write ∂L0 ∂L ∂ Λ˙ = + ∂qi ∂qi ∂qi and   d ∂L0 d ∂L ∂Λ = + . dt ∂ q˙i dt ∂ q˙i ∂qi It is clear that the shifts due to Λ in the Lagrange equations (4.11) cancel out. The same is true if Λ also depends on the velocities q˙i or on still higher time derivatives of qi (t). One should only have in mind that, for the Lagrangians involving higher derivatives (the shifted Lagrangian L0 does in this case), the form of the equations of motion is modified. For example, for L0 (¨ qi , q˙i , qi ) the equations of motion read     d2 ∂L0 d ∂L0 ∂L0 − + = 0. (4.25) dt2 ∂ q¨i dt ∂ q˙i ∂qi It is these equations that coincide in this case with (4.11). 4.1.1

Algebra of symmetries

Suppose now that the action has two different symmetries: the symmetries under the infinitesimal transformations δf qi = 1 fi (qj , q˙j ),

δg qi = 2 gi (qj , q˙j )

(which can, of course, be upgraded to finite ones).

(4.26)

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Definition 4.3. The variation δ[f,g] qi = (δf δg − δg δf )qi is called the Lie bracket of two transformations (4.26).

(4.27) 6

Consider the case when fi and gi do not depend on the velocities. Then   ∂fi ∂gi − gj ≡ 1 2 hi (qk ) . (4.28) δ[f,g] qi = 1 2 fj ∂qj ∂qj If the action is invariant under (4.26), it is also invariant under (4.28). The Lagrangian may be shifted under the transformations f and g by total derivatives. Suppose that these are the derivatives of the functions depending only on qi , but not on q˙i : d d δg L = 2 Λg (qi ) . (4.29) δf L = 1 Λf (qi ), dt dt Then d δ[f,g] L = 1 2 Λh (qi ), (4.30) dt where ∂Λf ∂Λg − gi . (4.31) Λ h = fi ∂qi ∂qi Consider now the Noether integrals of motion corresponding to the transformations f, g, h: Qf = pi fi − Λf ,

Qg = pi gi − Λg ,

Q[f,g] = pi hi − Λh . (4.32)

Theorem 4.8. The identity Q[f,g] = {Qf , Qg }P holds. This can be verified quite straightforwardly. The Lie bracket of the transformations f and h (and, correspondingly, the Poisson bracket {Qf , Qg }P ) may give still new symmetry of the system, and that is until the algebra of Lie brackets (resp. Poisson brackets) is closed. The simplest nontrivial algebra includes only three symmetries. Consider the Lagrangian depending on 3 coordinates x, y, z that is invariant under the infinitesimal rotations around the first and the second axes:   δ 1 y = 1 z δ 2 z = 2 x R1 : , R2 : . (4.33) δ1 z = −1 y δ2 x = −2 z 6 We define here the “physical” Lie bracket. The “mathematical” Lie bracket is defined as a commutator of abstract vector fields.

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The Lie bracket [R1 , R2 ] gives obviously the rotation R3 around the third axis. The Noether integrals are L1 = py z − pz y,

L2 = pz x − px z,

L3 = px y − py x , (4.34)

and the so(3) algebra {Li , Lj }P.B. = εijk Lk holds. A natural question is what happens if fi , gi , Λf , Λg depend not only on qj , but also on q˙j . As was mentioned, this is the case for the supersymmetry transformations—the main point of interest in our book. A conjecture is that Theorem 4.8 also holds in this general case. The argument is the following. Noether’s theorem guarantees the existence of the integrals of motion Qf and Qg . Then Theorem 4.1 tells us that {H, Qf }P = {H, Qg }P = 0. But then {H, {Qf , Qg }P }P also vanishes—this is a corollary of the Jacobi identity. Thus, Qh = {Qf , Qg }P is also an integral of motion. The only assertion that we cannot prove is that Qh is the Noether charge corresponding to the Lie bracket [f, g]. Well, being confronted with such a difficulty, the author can recall that he is not a mathematician caring about rigourous proofs, but a physicist, can pose a rhetoric question “What else can it be?” and close the discussion. Still, I would not mind to comprehend such a rigourous proof (which I was not able either to find in the textbooks, nor to cook up myself). One of the difficulties of proving this assertion is that the Lie bracket of generic f and g depends not only on the velocities, but also on accelerations. One should probably impose a restriction on f, g that such dependence in [f, g] is absent. Finally, I would like to point out that, as far as the supersymmetry transformations are concerned, the property Q[f,g] = {Qf , Qg }P holds in all the cases considered in this book. This can be verified explicitly. 4.2

Ordinary Quantum Mechanics

It is not proper to discuss here at length physical aspects of quantum mechanics. We concentrate on its mathematical structure. In quantum world, classical trajectories have no sense and the main object of interest is wave function Ψ(qi ). To find the wave functions Ψn (qi ) of the stationary states of a physical system, one has to solve the spectral problem ˆ Ψn (qi ) = En Ψn (qi ) H

(4.35)

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ˆ is an with appropriate boundary conditions. The quantum Hamiltonian H 2 elliptic differential operator acting in the L Hilbert space containing the square integrable functions Ψ(qi ). Besides square integrability, we require the existence of a well-defined inner product Z (4.36) hΨ|Φi = dµ Ψ(qi )Φ(qi ) , Q

where dµ is the measure on the space Q where the dynamical variables qi belong. We have used here Dirac’s notation |Φi for an element of our Hilbert space (ket-vector) and the notation hΦ| for the dual bra-vector. Definition 4.4. The integral Z

ˆ dµ Ψ(qi )AΦ(q i)

(4.37)

Q

will be called the matrix element of the operator Aˆ between the states Ψ ˆ and Φ and will be denoted as hΨ|A|Φi. The basic equation of quantum mechanics is not (4.35), but the timedependent Schr¨ odinger equation ∂ Ψ(qi , t) ˆ Ψ(qi , t) = H (4.38) ∂t (~ is the Planck constant). The equation (4.35) is derived from (4.38) by plugging there a stationary ansatz   iEn t Ψn (qi , t) = Ψn (qi ) exp − . (4.39) ~ i~

The quantum Hamiltonian is obtained from the classical Hamiltonian H(qi , pi ) by substituting the operators pˆi = −i~ ∂/∂qi for the classical momenta pi . Thus, the quantum counterpart of the classical Hamiltonian (4.9) is 2 2 ˆ p, q) = − ~ ∂ + U (q) . H(ˆ (4.40) 2m ∂q 2 For more complicated Hamiltonians that we will encounter later in the book, the transition from the classical to quantum Hamiltonian is not so trivial. It is not immediately clear, for example, what is the quantum counterpart for the classical Hamiltonian H cl = p2 q 2 . Is it

ˆ = pˆ2 q 2 , H

ˆ = q 2 pˆ2 H

(4.41)

or something else? (The two expressions above do not coincide because pˆ and q do not commute, [ˆ p, q] = −i~.)

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The problem of choice between different quantum expressions that correspond to the same classical one is the ordering ambiguity problem. To resolve this ambiguity, extra assumptions are needed, extra guiding principles. One such assumption is rather natural: the quantum Hamiltonian should be Hermitian, otherwise the energies of the eigenstates representing the solutions to the Schr¨ odinger equation (4.35) would become complex. It is unacceptable in physics and also leads to ugly nuissances in mathematics. Thus, the options (4.41) are excluded. On the other hand, the options ˆ = pˆq 2 pˆ or q pˆ2 q are not. H We will discuss it at length later, in Chaps. 5,6,8,9, but we would like to mention right now that among all ordering recipies, there is a distinguished one—the Weyl ordering procedure. We will give here the definition, while its relevance and niceness will be revealed later. Definition 4.5. Let A(pi , qi ) be a function in phase space.7 Consider its Fourier decomposition Z Y A(pi , qi ) = dαi dβi h(αi , βi )ei(αi pi +βi qi ) . (4.42) i

Then the operator ˆ pi , qi ) = A(ˆ

Z Y

dαi dβi h(αi , βi )ei(αi pˆi +βi qi ) .

(4.43)

i

is called the Weyl-ordered operator corresponding to the classical function A(pi , qi ). And the function A(pi , qi ) is called the Weyl symbol of the operˆ ator A. For the monoms like p2 q 2 , this amounts to the symmetric ordering: 1 2 2 p q + q 2 pˆ2 + pˆq 2 pˆ + q pˆ2 q + pˆq pˆq + q pˆq pˆ) . p2 q 2 → (ˆ 6 This follows from the simple fact that each term in the expansion of the exponential in (4.43) includes such symmetrized products. To find the Weyl symbol of an arbitrary polynomial operator, one should first represent it as a sum of symmetric combinations using [ˆ p, q] = −i~. For example, the Weyl symbol of pˆq is pq − i~/2 and the Weyl symbol of q pˆ is pq + i~/2. Note that the Weyl symbol of the product of two operators does not coincide with the product of their Weyl symbols. Instead of that, one has    ∂2 ∂2 i~ ˆ ˆ − AW (pi , qi )BW (Pi , Qi ) . (AB)W = exp − 2 ∂pi ∂Qi ∂Pi ∂qi p=P,q=Q (4.44) 7 Physicists

call such functions classical observables.

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(the so-called Gr¨ onewold-Moyal product [43]). It follows that the Weyl symbol of the commutator of two operators is    ∂2 ∂2 ˆ B] ˆ W = −2i sin ~ [A, − AW (pi , qi )BW (Pi , Qi ) 2 ∂pi ∂Qi ∂Pi ∂qi p=P,q=Q def

= −i~{AW , BW }GM .

(4.45)

In the classical limit ~ → 0, the Gr¨onewold-Moyal bracket {A, B}GM reduces to the Poisson bracket (4.5). For simple operators relevant for physical applications, the terms of order ∼ ~3 , etc. in (4.45) vanish, and it is the Poisson bracket that goes over to the commutator under quantization. An important corollary of the last statement and of Theorem 4.1 is that the quantum operator Aˆ corresponding to a classical integral of motion ˆ A] ˆ = 0. A(pi , qi ) commutes with the quantum Hamiltonian: [H, The final (in this section) remark is the following. When a mathematician writes (4.35), s/he assumes that the spectrum is pure point (discrete in physical language) and all the eigenstates Ψn are normalizable. In physics, one often discusses the problems with continuous spectrum. For example, one can take the Hamiltonian (4.40) with vanishing U (q) that describes free motion. In this case, the eigenfunctions represent plane waves ∼ eipq that are not normalizable. To make a mathematical sense of such a problem, one has to perform a regularization—introduce a parameter that slightly modifies the problem in such a way that the continuous spectrum becomes discrete and then explore the limit when the regularization is lifted. For the Hamiltonian like −∂ 2 /∂q 2 , one has to introduce a finite box—allow for the particle to move only in a finite range |q| < L and send L → ∞ afterwards. In our book, we will mostly study compact manifolds. In the associate dynamical systems, the motion is finite and the spectrum is discrete. And if the manifold is not compact (for example, the Taub-NUT manifold and the Eguchi-Hanson manifold studied in Chap. 3.1.1 are not compact), such regularization procedure will always be had in mind. 4.3

Grassmann Variables

The ordinary phase space discussed in the previous section was parameterized by coordinates and momenta representing ordinary commuting real numbers. One can, however, generalize this discussion and assume that the phase space of the system includes, besides ordinary commuting coordinates

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and momenta, also anticommuting Grassmann variables. Grassmann variables were introduced in physics half-a-century ago by Felix Berezin [44]. They are indispensible for a correct description of fermion fields in quantum field theory. In this book, we will not touch upon field theory issues, but Grassmann dynamical variables can also be introduced for classical and quantum mechanical systems. And that is what the rest of this chapter is about. To begin with, we introduce the mathematical notions of Grassmann numbers and Grassmann algebra. The main definitions are the following: • Let {ai } be a set of n basic anticommuting variables: ai aj + aj ai = 0. The elements of the Grassmann algebra can be written as the functions f (ai ) = c0 +ci ai +cij ai aj +. . . where the coefficents c0 , ci , . . . are ordinary (real or complex) numbers. The series terminates at the n-th term: the (n + 1) - th term of this series would involve a product of two identical anticommuting variables (say, a21 ), which is zero. Note that even though a Grassmann number like a1 a2 commutes with any other number, it still cannot be treated as an ordinary number but represents instead an even element of the Grassmann algebra. There are, of course, also odd anticommuting elements. The variables {ai } are called the generators of the algebra. • One can add Grassmann numbers, f (ai )+g(ai ) = c0 +d0 +(ci +di )ai +. . ., as well as multiply them. For example, (1 + a1 + a2 )(1 + a1 − a2 ) = 1 + 2a1 − 2a1 a2 (the anticommutation property of ai was used). • One can also differentiate the functions f (ai ) with respect to Grassmann def

def

variables: ∂/∂ai (1) = 0, ∂/∂ai (aj ) = δij , the derivative of a sum is the sum of derivatives, and the derivative of a product of two functions satisfies the Leibniz rule, except that the operator ∂/∂ai should be thought of as a Grassmann variable, which sometimes leads to a sign change when ∂/∂ai is pulled through to annihilate ai in the product. For example, ∂/∂a1 (a2 a3 a1 ) = a2 a3 , but ∂/∂a1 (a2 a1 a3 ) = −a2 a3 . • One can also integrate over Grassmann variables. In contrast to the ordinary case, the integral cannot be obtained here as a limit of integral sums, cannot be calculated numerically through “finite limits” (this makes no sense for Grassmann numbers) with Simpson’s method and so on. What one can do, however, is to integrate over a Grassmann variable in the “whole range” (“from −∞ to ∞” if you will, though this is, again, meaningless). The definition (due to Berezin) is Z ∂ def f (a) . (4.46) daj f (a) = ∂aj

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We have in particular Z dai aj = δij .

(4.47)

An obvious (and important for the following) corollary of (4.46) is Z ∂f daj = 0 (4.48) ∂aj (no summation over j). • If the Grassmann algebra involves an even number of generators 2n, one can divide them into two equal parts, {aj=1,...,2n } → {aj=1,...,n , a ¯j=1,...,n } and introduce an involution, aj ↔ a ¯j , which we will associate with complex conjugation. We will assume in particular that, simultaneously with the involution of the generators, the ordinary numbers c0 , etc. are replaced by their complex conjugates: ¯i + di ai + . . . (4.49) f (a) = c0 + ci ai + di a ¯i + . . . −→ f (a) = c0 + ci a It is also convenient to assume that, for any two elements f, g of the Grassmann algebra, (f g) = g¯f¯, as for the Hermitian conjugation (the order of the factors in (4.49) is, of course, not relevant). The following important relation holds: Z Y n dai d¯ ai exp{Mjk a ¯j ak } = det(M ) . (4.50) i=1

4.4

Grassmann Dynamics

The fact that the Grassmann variables can be treated in the same way as the ordinary ones, that one can describe their classical dynamics using the methods of analytical mechanics outlined in Sect. 5.1 and quantum dynamics by solving an approriate Schr¨odinger equation, was dicovered comparatively recently [45, 46], after the discovery of supersymmetry. But in our book the course of history is reversed. We will first discuss non-supersymmetric Grassmann systems, and supersymmetry will be introduced in the next chapter. As the simplest example, consider the classical Lagrangian ¯ , L = −iψ˙ ψ¯ + ω ψψ

(4.51)

where ψ and ψ¯ are anticommuting variables, ¯ 2 = 0, (ψ)2 = (ψ)

¯ = 0, ψ ψ¯ + ψψ

(4.52)

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R and ω is a real constant. The corresponding action S = Ldt is real (invariant under the involution defined above). For the second term in (4.51), it is seen immediately, and the conjugation of the first term gives ¯ ¯ The total time derivative in L ¯ does not contribute iψ ψ¯˙ = id(ψ ψ)/dt − iψ˙ ψ. in the action. The Lagrange equations have the same form as for the ordinary commuting variables:   d ∂L ∂L − = 0 (4.53) dt ∂ ψ˙ ∂ψ and the complex conjugate. This gives iψ˙ + ωψ = 0,

iψ¯˙ − ω ψ¯ = 0 .

(4.54)

The variables ψ and ψ¯ “parameterize” phase space (whatever the word “parameterize” means for a Grassmann coordinate). If the Lagrangian is written as in (4.51), involving time derivative of ψ ¯ it is natural to call ψ rather than ψ¯ a Grassmann coordinate but not of ψ, and Πψ = ∂L/∂ ψ˙ = −iψ¯ is then the canonical Grassmann momentum. The classical Hamiltonian is obtained from (4.51) by the Legendre transformation,8 ˙ ψ − L = ωψ ψ¯ . H = ψΠ

(4.55)

The Hamilton equations of motion are ∂H ˙ψ = −Π ∂ψ

(4.56)

and the complex conjugate. They coincide with (4.54). ¯ g(ψ, ψ) ¯ is now defined as The Poisson bracket of two functions f (ψ, ψ),  {f, g}P = −i

∂2 ∂2 + ¯ ¯ ∂ψ∂ Ψ ∂ ψ∂Ψ



¯ ¯ f (ψ, ψ)g(Ψ, Ψ)

. (4.57)

¯ Ψ ¯ ψ=Ψ, ψ=

We derive in particular ¯ P = {ψ, ¯ ψ}P = i , {ψ, ψ} ¯ ψ}P = −{ψ, ψ ψ} ¯ P = iψ , {ψ ψ, ¯ ψ} ¯ P = −{ψ, ¯ ψ ψ} ¯ P = −iψ¯ . {ψ ψ, 8A

(4.58)

reader-physicist has noticed a similarity between the Hamiltonian (4.55) and the expression H = ωa¯ a for the Hamiltonian of the harmonic oscillator expressed via the holomorphic variables. It is by no means a coincidence. In fact, the system that we are now studying is called the Grassmann oscillator.

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A generic expression for the Poisson bracket in a system involving several usual real commuting phase-space variables (qj , pj ) and several complex anticommuting phase-space variables (ψα , ψ¯α ) reads    ∂2 ∂2 ∂2 ∂2 + {f, g}P = − −i ¯α ∂pj ∂Qj ∂qj ∂Pj ∂ψα ∂ Ψ ∂ ψ¯α ∂Ψα ¯ ¯ p=P,q=Q, ψ=Ψ,ψ= f (p, q; ψ, ψ)g(P, Q; Ψ, Ψ)| (4.59) ¯ Ψ ¯ With this particular definition, the time evolution of an arbitrary phasespace function A(pj , qj , ψα , ψ¯α ) is still given by the relation (4.6). When the Poisson bracket {H, A}P vanishes, A(pj , qj , ψα , ψ¯α ) is an integral of motion. We can now observe that • The Poisson bracket (4.59) satisfies the generalized Jacobi identity similar to that in (4.7), but with the signs depending on whether A, B and C are even or odd. Its corollary, Theorem 4.2 is still valid. • If at least one of the functions f, g represents an even element of the Grassmann algebra generated by ψα , ψ¯α , the expression (4.59) is antisymmetric under exchange f ↔ g. When going from classical to quantum mechanics, the Poisson bracket goes in this case to the commutator of two operators.9 {f, g}P → i[fˆ, gˆ] . (4.60) • When both f and g are odd anticommuting elements of the Grassmann algebra, the expression (4.59) is symmetric under exchange f ↔ g and goes over to the anticommutator under quantization:10 {f, g}P → i{fˆ, gˆ} . (4.61) We derive in particular that ¯β } = δαβ {ψˆα , ψˆ {ψˆα , ψˆβ } = 0, {ψˆ¯α , ψˆ¯β } = 0 . (4.62) ˆ A nontrivial anticommutator {ψˆα , ψ¯β } tells us that the quantum operators corresponding to classical Grassmann variables satisfy the Clifford algebra: “Grassmann” becomes “Clifford” after quantization. For the toy model discussed above, the only nontrivial anticommutator ˆ ¯ ψ} ˆ = 1. A natural representation of this quantum algebra is is11 {ψ, ¯ = ∂/∂ψ. ˆ ψ = ψ and ψˆ 9 We

have set now ~ = 1 and will mostly stick to this choice throughout the book. is why Grassmann variables are so interesting for physicists. It has been known since the old works of Fierz and Pauli [47] that the fermion field operators anticommute. 11 Under this convention, the classical canonical momentum Π = −iψ ¯ derived from the ψ ˆ Lagrangian (4.51), goes over to Πψ = −i∂/∂ψ. 10 That

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The quantum counterpart for the classical Hamiltonian (4.55) reads ˆ = ωψ ∂ , H (4.63) ∂ψ The Hilbert space where it acts is very simple. It involves the wave functions Ψ(ψ) which can be written as Ψ(ψ) = b + aψ . (4.64) All the higher-order terms in the Taylor expansion of Ψ(ψ) vanish due to the property ψ 2 = 0! The Hamiltonian (4.63) has only two eigenfunctions: Ψ(ψ) = 1 with the eigenvalue E = 0 Ψ(ψ) = ψ with the eigenvalue E = ω . (4.65) Thus, our Grassmann oscillator is simpler than the ordinary harmonic oscillator: instead of the infinite tower of equidistant states, it has only two states. Note that the Hamiltonian (4.63) also admits a matrix representation. One can trade the wave functions (4.64) of a holomorphic Grassmann variable ψ for the 2-component vectors12   a Ψ = . (4.66) b ¯ and the Hamiltonian H ˆ are represented by the Then the operators ψˆ and ψˆ matrices ψˆ =



01 00

 = σ+ ,

ψˆ¯ =



00 10

 = σ−

 1 + σ3 10 . (4.67) = ω 00 2 The matrix representation has been habitually used in quantum mechanics to describe e.g. the electron motion in a magnetic field (see the next chapter), but it is not convenient for the systems involving many ordinary and Grassmann degrees of freedom. For such systems, the Hilbert space includes the wave functions Ψ(qj , ψα ) = Ψ(0) (qj ) + Ψ(1) (4.68) α (qj )ψα + · · · The highest term in this expansion includes the product of all Grassmann variables available. An alternative (a less convenient and rarely used) description would involve wave functions representing tensor products of a large number of spinors. ˆ = ω H



12 Physicists prefer to call the objects (4.66) spinors. There are certain reasons for that, but in fact, as far as the quantum system (4.63) is concerned, they are neither spinors, nor vectors—they do not transform under rotations in any space.

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Chapter 5

Supersymmetry

5.1

Basic Definitions

Definition 5.1. A quantum system is called supersymmetric if its Hamiltonian can be represented as the anticommutator of two nilpotent mutually conjugated operators called supercharges: ˆ 2 = (Q ˆ † )2 = 0, ˆ Q ˆ†} = H ˆ. Q {Q, (5.1) In the Introduction, we already displayed this algebra [see Eq. (0.4)] and ˆ H] ˆ = [Q ˆ † , H] ˆ = talked about it. As was mentioned there, it implies that [Q, ˆ Q ˆ † are the integrals of motion. 0, i.e. the supercharges Q, ˆ as the sum of its real and Representing the complex supercharge Q imaginary parts, ˆ ˆ ˆ = Q1 + iQ2 (5.2) Q 2 ˆ 1,2 , one can rewrite (5.1) as with Hermitian Q ˆ 21 = Q ˆ 22 = H ˆ, ˆ1, Q ˆ2} = 0 . Q {Q (5.3) In other words, the Hamiltonian admits in this case two different anticommuting Hermitian square roots. Note that the existence of a single square root is a trivial property of any Hamiltonian whose spectrum is bounded from below. To be more precise, one has first to bring the ground state energy to a positive or zero value by adding, if necessary, a constant to the Hamiltonian and to extract the square root afterwards. But the presence of two different square roots imposes nontrivial constraints on the spectrum. Theorem 5.1. The eigenstates of a supersymmetric Hamiltonian have non-negative energies. If a state Ψ0 with zero energy exists, it is annihilated by the action of the supercharges: ˆ Ψ0 = Q ˆ † Ψ0 = 0 . Q (5.4) 97

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ˆ 1 Ψ0 6= 0 or Q ˆ 2 Ψ0 6= 0. Proof. Suppose that (5.4) does not hold. Then Q ˆ 1 Ψ0 6= 0. The Hermitian operator Q ˆ 1 commutes with the Hamiltonian Let Q and hence these two operators can be simultaneously diagonalized. Hence ˆ 1 . Then Q ˆ 1 Ψ0 = λΨ0 with Ψ0 can be chosen to be an eigenstate of Q ˆ 0= nonzero λ. And then it follows from the first relation in (5.3) that HΨ 2 1 λ Ψ0 and the energy is strictly positive. ˆ with E 6= 0 are double degenerate. Theorem 5.2. All eigenstates of H Proof. Choose the basis in our Hilbert space including the states that are ˆ and simultaneously the eigenstates of Q ˆ 1 . Pick up one eigenstates of H ˆ of such states with positive √ energy E. The eigenvalue √ of Q1 may in this case be equal √ to λ = ± E. Let for definiteness λ = E. Then0 the state √ ˆ 2 Ψ/ E is also an eigenstate of Q ˆ 1 with the eigenvalue λ = − E. Ψ0 = Q ˆ 1 and Q ˆ 2 anticommute, we derive Indeed, bearing in mind that Q ˆ ˆ √ ˆ 1 Ψ0 = − Q2√Q1 Ψ = − E Ψ0 . Q E

(5.5)

ˆ 1 (and The states Ψ and Ψ0 have different eigenvalues of the operator Q hence they are orthogonal), but the same energies and the same norm. ˆ 2 (note that They are√obtained from one another by the action of Q ˆ 2 Ψ0 = E Ψ) and represent a double degenerate pair in the spectrum Q of the Hamiltonian. Had we picked up in the beginning another eigenstate ˆ 1 , we would obtain another such degenerate pair (or maybe exactly the of Q same if the state Ψ0 were chosen). Consider the states Ψ± = Ψ ± iΨ0 . It is easy to see that √ ˆ Ψ− = E Ψ+ , ˆ † Ψ− = 0 , Q Q √ ˆ Ψ+ = 0 , ˆ † Ψ+ = E Ψ− . Q Q 1 One

(5.6)

can make here a following side remark. It was essential for the proof that the opˆ and Q ˆ † entering (5.1) are Hermitially conjugate to one another so that Q ˆ 1 and erators Q ˆ Q2 are Hermitian. If this condition is not fulfilled, the spectrum may include negative energies. It seems at the first sight that in that case one would obtain a non-Hermitian Hamiltonian that can have any eigenvalues, not necessarily real. Unexpectedly, there exist a certain special generalized supersymmetric system whose algebra includes the ˆ ¯ 6= Q ˆ † . Seemingly, its Hamiltonian is not Hermitian, but actually ˆ and Q supercharges Q it belongs to the class of so-called pseudo-Hermitian or, better to say, crypto-Hermitian Hamiltonians [48]. Its spectrum is real, but it includes both positive and negative energies. This system represents an example of the system with benign ghosts: a ground state is absent, but still the evolution operator is unitary [49].

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In other words, all the excited states can be divided into two sectors. The ˆ † and go over to Ψ+ under the states Ψ− are annihilated by the operator Q ˆ ˆ and go over action of Q. The states Ψ+ are annihilated by the operator Q † ˆ (Fig. 5.1). to Ψ− under the action of Q

Fig. 5.1

A supersymmetric doublet.

ˆ and Q ˆ † and are And the zero-energy states are annihilated by both Q not paired. As we will see soon, it often makes sense to attribute some of these states to the sector |−i and some others to the sector |+i, but to do so, one has to strengthen our discussion, to write down supersymmetric Hamiltonian for some particular systems and mark out their peculiarities and common features. Before doing that, we will prove a simple general theorem (the inverse of Theorem 5.2): Theorem 5.3. If all the excited eigenstates of the Hamiltonian are double degenerate, the system is supersymmetric. Proof. Divide all the eigenstates into two subsets such that each subset ˆ and Q ˆ † can includes one member of the degenerate pair. The operators Q be defined via their matrix elements as in (5.6). It is easy then to check that the algebra (5.1) is fulfilled. 5.2

Supersymmetric Oscillator

The simplest ordinary quantum system is the harmonic oscillator. And the simplest supersymmetric quantum system is the supersymmetric oscillator [50].2 It is convenient to use the holomorphic representation where the wave functions of the ordinary oscillator depend on the holomorphic variable ωq − ip a ¯ = √ (5.7) 2ω and the Hamiltonian reads3   ∂ ∂ ˆ osc = ω a H ¯ + a ¯ (5.8) 2 ∂¯ a ∂¯ a 2 In fact, one can supersymmetrize any ordinary system—supersymmetry is like a universal robe that can cover any body. 3 Here a ¯ and ∂/∂¯ a represent the creation and annihilation operators a ˆ† and a ˆ.

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Its normalized eigenstates are (¯ a)n Ψn (¯ a) = √ . n!

(5.9)

Its spectrum En

  1 = n+ ω 2

(5.10)

is very well known. To write down the Hamiltonian of the supersymmetric oscillator, one simply has to add to (5.8) the Hamiltonian (4.63) for the Grassmann oscillator, shifted by an appropriately chosen constant:   ∂ ∂ ˆ sup−osc = ω a +ψ . (5.11) H ¯ ∂¯ a ∂ψ The wave functions depend now on a ¯ and the Grassmann variable ψ. The eigenfunctions and the eigenvalues read (¯ a)n Ψn (¯ a, ψ) = √ , n! (¯ a)n ˜ n (¯ Ψ a, ψ) = √ ψ , n!

E = ωn , E = ω(n + 1) .

(5.12)

We see two towers of states. There are the states that do not involve the Grassmann factor ψ and there are the states that do. For a physicist, it is natural to call the former bosonic and the latter fermionic. The ground state with zero energy is bosonic.4 All excited states come in degenerate pairs including a bosonic and a fermionic state. The system is thus supersymmetric, and it is easy to write down explicit expressions for the supercharges: ˆ= Q

√

ωψ

∂ ∂¯ a

ˆ† = Q

√

ωa ¯

∂ . ∂ψ

(5.13)

The supercharges (5.13) are nilpotent, and their anticommutator gives the Hamiltonian (5.11). 4 It is also possible to change signs in the appropriate places and consider the system where the ground state is fermionic. This is the convention that we used in [51] to illustrate physical Dirac’s picture where the vacuum state involves an infinite number of fermions occupying the negative-energy levels in the Dirac sea. But in this book, we have preferred the convention that illustrates better the mathematical structure of the theory.

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Supersymmetry

5.3

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101

Electrons in a Magnetic Field

Historically, it was the first supersymmetric system constructed and studied. This was done by Landau back in 1930 [52].5 This section is written in a different style, compared to the rest of the book. It is mainly addressed to a reader-physicist who knows this system from kindergarten and might be surprised to find that it is in fact supersymmetric. A mathematician may skip it [maybe after having a brief look at Fig. 5.2 and Eq. (5.25)] and go directly to Sect. 5.4. The generic Hamiltonian describing the motion in a magnetic field (the Pauli Hamiltonian) reads h  i2 ˆ − eA σ· P c ˆ = , (5.14) H 2m where e is the electron charge, c is the speed of light, A is the vector potential, and σ are the Pauli matrices (3.12). The latter satisfy the algebra σj σk = δjk 1 + ijkl σl .

(5.15)

Bearing this and the definition Pˆj = −i~ ∂/∂xj in mind, we may rewrite (5.14) as 2  ˆ − eA P c e~ ˆ = H − σ·B , (5.16) 2m 2mc where B = ∇ × A is the magnetic field. The Hamiltonians (5.14) and (5.16) act on spinor wave functions   a+ (x) Ψ(x) = . (5.17) a− (x) The second term in (5.16) describes the interaction of the Dirac magnetic moment of an electron with an external magnetic field. The first term in (5.16) is the quantum Hamiltonian of a spinless particle in a magnetic field. Its classical counterpart [the classical counterpart of the full Hamiltonian (5.16) also exists, we will present it at the end of the chapter] is (P − ec A)2 /2m, where P is the canonical momentum, to distinguish from the kinetic momentum p = mx˙ = m ∂H/∂P . 5 Of course, Landau did not call it supersymmetry—the word did not exist yet. Landau can be compared in this respect to Mr. Jourdain, the bourgeois gentilhomme, who spoke prose without knowing what prose is.

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Generically, the Hamiltonian (5.14) is not supersymmetric. Let us assume, however, that the direction of the magnetic field is fixed. We may choose it to lie along the third axis: B = [0, 0, B(x, y)] .

(5.18)

Landau assumed that the field is constant. But that is not necessary for supersymmetry; one can allow it to depend on6 x and y. If the magnetic field has the form (5.18), one can choose the vector potential A to have only the x and y components. In that case, the Hamiltonian represents a sum of two terms. The first term describes a nontrivial motion in the (x, y) plane, while the second term has the form Pˆ32 /2m and describes free motion along the direction of the field (indeed, when vkB, the Lorentz force is zero). Consider only the first nontrivial part. It coincides with (5.14), only now ˆ and A lie in the (x, y) plane. we assume that the vectors P Consider the operators   ˆ − e A 1 − σ3 ˆ = √1 σ· P (5.19) Q c 2 2m and   ˆ − e A 1 + σ3 , ˆ † = √1 σ· P Q c 2 2m

(5.20)

ˆ , A and σ. Capitalizing on the fact that σ1,2 antiwith 2-dimensional P commute with σ3 . it is easy to check that these supercharges satisfy the algebra (5.1). The problem is supersymmetric for any field (5.18). But for the homogeneous magnetic field, one can solve the Schr¨odinger problem exactly. In the calculations below we make a simplification, getting rid of all dimensional constants ~, c, m, |e| (so that the electron charge e goes over to −1). We have written them before to make it clear that the problem under consideration is not just a mathematical construction, but a real physical problem. Its solution gives an explanation of a real physical phenomenon, the so-called Landau diamagnetism.7 But it is the mathematical structure of the problem that we are now mostly interested in, and keeping ~, etc. would obscure it. 6 But not on the third coordinate, otherwise the divergence ∇ · B would be different from zero, which is not allowed by Maxwell’s equations. 7 Electrons rotating in an external magnetic field B induce their own small magnetic field in the direction opposite to B.

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Now the vector potential can be chosen as A = (−By/2, Bx/2, 0). The Hamiltonian and the supercharges acquire the form " 2  2 # 1 ∂ B ∂ By Bx ˆ H = + −i + σ3 , −i − + (5.21) 2 ∂x 2 ∂y 2 2  B z¯ ∂ + σ+ , ∂z 2   ∂ Bz = −i − σ− , ∂ z¯ 2

ˆ = −i Q ˆ† Q



(5.22)

√ where8 z = (x + iy)/ 2 and σ± were defined in (4.67). In agreement with general theorems, the whole Hilbert space (5.17) splits into two subspaces: subspace |+i with positive electron spin projection along the third axis and subspace |−i with negative spin projection. The spectrum of positive-energy states splits into degenerate doublets and the action of the supercharges follow the pattern displayed in Fig. 5.1. In both sectors the Hamiltonian (5.21) reduces to the Hamiltonian of the harmonic oscillator. The spectrum is thus equally spaced, the spacing between the levels being equal to B (we assume, for definiteness, that B > 0). All the levels are infinitely degenerate due to the symmetry under translations in the (x, y) plane. The difference between the two sectors is that the spin-down states are shifted down with respect to the spin-up states by B, which exactly coincides with the spacing between the oscillator levels (Fig. 5.2). This is the physical reason for the double degeneracy of all excited levels. [And we remember: the mathematical reason unsuspected by Landau is the supersymmetric algebra (5.1)!] The left tower in Fig. 5.2 involves the vacuum zero-energy states. To find their wave functions is simpler than to find the whole spectrum. To ˆ = 0 (Q ˆ † Ψ = 0 is satisfied this end, one only has to impose the condition QΨ automatically), which gives the first-order equation   B z¯ ∂ + a− = 0 (5.23) ∂z 2 with the solutions   B z¯(z + c) a− (z, z¯) = exp − . (5.24) 2 Different complex c correspond in the classical limit to different positions of the center of the circular electron orbits in the (x, y) plane. 8 We hope that the reader will not confuse this complex coordinate z with the third real spatial coordinate that we left unnamed!

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Fig. 5.2 levels.

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Spectrum of the Landau Hamiltonian (B > 0). n = 0, 1, . . . mark the oscillator

If B is negative, the picture is opposite. There are no zero-energy states in the spin-down sector and infinitely many of them in the spin-up sector. If the magnetic field in (5.18) is not homogeneous, but depends nontrivially on x, y, the problem is still supersymmetric, and even though one cannot solve it analytically any more, the generic pattern of the spectrum is the same as for the homogeneous field—some number of supersymmetric zero-energy states and a set of degenerate supersymmetric doublets. In the next chapter, we will prove the theorem: Theorem 5.4. The number of the normalized zero energy states (alias zero modes) in the Landau problem is proportional to the magnetic flux: Z |Φ| 1 nE=0 = = B(x, y)dxdy . (5.25) 2π 2π The flux Φ must be an integer multiple of 2π—otherwise supersymmetry is lost; see also Chap. 13.2.1. For a homogeneous field, the flux is infinite, and the number of the zero modes is infinite too. If the flux is finite, so is the number of the zero modes.

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The latter are spin-down states if the flux is positive and spin-up states if the flux is negative. “And what if the flux is zero”, the reader may ask. Well, in that case there are no normalized supersymmetric ground states annihilated by both supercharges whatsoever, and all the states are paired into supersymmetry doublets. The absence of supersymmetric ground states means that the supersymmetry is spontaneously broken—the vacuum states are not invariant under supersymmetry transformations.9 We will talk more about spontaneous supersymmetry breaking in the next section and then in the next chapter where a general theory of this phenomenon will be discussed. When we described the Grassmann oscillator in Chap. 4.4, we mentioned that the same dynamics can be described in the matrix language. Obviously, this applies also to the supersymmetric oscillator treated in the preceding section. The dictionary of translations from one language into another was given in Eq. (4.67). This dictionary works both ways, and we can easily write the quantum supercharges of the Landau problem in the Grassmann form, simply replacing the matrices σ+ and σ− in (5.22) by ψ ¯ = ∂/∂ψ, respectively. We obtain the expressions and ψˆ   B z¯ ∂ + ψ, Q = −i ∂z 2   ∂ Bz ∂ Q† = −i − , (5.26) ∂ z¯ 2 ∂ψ and the Hamiltonian is given by the expression (5.21) where the matrix σ 3 is traded for σ3 = [σ+ , σ− ] −→ ψ

∂ ∂ − ψ. ∂ψ ∂ψ

The wave functions depend on z, z¯ and the holomorphic variable ψ. There are the states of the form Ψ(z, z¯; ψ) = a− (z, z¯) (they are what we called before spin-down states) and the states of the form Ψ(z, z¯; ψ) = ψa+ (z, z¯) (the spin-up states). 5.4

Witten’s Model

When Enrico Fermi met a nontrivial physical problem, he used to ask himself and his students and colleagues—what is the hydrogen atom in this 9 The reader has the right to be confused at this point, because we have not explained yet what the supersymmetry transformations are. But we will do so soon.

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case? He meant to ask whether there exists a simple model bearing all the essential features of the complicated phenomenon he was set to understand. The model of the supersymmetric oscillator treated in Sect. 6.2 is simple, indeed, but it is probably too simple and too rigid. The Landau problem discussed in the preceding section is both simple and complicated enough to play the role of such hydrogen atom, but there is an even better candidate for this role: the model suggested by E. Witten [53]. The wave functions in this model depend on a single ordinary variable x and a holomorphic Grassmann variable ψ. The supercharges and Hamiltonian read ˆ = √1 ψ [Pˆ − iW 0 (x)] , Q 2 1 † ˆ = √ ψˆ¯ [Pˆ + iW 0 (x)] ; Q 2

(5.27)

0 2 ˆ2 ˆ¯ . ˆ = P + [W (x)] + 1 W 00 (x)(ψ ψˆ¯ − ψψ) H (5.28) 2 2 ¯ are the ordinary and Grassmann momentum operators— where Pˆ and ψˆ ¯ = ∂/∂ψ—and W (x) is an arbitrary real function called ˆ P = −i∂/∂x and ψˆ the superpotential. In the matrix formulation, the Hamiltonian (5.28) describes the one-dimensional motion of a spin 21 particle in the external potential V = 12 [W 0 (x)]2 and the magnetic field W 00 (x). In the particular case W (x) = ωx2 /2, the Hamiltonian acquires the form 2 2 ˆ2 ˆ¯ . ˆ = P + ω x + ω (ψ ψˆ¯ − ψψ) H (5.29) 2 2 It is nothing but the Hamiltonian (5.11) of the supersymmetric oscillator discussed before, but expressed in the ordinary representation: the operator (5.29) acts on the wave functions Ψ(x, ψ) rather than Ψ(¯ a, ψ). For a generic W (x), one cannot solve the complete spectral problem analytically, but one can always find explicit expressions for the wave function of the zero-energy state if such a state exists. As was the case also for two preceding problems, the Hilbert space is divided into two sectors: the “fermionic” states involving the factor ψ in Ψ(x, ψ) and the “bosonic” states without such factor.

Theorem 5.5. A normalized bosonic zero-energy solution to the Schr¨ odinger equation with the Hamiltonian (5.28) exists iff lim W (x) = ∞ .

x→±∞

(5.30)

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A normalized fermionic zero-energy solution to the Schr¨ odinger equation with the Hamiltonian (5.28) exists iff lim W (x) = −∞ .

x→±∞

(5.31)

ˆ † and all the Proof. All the bosonic states are clearly annihilated by Q ˆ fermionic states are annihilated by Q. A zero-energy state in the bosonic ˆ This gives the equation sector should in addition be annihilated by Q.   ∂ + W 0 (x) ΨB (5.32) 0 (x) = 0 . ∂x Its formal solution reads ΨB 0 (x) = exp {−W (x)} .

(5.33)

The condition (5.30) is the condition for its normalizability. F Likewise, the fermion zero modes ΨF 0 (ψ, x) = ψΨ0 (x) should satisfy the equation   ∂ − W 0 (x) ΨF (5.34) 0 (x) = 0 ∂x with the formal solution ΨF 0 (x) = exp {W (x)} .

(5.35)

It is normalizable iff the condition (5.31) is fulfilled. If W (∞) = ±∞, but W (−∞) = ∓∞, the zero-energy states are absent.10 The simplest example is given by the cubic superpotential W (x) = λx3 . The ground states of the Hamiltonian have in this case nonzero energy. They are paired in supersymmetric doublets, like in Fig. 5.1. It is the same phenomenon of spontaneous supersymmetry breaking that we met in the previous section for the motion in a magnetic field with zero net flux (5.25). 10 A special case is when W (∞) or W (−∞) is a constant. In this case, the spectrum is continuous, the spectral problem should be regularized (see the discussion at the end of Chap. 4.2) and zero modes can appear as a result of this regularization.

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Extended Supersymmetry

The algebra of the systems considered in three preceding sections involved two different Hermitian supercharges. By the convention adopted now in the literature, such systems are called N = 2 supersymmetric quantum mechanical (SQM) systems. This terminology is somewhat misleading. As we noted before, one can always extract a Hermitian square root of a Hermitian Hamiltonian with positive-definite spectrum and the algebra (5.1) is the minimal nontrivial supersymmetry algebra. Thus, it is better to count not real, but complex supercharges, in which case the algebra (5.1) would be called N = 1. But a universally adopted convention is a universally adopted convention, and we have to follow it, as we did earlier when adopting the term HKT, hyper-K¨ahler with torsion, for the manifolds that were not at all hyper-K¨ ahler and not even K¨ahler. Anyway, the symmetry of the system can be richer than (5.1). A supersymmetric system may involve several pairs of supercharges. The corresponding algebra is11 ˆ¯ i , Q ˆ¯ j } = 0 , ˆi, Q ˆ j } = {Q {Q ˆ j } = δ j H, ˆi, Q ¯ ˆ {Q i, j = 1, . . . N /2 i

(5.36)

ˆ ¯ i def ˆ i )† . with Q = (Q The indices i, j distinguishing different complex supercharges and the associated Grassmann variables are written at the different levels to highˆ¯ i belong to the different—fundamental and ˆ i and Q light the fact that Q antifundamental—representations of the U (N /2) group, a subgroup of the full automorphism group SO(N ) of the algebra (5.36), which rotates the complex supercharges and Grassmann variables. In all the extended models considered in this book, this symmetry (called R-symmetry) will play an important role. In many (though not in all) cases, it is a symmetry of the Hamiltonian. N ≥ 2 is an even number. The minimal extention is N = 4. In that case, the R-symmetry group is U (2), its fundamental and the antifundamental representations coincide and the indices may be raised and lowered by the invariant SU (2) tensors εij = −εij : Xi = εij X j

and

X i = εij Xj .

(5.37)

The convention ε12 = 1 is chosen. 11 There are also SQM systems with more intricate extended algebras [54], but they do not play a role in geometrical applications and we will not talk about them in this book.

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We give here one example of an N = 4 SQM system. It describes the motion on the complex plane, but, in contrast to the Landau problem, involves two Grassmann variables ψ1,2 and the corresponding momenta ¯1,2 . Π1,2 = −i∂/∂ψ1,2 ≡ −iψˆ The supercharges are12 ˆ =ψ π ˆ = ψ π Q ˆ + i(¯ z )2 ψˆ¯2 , Q ˆ − i(¯ z )2 ψˆ¯1 , 1

1

2

ˆ1 π ˆ ¯ 1 = ψ¯ Q ˆ † − i(z)2 ψ2 ,

2

ˆ¯ 2 = ψ¯ˆ2 π Q ˆ † + i(z)2 ψ1 ,

(5.38)

†

with π ˆ = −i∂/∂z, π ˆ = −i∂/∂ z¯. The Hamiltonian ˆ = π H ˆ†π ˆ + (z z¯)2 − 2zψ1 ψ2 − 2¯ z ψˆ¯2 ψˆ¯1

(5.39)

is not quadratic and describes a nontrivial interaction. More examples will follow later. For geometrical applications, certain N = 4 and N = 8 systems are relevant. But the N = 6 systems and more complicated, up to N = 32, exist and have been studied. 5.6

Classical Supersymmetry

In the beginning of this chapter, we defined what quantum supersymmetric system is. We considered then several particular quantum supersymmetric Hamiltonians, but have not posed yet a question: “What are their classical counterparts?” If one is interested only in quantum mechanics, using the Grassmann variables is convenient (especially in complicated systems), but not necessary. As we noted, one can well work with the Hamiltonians represented in the matrix form. But Grassmann variables are undispensible if one wishes to write down a classical Hamiltonian or a classical Lagrangian. In all examples considered heretofore, supersymmetric classical Hamiltonians can be written very easily. Let us do so first for the supersymmetric oscillator. Take the quantum ¯ Hamiltonian (5.11) and replace ∂/∂¯ a → iΠa¯ ≡ a and ∂/∂ψ → iΠψ ≡ ψ. We obtain ¯ H sup−osc = ω(¯ aa + ψ ψ) The Hamilton equations of motion are ∂H ∂H = −iωa , ψ˙ = −i ¯ = iωψ a˙ = −i ∂¯ a ∂ψ and complex conjugates.

(5.40)

(5.41)

12 Do not confuse these complex N = 4 supercharges with Q 1,2 the real and imaginary parts of the N = 2 supercharge.

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The classical supercharges are derived in a similar way from (5.13): Q =

√

ω aψ

¯ = Q

√

ωa ¯ψ¯ .

The quantum supersymmetry algebra (5.1) transforms to ¯ Q} ¯ P = 0, ¯ Q}P = iH {Q, Q}P = {Q, {Q,

(5.42) (5.43)

or else13 {Q1 , Q1 }P = {Q2 , Q2 }P = 2iH ,

{Q1 , Q2 }P = 0 , (5.44)

where {A, B}P is the Poisson bracket defined in (4.59). It follows from the Jacobi identity (its Grassmann version) that the Poisson brackets {Q, H}P ¯ H}P vanish, which means that Q and Q ¯ are classical integrals of and {Q, motion. The classical Lagrangian is obtained from H by the Legendre transformation: ˙ ψ −H. L = a ¯˙ Πa¯ + ψΠ (5.45) (This particular order of the Grassmann factors follows from the definition ˙ Up to a total time derivative,14 we obtain Πψ = ∂L/∂ ψ.) ¯ − ω(¯ ¯ . L = i(a¯ ˙ a − ψ˙ ψ) aa + ψ ψ) (5.46) By Noether’s theorem, the existence of classical integrals of motion implies the invariance (up to a total derivative) of the Lagrangian under certain associated transformations. In our case, these are supertransformations. They read δa = iψ¯ , δ¯ a = i¯ ψ , δψ = −i¯ a ,

δ ψ¯ = ia¯ ,

(5.47)

where  is a Grassmann transformation parameter. For Witten’s model, the classical expressions for the supercharges and the Hamiltonian can be written immediately. They simply coincide with (5.27) and (5.28), where now P and ψ¯ should be understood as the phase space variables rather than differential operators. The corresponding classical Lagrangian is x˙ 2 − [W 0 (x)]2 − iψ˙ ψ¯ − W 00 (x)ψ ψ¯ . (5.48) L = 2 13 One may note here that, while for the ordinary phase space the relation {A, A} = 0 holds for any function A(pj , qj ), this is not so when the phase space includes also Grassmann variables. 14 We have seen (see p. 86) that the Lagrangians that differ by a total derivative give the same equations of motion and are equivalent.

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The Lagrangian is invariant up to a total derivative under ¯ , δx = ψ + ψ¯ δψ = ¯[−ix˙ + W 0 (x)] , δ ψ¯ = [ix˙ + W 0 (x)] .

(5.49)

For the Landau problem, the classical supercharges are derived from the quantum expressions (5.26) by substituting the classical momenta π, π ¯ and Πψ = −iψ¯ for −i∂/∂z, −i∂/∂ z¯ and −i∂/∂ψ:   iB z¯ ψ, Q = π− 2   iBz ¯ ¯ = π Q ¯+ ψ. (5.50) 2 Their Poisson bracket gives the classical Hamiltonian,    iB z¯ iBz H = π− π ¯+ + Bψ ψ¯ . 2 2

(5.51)

The corresponding classical Lagrangian reads iB (z˙ z¯ − z z¯˙ ) − iψ˙ ψ¯ − Bψ ψ¯ . 2 It is invariant up to a total derivative under L = z˙ z¯˙ +

δz = ψ , δ¯ z = −¯ ψ¯ ,

δψ = −i¯ z˙ ¯ δ ψ = iz¯˙ .

(5.52)

(5.53)

In the systems considered so far, there were no ordering ambiguities and natural expressions for the classical supercharges could be easily written. But if the ordering ambiguities are present, the questions: (1) What particular expressions for the classical supercharges should be written? (2) Is it always possible to do so in such a way that the quantum supersymmetry algebra would be preserved at the classical level, with the Poisson brackets taking over the role of commutators and anticommutators? become nontrivial. The answers are also non-trivial. The answer to the second question is “Not always!”. The first example of a system where the quantum supersymmetry algebra cannot be preserved at the classical level was constructed in [55]. A simpler example was given in [56]. The latter concerns a supersymmetric system describing the motion

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of a spinorial particle in Euclidean space of even dimension D equipped by an external gauge field. In this system, the Dirac operator D / and also the operator D / γ D+1 (with γ D+1 being the multidimensional counterpart of the familiar to the physicists matrix γ 5 ) play the role of the quantum supercharges. The operator D / has a classical counterpart, but the operator D+1 D /γ has not (see Chap. 13 for detailed discussion). In quantum field theories we are used to have quantum anomalies when a classical symmetry cannot be preserved at the quantum level (the chiral anomaly, the conformal anomaly. . . ). And here we are dealing with the opposite situation: supersymmetry that is realized at the quantum, but not the classical level. The first question in the list above was studied in [57]. We analyzed there different supersymmetric systems where the supercharges and the Hamiltonians include products of canonical coordinates and momenta, and the ordering problem exists. Based on observations performed for various supersymmetric systems involving ordering ambiguities, we formulated the following conjecture: (1) Consider a system enjoying an ordinary or extended classical supersymmetry algebra. Take the classical supercharges and write down the corresponding quantum operators using the Weyl ordering prescription (4.43). Then these operators satisfy the quantum supersymmetry algebra.15 (2) Inversely: if the quantum supersymmetry algebra can be preserved at the classical level, this can always be achieved by choosing the classical supercharges as the Weyl symbols of the quantum ones. This conjecture looks natural. After all, we know [see Eq. (4.45)] that the Weyl symbol of the commutator of two ordinary operators coincides with the Gr¨ onewold-Moyal bracket of their Weyl symbols. This result can be generalized on the case when the phase space involves Grassmann variables: the Weyl symbol of the commutator of two operators or their anticommutator if both operators are Grassmann-odd coincides with the GM 15 The observation that the classical supersymmetry algebra holds at the quantum level under a particular ordering prescription was also made in [58]. But the universal recipe, Take the classical supercharges and order the operators in their quantum version according to (4.43), was not formulated there.

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bracket of their Weyl symbols, defined as      ∂2 1 ∂2 ∂2 i ∂2 + {A, B}GM = 2 sin − − ¯α 2 ∂pi ∂Qi ∂Pi ∂qi 2 ∂ψα ∂ Ψ ∂ ψ¯α ∂Ψα ¯ ¯ A(p, q; ψ, ψ)B(P, Q; Ψ, Ψ) (5.54) ¯ ¯ p=P,q=Q; ψ=Ψ,ψ=Ψ

(~ = 1). This means that the quantum algebra is always preserved at the classical level if the GM bracket is chosen as the binary operation. It is not counterintuitive to expect that whatever holds for the GM brackets should also hold in most cases for the Poisson brackets. And if this rule of thumb fails, nothing helps. But, of course, it would be interesting to rigourously prove (or disprove) it.

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Chapter 6

Path Integrals and the Witten Index

This chapter will provide the reader with the necessary prerequisits to understand the second half of Chap. 8 and also Chap. 14, where we will prove the famous Atiyah-Singer index theorem by supersymmetric methods. It was originally proven by Atiyah and Singer using traditional mathematical tools [59]. But the physical way of deriving this result by analyzing the spectrum of certain SQM systems appears to be both more simple and more beautiful [60]. We have to emphasize that the quantum systems to be considered in this chapter have all a discrete spectrum. Otherwise, the notion of index is ill-defined.

6.1

Path Integrals in Quantum Mechanics

Classical mechanics has two basic ways of description—the Lagrange formalism and the Hamilton formalism. The same concerns quantum mechanics. Historically, it was first formulated in the framework of the quantum Hamiltonian formalism (see Chap. 4.2). The quantum counterpart of the Lagrange formalism is the path integral approach developed by Feynman. We will give here its basics addressing the reader to the book [61] for details. Consider the time-dependent Schr¨odinger equation (4.38). To focus on its mathematic structure we will set ~ = 1. To make things still simpler, we assume the presence of only one dynamical real variable q ∈ (−∞, ∞) and the natural measure dµ = dq on this line. But all the results that we will derive can be generalized to systems with several dynamical degrees of freedom and a more complicated measure. To solve this equation, one should fix the initial conditions—the values of Ψ(q, t) at some t = tin . Then the formal solution of the equation (4.38) 115

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reads ˆ (t − tin )Ψ(q, tin ) , Ψ(q, t) = U

(6.1)

n o ˆ (t − tin ) = exp −i(t − tin )H ˆ U

(6.2)

where

is the evolution operator. We now consider the kernel of the evolution operator, that is, the matrix element ˆ (tf − tin )|qin i ≡ hqf , tf |qin , tin i K(qf , qin ; tf − tin ) = hqf |U

(6.3)

(tf − tin ≡ ∆t > 0). It describes the probability amplitude that the system will find itself at the position qf at t = tf provided it was located1 at q = qin at t = tin . In terms of (6.3), the solution (6.1) is expressed as Z Ψ(qf , tf ) = dqin K(qf , qin ; tf − tin ) Ψ(qin , tin ) . (6.4) The quantity K plays a fundamental role in the whole approach. Theorem 6.1. The kernel K is given by the sum (spectral decomposition) X (6.5) K(qf , qin ; ∆t) = Ψk (qf )Ψk (qin )e−iEk ∆t , k

ˆ with eigenvalues Ek . where Ψk (q) are the eigenstates of H Proof. This follows from (6.4), from orthogonality, Z ¯ k (q)Ψl (q) dq = δkl , hk|li = Ψ

(6.6)

and from the fact that the time evolution of an eigenfunction Ψk (q, t) amounts to the multiplication by the phase factor e−iEk ∆t . The reader can recognize in Eq. (6.5) the fundamental solution of the hyperbolic equation (4.38). Indeed,   ∂ ˆ K(qf , qin ; ∆t) = 0 , i −H (6.7) ∂t 1 Note that, in the definition above, the bra-vector on the left describes the final state of the particle, while the ket-vector on the right describes its initial state. This unnatural for an air traveller convention follows from the definition of a matrix element in (4.37). It conforms with (6.4) and (6.6).

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ˆ acts on the first argument in K. And in the limit where the operator H when ∆t vanishes, the kernel acquires the form X K(qf , qin ; 0) = Ψk (qf )Ψk (qin ) = δ(qf − qin ) . k

In simple cases, the kernel can be found explicitly. For a free particle of unit mass m = 1 moving along a line,   1 i(qf − qin )2 K(qf , qin ; ∆t) = √ exp . (6.8) 2∆t 2πi∆t An exact analytical expression for the kernel can also be derived for the Hamiltonian of the harmonic oscillator [61]. But how to solve the problem in the general case? We note first that the matrix element (6.3) satisfies the completeness relation Z ∞ hqf , tf |qin , tin i = dq∗ hqf , tf | q∗ , t∗ i hq∗ , t∗ | qin , tin i , (6.9) −∞

where t∗ is any time moment between tin and tf . The relation (6.9) has a transparent physical meaning: the probability amplitude for the particle to go from the point qin at the initial moment tin to the point qf at the final moment tf is given by the convolution of the probability amplitudes to go first to an intermediate point q∗ at t∗ and then to the final point.2 We divide now the time interval ∆t in a large number n of equal tiny time slices and write hqf , tf |qin , tin i = Z ∞ n−1 Y dqj hqf , tf |qn−1 , tf − i · · · hq1 , tin + |qin , tin i

(6.10)

−∞ j=1

with  = ∆t/n = (tf − tin )/n. The integrand involves the product of a large number of factors. To evaluate each such factor, we use (6.2) and (6.3). For example, ˆ in i hq1 , tin + |qin , tin i = hq1 | exp{−iH}|q Z dp ˆ hq1 |pihp|e−iH |qin i . = 2π

(6.11)

2 The reader-physicist may recall the Huygens-Fresnel principle in optics, which tells basically the same, but for the amplitude of an electromagnetic wave. The analogy between quantum mechanics and optics is very deep, though it is, of course, outside the scope of our book and we will not pursue it here.

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We used here the completeness relation again, inserting the intermediate states |pi that represent the eigenstates of the momentum operator pˆ = −i∂/∂q. They are plane waves: hq1 |pi = Ψp (q1 ) = eipq1 ,

hp|qin i = Ψp (qin ) = e−ipqin .

(6.12)

The factor 1/2π in the measure provides the correct normalization: Z dp ip(q1 −qin ) hq1 |qin i = e = δ(q1 − qin ) . 2π Suppose that the Hamiltonian has a simple form 2 ˆ = pˆ + V (q) . H 2 We may then write for small  ˆ

hp|e−iH |qin i ≈ e−iH(p,qin ) hp|qin i = e−ipqin −iH(p,qin ) ,

(6.13)

(6.14)

where H(p, q) is the classical Hamiltonian. Plugging it in (6.11), we derive Z dp hq1 , tin + |qin , tin i ≈ exp{ip(q1 − qin ) − iH(p, qin )} . (6.15) 2π For a generic Hamiltonian mixing the coordinates and momenta, the situation is more complicated. As we discussed earlier, the order in which pˆ and ˆ matters. This ordering ambiguity matches the ambiguity in the q enter H path integral definition. Generically, it matters whether we write H(p, qin ),
in H(p, q1 ) or maybe H p, q1 +q in the integrand in Eq. (6.15). For exam2 ˆ = pˆ2 q 2 , but not for H ˆ = q 2 pˆ2 . We ple, the relation (6.14) holds for H address the reader to Ref. [62] where this question is thouroughly studied and pedagogically explained. We will see later, however, that this ordering uncertainty is irrevant for the final result we are interested in—the integral representation of the Witten index. Doing the same transformation for each factor in (6.10) and sending n to infinity, we obtain the evolution kernel at finite time interval in the following form: Z K(qf , qin ; tf − tin ) = lim exp{ipn (qf − qn−1 ) + . . . + ip1 (q1 − qin ) n→∞

dp1 dq1 dpn dpn−1 dqn−1 ··· . (6.16) 2π 2π 2π Trading  for dt, this can be formally written as Y  Z tf Z dp(t)dq(t) K(qf , qin ; tf − tin ) = exp i [pq˙ − H(p, q)]dt (6.17) 2π tin t − i[H(pn , qn−1 )+ . . . +H(p1 , qin )]}

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with the boundary conditions q(tin ) = qin ,

q(tf ) = qf ,

(6.18)

while the function p(t) is quite arbitrary and is integrated over. For the Hamiltonian (6.13), the integral over all dpn in Eq. (6.16) is Gaussian and can be easily done: Z n−1 Y K(qf , qin ; tf − tin ) = N lim dqj n→∞

j=1

   (q1 − qin )2 (qf − qn−1 )2 + ··· + − V (qn−1 ) · · · − V (qin ) exp i 2 2    Z tf  2 Z Y q˙ − V (q) dt . (6.19) ∝ dq(t) exp i 2 tin t An infinite normalization factor N plays no role for our purposes. The third line in Eq. (6.19) was derived from the second one by replacing q(t1 ) − q(tin ) → q(t ˙ in ), etc. The representation (6.19) is the original Feynman form of the path integral where the integral is done over the trajectories in configuration space rather than in phase space as it is the case in (6.17). The boundary conditionsR (6.18) are assumed. The integral in the exponent is just the action S = Ldt on a trajectory q(t) connecting the points qin , tin and qf , tf . The (semi-)classical limit of quantum theory corresponds to large action (in units of ~). In this case, the main contribution to the integral (6.19) stems from the stationary points (minima) of the action where many trajectories are added up with nearly identical phase factors. That is how the classical principle of least action emerges. The path integral has been now used in physics for more than half a century. But not in mathematics — mathematicians still do not like it and consider it not to be a rigorously defined object. To be more precise, they have nothing against the Euclidean path integral where the time is considered to be imaginary, t = −iτ . In that case, the function to be integrated in (6.19) goes over to #) ( Z "   2 1 dq + V (q) . (6.20) exp − dτ 2 dτ If the potential grows at large q, the contribution of the trajectories with wildly deviating q(τ ) is exponentially suppressed. Mathematicians call such integrals the integrals with Wiener measure and can work with them. A

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good news for a reader-mathematician is that we will also only work with Euclidean path integrals in this book! It is not so relevant for us, but we can still mention that the main problem with the real-time path integral is that it is difficult to prove the existence of the limit of a finite-dimensional integral (6.10), when the number of points goes to infinity and the time interval  between the adjacent points goes to zero. I am not myself a mathematician and cannot even try to prove it. The only thing that I can say is that, for simple-minded but nontrivial quantum mechanical problems like the anharmonic oscillator, the real-time path integrals were calculated numerically and the existence of the limit  → 0 was established experimentally, if you will [63].3 Let us concentrate on the Euclidean path integral (6.20). In the Minkowski case, we were interested in the kernel (6.3) of the evolution operator that depended on the initial and final coordinates and the time interval ∆t. But the object of principal interest for us in this book is the partition function and especially its supersymmetric generalization to be discussed soon. Definition 6.1. The partition function of a quantum system is defined as the following sum over the spectrum of its Hamiltonian: X Z(β) = e−βEk , (6.22) k

where β is the parameter whose physical interpretation is the inverse temperature, β = 1/T . Note that, for large enough β (small enough temperatures), the main contribution in the sum (6.22) comes from the ground state in the spectrum, while the contribution of the excited states is suppressed. Theorem 6.2. The partition function is given by the integral Z Z = dq K(q, q; −iβ) , 3 In

(6.23)

this paper, the Fourier image ∞

Z G(q, E) =

dT eiET K(q, q; T )

(6.21)

0

of the evolution operator (6.19) with just 2–3 intermediate points was evaluated and the peaks at the right places—at the eigenvalues Ek of the Hamiltonian—were observed, in agreement with the spectral decomposition (6.5). The actual calculations were made assuming that the parameter E in (6.21) has a small positive imaginary part (a little step in the Euclidean direction). This makes the integrals well-defined and convergent.

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where K(q, q; −iβ) is the Euclidean evolution operator over the imaginary time −iβ with coinciding initial and final points. Proof. Simply substitute −iβ for ∆t in Eq. (6.5), set qin = qf → q and integrate over q. Another name for the Euclidean evolution operator K(qf , qin ; −iβ) is the heat kernel. It is called so because it represents the fundamental solution of the diffusion equation or heat equation in the same way as K(qf , qin ; ∆t) represents the fundamental solution of the Schr¨odinger equation. 6.2

Grassmann Evolution Kernel

We now go over to the systems including Grassmann dynamical variables. As a toy model, consider first the Grassmann oscillator of Chap. 4.4. It is described by a single holomorphic Grassmann variable ψ, and the Hilbert space includes only two states: the state |1i with Ψ(ψ) = 1 and the state |ψi with Ψ(ψ) = ψ. To proceed, we need to define the inner product in this space. We should have h1|1i = hψ|ψi = 1 and h1|ψi = hψ|1i = 0. This follows from the nice formulaZ hf |gi =

¯

dψdψ¯ e−ψψ f (ψ)g(ψ) .

(6.24) ¯

As we see, the measure includes now the factor e−ψψ . Its appearance is not so surprising—an exponential factor in the measure arises also for the ordinary oscillator in the holomorphic representation. The states in the first line in Eq. (5.12) form an orthonormal basis with the measure dad¯ a −a¯a e . (6.25) dµ = 2π We will later discuss the systems including many Grassmann dynamical variables ψα , in which case the fermion measure is Y ¯ dµG = dψα dψ¯α e−ψα ψα . (6.26) α

The states |1i and |ψi are the eigenstates of the Hamiltonian (4.63) : ˆ ˆ H|1i = 0 and H|ψi = ω|ψi. The kernel of the evolution operator has now the form ˆ hf |e−iH∆t |ii = K(ψf , ψ¯in ; ∆t) . (6.27) Then the wave function at a later moment tf is given by the convolution Z ¯ Ψ(ψf , tf ) = dψin dψ¯in e−ψin ψin K(ψf , ψ¯in ; ∆t) Ψ(ψin , tin ) . (6.28)

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It involves the integral over the initial phase space with the measure prescribed in (6.24). For the Grassmann oscillator or any quantum problem where the spectrum is explicitly known, an explicit expression for the evolution kernel can be written. This is given by the spectral decomposition (Theorem 6.1, which can be easily generalized for Grassmann systems) that has the form X (6.29) K(ψf , ψ¯in ; ∆t) = Ψk (ψf )Ψk (ψin )e−iEk ∆t . k

We derive for the oscillator: K(ψf , ψ¯in ; ∆t) = 1 + ψf ψ¯in e−iω∆t .

(6.30)

The Euclidean evolution operator is given by (6.30) where ∆t is substituted for −iβ: K(ψf , ψ¯in ; −iβ) = 1 + ψf ψ¯in e−ωβ . Consider a natural generalization of (6.23): Z ¯ ¯ −iβ) , ˜ Z(β) = dψdψ¯ e−ψψ K(ψ, ψ;

(6.31)

(6.32)

where the measure (6.24) was used again. Doing the integral, we obtain ˜ Z(β) = 1 − e−βω .

(6.33)

This resembles the partition function (6.22), but the second term, the contribution of the excited oscillator state, enters with the negative sign! Remember this remarkable fact and look again at the kernel (6.31). For ¯ 0) = 1 + ψ ψ¯ = β = 0 (when ψin and ψf coincide), it is reduced to K(ψ, ψ; ¯ ψψ −1 e . For small β  ω , the kernel can be written as ¯ ¯ K(ψf , ψ¯in ; −iβ) ≈ eψf ψin −βH(ψf ,ψin ) ,

(6.34)

¯ is the classical Hamiltonian of the Grassmann oscillator where H(ψ, ψ) (4.55). The sign ≈ in Eq. (6.34) means the following. For small β, we keep in ¯ but neglect the terms of order K(ψf , ψ¯in ; −iβ) the terms of order ∼ βψ ψ, n≥2 ¯ ψ ψ (and would neglect the terms of order ∼ β n≥1 not involving ∼ β ¯ were such terms present). The meaning of this selective the factor ψ ψ, neglection will be clarified by the end of the chapter: we will see that the neglected terms are irrelevant for the Witten index calculation. See also the discussion on p. 321.

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The result (6.34) can be generalized for systems with an arbitrary number of Grassmann variables ψα=1,...,n . The inner product is then defined with the measure (6.26). ˆ α , ψˆ¯α ) is the operator Definition 6.2. A normal ordered operator A(ψ where all the factors ψα stay on the left and the factors ψˆ¯α = ∂/∂ψα stay on the right. It is clear that any operator can be brought to the normal ordered form by ¯α to the right and using {ψˆ¯α , ψβ } = δαβ . pulling the factors ψˆ Definition 6.3. The function AN obtained from Aˆ by lifting the hats is ˆ called the normal symbol of the operator A. Thus, the normal symbol of ψα ψˆ¯β is ψα ψ¯β and the normal symbol of ψˆ¯β ψα is δαβ − ψα ψ¯β . ˆ defined according Theorem 6.3. The kernel KA (ψα , ψ¯α ) of any operator A, 4 to Z Y ˆ (AΨ)(ψ) = dχα dχ ¯α e−χα χ¯α KA (ψα , χ ¯α ) Ψ(χ) (6.35) α

is related to its normal symbol AN (ψα , ψ¯α ) by the formula ¯ KA (ψα , ψ¯α ) = eψβ ψβ AN (ψα , ψ¯α ) ,

(6.36)

the summation over β being assumed. Proof. A normal ordered operator Aˆ can be decomposed as Aˆ = C(ψ) + Aα (ψ)ψˆ¯α + Aαβ (ψ)ψˆ¯α ψˆ¯β + . . .

(6.37)

In the simplest case, the operator does not depend on ψˆ¯ and its action boils down to the multiplication by C(ψ). To prove the theorem in this case, we have to prove that Z Y dχα dχ ¯α e−χα χ¯α eψα χ¯α Ψ(χ) = Ψ(ψ) (6.38) α

for any wave function Ψ. And this is true for any monomial, like Ψ(χ) = χ1 χ3 χ7 , due to the properties Z Z dχdχ ¯ e(ψ−χ)χ¯ = 1, dχdχ ¯ e(ψ−χ)χ¯ χ = ψ . (6.39) 4 Cf.

Eq. (6.28) defining the kernel of the evolution operator.

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Consider now the second term Aα (ψ)ψˆ¯α . We have to prove that Z Y ∂ dχα dχ ¯α e(ψα −χα )χ¯α Aβ (ψ)χ ¯β Ψ(χ) = Aβ (ψ) Ψ(ψ) (6.40) ∂ψβ α or Z Y α

dχα dχ ¯α e(ψα −χα )χ¯α χ ¯β Ψ(χ) =

∂ Ψ(ψ) . ∂ψβ

(6.41)

To prove the validity of the latter relation, it is sufficient to notice that, for the integral not to vanish, the factor χ ¯β in the integrand should be paired with some factor χγ in the expansion of Ψ(χ). And such a pairing amounts to differentiating ∂Ψ(χ)/∂χβ . Using (6.38), we derive (6.41). The same reasoning works for the higher terms of the expansion (6.37). For example, the validity of the theorem for the term including two opera¯ follows from the relation tors ψˆ Z Y ∂2 Ψ(ψ) . (6.42) dχα dχ ¯α e(ψα −χα )χ¯α χ ¯β χ ¯γ Ψ(χ) = ∂ψβ ∂ψγ α

ˆ

Applying this theorem to the Euclidean evolution operator e−β H , we derive   ˆ K(ψαf , ψ¯αin ; −iβ) = exp{ψαf ψ¯αin } e−β H . (6.43) N

Now, for small β the normal symbol of the exponential can be approximated by the exponential e−βH of the classical Hamiltonian. The latter can be chosen as the normal symbol of the quantum one, but it is not necessary, and actually we cannot make a particular choice of H—it is beyond the validity of our approximation. We obtain f ¯in K(ψαf , ψ¯αin ; −iβ) ≈ exp{ψαf ψ¯αin } e−βH(ψα ,ψα ) .

(6.44)

The reader may look again at Eq. (6.14) for the bosonic infinitesimal evolution kernel. It could also be expressed in the language of symbols. The ˆ in i that we were interested in is in fact the soquantity hp| exp{−iH}|q ˆ And for small , it can called p-q symbol [62] of the operator exp{−iH}. be approximated by the exponential exp{−iH(p, q)}, where H(p, q) is the ˆ p-q symbol of H.

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Witten Index

Consider now an SQM system including both ordinary variables qj and Grassmann variables ψα . Consider the infinitesimal Euclidean kernel Kinf (qj , ψα ; qj , ψ¯α ; −iβ) = hqj , ψα ; −iβ|qj , ψα ; 0i with the same values of the variables at t = 0 and t = −iβ. Combining the results (6.15) (with  → −iβ ) and (6.44), we derive Z Y dpj −βH(pj ,qj ; ψα ,ψ¯α ) ¯ e Kinf (qj , ψα ; qj , ψ¯α ; −iβ) ≈ eψα ψα , (6.45) 2π j where H(pj , qj ; ψα , ψ¯α ) is the classical Hamiltonian in some ordering prescription. The expression above is written in the small β approximation. The evolution operator at finite β may be calculated as a path integral involving the convolution of an infinite number of the infinitesimal evolution operators: K(qj , ψα ; qj , ψ¯α ; −iβ) =   Z n−1 Y Y (m) Y (m) ¯(m)  dqj dψα(m) dψ¯α(m) e−ψα ψα  lim n→∞

 Kinf

m=1

α

j (1)

qj , ψα ; qj , ψ¯α(1) ;

−iβ n



 · · · Kinf

(n−1)

qj

, ψα(n−1) ; qj , ψ¯α ;

 −iβ . (6.46) n

But a very good news is that in most cases (though not in all cases—see the discussion in Chap. 14) we do not need to evaluate this complicated path integral. For our present purposes, it is sufficient to know the expression (6.45) for the evolution kernel at small β and its spectral decomposition at an arbitrary β. The spectral decomposition of the bosonic evolution kernel was written in Eq. (6.5) and, for the kernel in the toy Grassmann oscillator model, in Eq. (6.29). Performing an obvious generalization and setting also “in” = “f”, we may write X K(qj , ψα ; qj , ψ¯α ; −iβ) = Ψk (qj , ψα )Ψk (qj , ψα )e−βEk . (6.47) k

As a generalization of (6.32), we consider the integral Z Y Y ¯ ˜ Z(β) = dqj dψα dψ¯α e−ψα ψα K(qj , ψα ; qj , ψ¯α ; −iβ) . (6.48) j

α

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We substitute there the spectral decomposition (6.47) and, bearing in mind the normalization condition hΨk |Ψk i = 1 with the measure (6.26), we derive X ˜ Z(β) = ηk e−βEk , (6.49) k

where ηk = 1 for the bosonic states with Grassmann-even wave functions Ψk and ηk = −1 for the fermionic states with Grassmann-odd wave functions. The quantity Z˜ is called the supersymmetric partition function or simply the Witten index. The path integral (6.48) for Z˜ can be presented in the following symbolic form [cf. (6.19)]: Z Y Y ˜ Z(β) = dqj (τ ) dψα (τ )dψ¯α (τ ) ατ

jτ

( Z ) β ¯ exp − LE [qj (τ ), ψα (τ ), ψα (τ )]dτ ,

(6.50)

0

where LE is the Euclidean Lagrangian and both the ordinary and Grassmann variables satisfy the periodic boundary conditions: qj (β) = qj (0);

ψα (β) = ψα (0);

ψ¯α (β) = ψ¯α (0) .

(6.51)

The reader may wonder whether it is possible to define in a similar way the conventional partition function (6.22) where all the states, both bosonic and fermionic, contribute in the sum with the positive sign. The answer is affirmative. One just has to write Z Y Y ¯ Z(β) = dqj dψα dψ¯α e−ψα ψα K(qj , ψα ; qj , −ψ¯α ; −iβ) . (6.52) j

α

The minus before the second Grassmann argument in K brings about an extra minus in the spectral sum for Grassmann-odd wave functions, which compensates the factor ηk = −1 in (6.49).5 Going back to the alternating sum (6.49), it has interesting properties. You remember that a supersymmetric quantum system is characterized by the double degeneracy of all excited states—for any bosonic state there is 5 It is not so relevant for us, but one can mention that the conventional partition function is interesting for a physicist when s/he studies thermal properties of the system. For a finite β, the evolution kernel and the partition function are described by a path integral. The appearance of the extra minus sign in the second argument of K for the fermionic, but not bosonic variables means that one should impose in the path integral (6.50) periodic boundary conditions in imaginary time for the bosonic variables and antiperiodic ones for the fermionic variables. Physicists say that the symmetry between bosons and fermions, i.e. supersymmetry, is broken by finite temperature effects.

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a fermionic partner with the same energy. This means that the excited states do not give any contribution to the sum (6.49), only the ground alias vacuum states having zero energy, which are not necessarily paired, ˜ contribute. And if Ek = 0, e−βEk = 1. We see that Z(β) does not depend on β and is given by a remarkable formula [53, 64] ˜ Z(β) ≡ IW = nE=0 − nE=0 , B F

(6.53)

where nE=0 is the number of bosonic vacuum states and nE=0 is the number B F of fermionic vacuum states. Obviously, the Witten index is an integer ˜ number. But if Z(β) does not depend on β, one may calculate it for any β, and it is convenient to do so when β is small. In this case, we can substitute the approximate expression (6.45) for the integrand in (6.48) and calculate the integral. We obtain [65] Z Y dpj dqj Y ¯ dψα dψ¯α e−βH(pj ,qj ;ψα ,ψα ) , (6.54) IW = lim β→0 2π α j This is an ordinary finite-dimensional integral, and it can be evaluated analytically for many SQM systems. One can also represent the result (6.54) in an alternative form when the Grassmann dynamical variables are not introduced, but the supersymmetric Hamiltonian has matrix nature. For example, for the Witten model, we may write 2 0 2 ˆ = P + [W (x)] + 1 W 00 (x)σ3 . (6.55) H 2 2 ˆ is a matrix, but it is not the Here the hat over H only signifies that H full-scale quantum Hamiltonian representing a differential operator. The momentum P is a classical phase space variable. In this case, the Euclidean evolution kernel also has matrix nature. The functional trace of this kernel including the ordinary matrix trace and the integration over coordinates, as in (6.52), gives us the partition function of the system: Z Y ˆ j ; qj ; −iβ)} . Z(β) = dqj Tr{K(q (6.56) j

To find the supersymmetric partition function, we have to recall that the spectrum of a supersymmetric Hamiltonian includes two towers of degenerate states, as in Fig. 5.2—bosonic and fermionic states. We introduce ˆ then the operator (−1)F which commutes with the Hamiltonian and whose eigenvalues are +1 for the bosonic states and −1 for the fermionic states.

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To understand the commonly used notation (−1)F for this operator, one should go back to the Grassmann representation. Then Fˆ is the operator of the fermion charge, Fˆ = ψj ψˆ¯ j . (6.57) It simply counts the number of the Grassmann factors in the wave function. The operator Fˆ may commute with the Hamiltonian, but it is not necessary. ˆ ˆ always holds, however. Indeed, the supercharge The property [(−1)F , H] operators always include an odd number of Grassmann factors ψ or ψˆ¯ and they transform the states with even F to the states with odd F and vice versa. Thus, in the Grassmann language, the two towers of states mentioned in the previous paragraph are the states of even and odd fermionic charges. It is natural to call “bosonic” the states with even F and “fermionic” the states with odd F . The supersymmetric partition function can thus be represented as Z Y ˆ ˆ ˜ Z(β) = dqj Tr{(−1)F K(q (6.58) j ; qj ; −iβ)} . j

The formula (6.54) acquires the form Z Y n o dpj dqj ˆ ˆ IW = lim Tr (−1)F e−β H(pj ,qj ) . β→0 2π j

(6.59)

The importance of the Witten index notion is the following. Suppose that we have a complicated supersymmetric system (not necessarily an SQM system, it can be a field theory system) and we want to know whether the spectrum of this system involves or not supersymmetric zero-energy states annihilated by the action of supercharges. In other words, we want to know whether supersymmetry is spontaneously broken or not. To answer this question by explicitly solving the Schr¨odinger equation or the equation ˆ ˆ = QΨ ¯ = 0 may be a very difficult task. QΨ But there is an alternative to this brute force approach: one can determine IW by calculating the integral (6.54). If this integral is still complicated in the theory in question, one can modify the theory by a smooth change of parameters. The Witten index is an integer, and it cannot change abrubtly under a smooth parameter change. For example, one can modify a field theory by putting it into a finite spatial box and then sending the size of the box to zero, so that only zero Fourrier modes of the fields are left, and the problem is reduced to a SQM problem [64]. We hasten to comment here that the method just described has its limits of applicability.

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• To begin with, the very notion of the Witten index applies only to the systems with discrete spectrum. As we already mentioned at the end of Chap. 4.2, only such systems are mathematically well defined. If the spectrum is continuous and the states are not normalizable, most methods of the functional analysis do not work. And even a physicist who is brave enough to consider systems with continuous spectrum (like the free motion along a line), always has in mind a finite-volume regularization making the spectrum discrete. For systems with continuous spectrum, the integrals (6.54) often have fractional values, which does not make much sense.6 • Even for the systems with discrete spectrum, the integral (6.54) gives in some cases wrong fractional numbers. The reason for that is that the approximation (6.45) does not work there and one-loop effects come into play. We will meet and treat such systems in Chap. 14. Once IW is calculated, we can answer the question that we posed. Strictly speaking, we can only give a definite answer if IW 6= 0. Then a supersymmetric vacuum exists and supersymmetry is not broken. And may still be and nE=0 if IW = 0, one cannot issue a judgement: nE=0 F B nonzero, but equal. In most cases, however, the vanishing of IW signalizes . If these numbers do not vanish, there also the vanishing of nE=0 = nE=0 F B should be a special reason for that. And in such a case, it is usually possible to define a modified elaborated version of IW that does not vanish. 6.3.1

Examples

We will now show how this general machinery works for the simple SQM models discussed in the previous chapter. Consider first the Witten model with the classical Hamiltonian P 2 + [W 0 (x)]2 + W 00 (x)ψ ψ¯ . (6.60) 2 Substitute this into (6.54). Integrating over dψdψ¯ and over momenta, we derive r   Z ∞ β β[W 0 (x)]2 IW = lim dx W 00 (x) exp − . (6.61) β→0 2π −∞ 2 H =

6 There are certain rather complicated supersymmetric gauge models where the spectrum is continuous, but on top of that, ground zero-energy normalizable states exist and Witten index can still be defined [66]. But such systems are beyond the scope of this book.

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Changing the variable y = W 0 (x), we obtain r   Z W 0 (∞) βy 2 β dy exp − . IW = lim β→0 2π W 0 (−∞) 2

(6.62)

We are interested only in the systems where the potential U (x) = [W 0 (x)]2 /2 grows at infinity. Otherwise, the spectrum of the Hamiltonian would not be discrete and the notion of the index would not be well defined. Let W (x) be a polynomial. There are three possibilities. (1) The highest power in W (x) is odd. Then W 0 (∞) = W 0 (−∞), the limits in the integral (6.62) coincide and IW = 0. This signalizes the spontaneous supersymmetry breaking, which agrees with the explicit analysis of the preceding chapter. (2) The highest power in W (x) is even and the coefficient in front is positive. Then W 0 (±∞) = ±∞ and the integral in (6.62) does not depend on β, giving IW = 1—there is one bosonic zero-energy eigenstate in the spectrum7 in agreement with the explicit analysis in Chap. 5.4. (3) The higher power in W (x) is even and the coefficient in front is negative. Then W 0 (±∞) = ∓∞ and the integral (6.62) gives IW = −1—there is one fermionic zero-energy eigenstate in the spectrum. Again, this agrees with Chap. 5.4. Consider now the generalized Landau problem when the magnetic field B has the form (5.18). The classical Hamiltonian then reads 1 1 (Px + Ax )2 + (Py + Ay )2 + B(x, y)ψ ψ¯ , (6.63) 2 2 where Ax,y are the components of the vector potential. We have Z dPx dx dPy dy IW = lim dψdψ¯ β→0 2π 2π   β exp − [(Px + Ax )2 + (Py + Ay )2 ] − βB(x, y)ψ ψ¯ . (6.64) 2 H =

¯ we obtain Doing the integral over dPx dPy dψdψ, Z 1 Φ IW = dxdy B(x, y) = , 2π 2π

(6.65)

7 Strictly speaking, we can only say that nE=0 − nE=0 = 1, which does not exclude the B F possibility for the system to have e.g. two bosonic and one fermionic vacua. But one can make here the same comment that we did when discussing the case IW = 0 : there should be a special reason for the existence of extra vacuum states not dictated by the index considerations. For the Witten model, there is no such special reason.

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where Φ is the magnetic flux. We have reproduced thereby the relation (5.25) with an additional specification: the sign of the index coincides with the sign of the flux. And this conforms with the analysis of Chap. 5.3: if B > 0, the zero-energy states are spin-down in the spinorial language and bosonic in the Grassmannian language. It is instructive to reproduce the result (6.65) using the matrix formula ˆ (6.59). In this case, (−1)F ≡ −σ3 . Indeed, in the quantum Hamiltonian, ˆ ˆ ¯ ψ ψ¯ → (ψ ψ¯ − ψψ)/2. According to our convention, ψ ≡ σ+ , ψˆ¯ ≡ σ− with the commutator σ3 , and bosonic states are characterized by the negative spin projection. We derive IW = lim

β→0

1 2πβ

Z

n o dxdy Tr −σ3 e−βσ3 B(x,y)/2 ,

(6.66)

which coincides with (6.65). By definition, the index IW is an integer number. It follows that the flux Φ should be quantized, Φ = 2πn. In Chap. 13.2.1, we will see what is wrong with the systems where Φ/2π is not an integer. In contrast to what is usually assumed, one can in this case define the spectral problem for the Hamiltonian, but this Hamiltonian is not supersymmetric any more. In the most part of this book we will assume, when talking about the gauge fields, that the flux (6.65) and its generalizations for more complicated systems are quantized. The reader has noticed that in the examples considered above the integral did not depend on β and taking the limit β → 0 was superfluous. Actually, this was due to the clever choice of the classical Hamiltonians (6.60), (6.63) which are given (with the factor −i) by the Poisson bracket of the classical supercharges obtained from (5.27) and (5.26) by replacing Pˆ → P . In other words, the classical supersymmetry algebra (5.43) holds. In the expressions (5.27) and (5.26), the coordinates and momenta are not entangled and there is no ordering problem. We mentioned at the end of Chap. 5 that, in the complicated systems where such a problem exists (we will meet these systems in the third part of the book), classical supercharges should be taken as the Weyl symbols of the quantum ones. We can formulate a conjecture: ˆ ˆ Q ¯ and H ˆ be quantum operators satisfying the supersymmetry Let Q, ˆ be discrete. Let Q and Q ¯ be the algebra (5.1). Let the spectrum of H ˆ ˆ ¯ Weyl symbols of Q and Q and let the Poisson bracket {Q, Q}P and hence

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¯ Q} ¯ P vanish.8 Define the classical Hamiltonian as H = −i{Q, Q} ¯ P. {Q, Then the integral in (6.54) does not depend on β. We do not know a general proof of this statement, but it holds in all the SQM systems that we know. If H is chosen sloppily, but the quantum Hamiltonian is still derived from H by some ordering prescription, the integral in (6.54) may depend on β, but its value in the limit β → 0 should be the same in all definitions. Again, we cannot give a mathematical proof of this assertion, but a heuristic physical reasoning is the following. The ordering ambiguities is a quantum effect. On the other hand, small β correspond to high temperatures when quantum effects are not important and the dynamics is classical. Then the integral in (6.54) should not be sensitive in this limit to a particular way to resolve these ambiguities. What we can do is to illustrate this general property in a particular example. Let us concentrate on Witten’s model. We note that: (1) As the coordinates and momenta are not entangled, all the symbols of the quantum supercharges (5.27) coincide, having the same functional form as in (5.27). ¯ P . [One can (2) The classical Hamiltonian (6.60) coincides with −i{Q, Q} also notice that it coincides in this case with the Weyl symbol of the quantum Hamiltonian (5.28), but it is not so for more complicated systems]. (3) With this choice, the integral for the index is β-independent, indeed. (4) If another form of H were chosen—for example, the normal symbol of (5.28) instead of the Weyl one —the integrand in (6.61) would acquire the extra factor exp{βW 00 (x)/2}. (5) For the motion to be finite and the spectrum discrete (otherwise, the index is ill-defined), the potential [W 0 (x)]2 /2 and hence the function W 0 (x) should grow at infinity. Then for small β, the integral in (6.61) is saturated by large x. And under these conditions, [W 0 (x)]2 /2 dominates over W 00 (x)/2 and the factor exp{βW 00 (x)/2} is irrelevant—it does not affect the integral for the index in the limit β → 0.

8 This reservation is necessary to exclude the systems suffering from the classical supersymmetry anomaly, see p. 111.

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Chapter 7

Superspace and Superfields

Supersymmetric Lagrangians (5.46), (5.48), (5.52) constructed in Chap. 5 are invariant up to a total time derivative under the supersymmetry transformations (5.47), (5.49), and (5.53). These are the transformations mixing the ordinary and Grassmann variables, which can be compared to rotations mixing different vector components. In the latter case, we know that the description of a vector as a set of its components is possible, but it is much more convenient to use the vector notation and to write instead of many different scalar equations, a single vector equation, like B = ∇×A instead of Bx = ∂y Az − ∂z Ay , etc. One can ask, “Is there an analog of such vector notation in supersymmetry? Can one introduce a single object including the ordinary and Grassmann variables as different components, so that the supersymmetric actions and equations of motion can be written in a more compact and transparent way?” The answer is positive. Such an object exists, it is called the superfield—a function depending on superspace coordinates. At this point, we should complain again on the universally accepted terminology. Historically, the concept of superspace was first introduced for supersymmetric field theories [67]. In this book, we are interested only in supersymmetric quantum mechanics, and the objects that we are about to introduce should better be called dynamical supervariables depending on supertime. But people do not do so. They still use the terms “superfield” and “superspace” (adding sometimes a specification “one-dimensional”). And that is what we also will do in our book.

133

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N =1

Definition 7.1. The simplest N = 1 one-dimensional superspace includes ¯ The transforon top of time t a single real Grassmann-odd variable θ = θ. mations θ → θ + , t → t + iθ .

(7.1)

with a real Grassmann-odd  play the role of “rotations”. Consider the composition of two different transformations (7.1). We obtain θ 1 = θ + 1 , t1 = t + i1 θ ,

θ2 = θ1 + 2 = θ + 1 + 2 , t2 = t1 + i2 θ1 = t + i(1 + 2 )θ + i2 1 .

(7.2)

In other words, the composition of two supertransformations characterized by the parameters 1 and 2 is a supertransformation with the parameter 1 +2 supplemented by a shift of time by a real Grassmann-even parameter α = i2 1 . (α ¯ = −i1 2 = i2 1 = α.) Theorem 7.1. The set of all transformations (7.1) supplemented by time shifts t → t + α and characterized by a Grassmann-even parameter α and a Grassmann-odd parameter  form a group. Proof. We leave it to the reader. Such groups, characterized by both Grassmann-even and Grassmann-odd parameters and acting on superspaces, are called supergroups. Calling an element of this supergroup g(α, ), the law (7.2) can be rewritten as g(0, 2 )g(0, 1 ) = g(i2 1 , 2 + 1 ). We see that this group is not commutative. The group commutator  θ→θ g(0, 2 )g(0, 1 )g(0, −2 )g(0, −1 ) = g(2i2 1 , 0) : (7.3) t → t + 2i2 1 is not trivial and amounts to a pure time shift. A group element is the exponential of a linear combination of its generators. In our case, we may write ˆ + Q)} ˆ , g(α, ) = exp{i(αH

(7.4)

ˆ = −i∂/∂t H

(7.5)

where

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is the generator of time shifts, i.e. the Hamiltonian, and   ˆ = −i ∂ + iθ ∂ Q ∂θ ∂t

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(7.6)

is none other than the supercharge! Indeed, the relations ˆ2 = H ˆ Q

and hence

ˆ H] ˆ =0 [Q,

(7.7)

(the N = 1 superalgebra) hold. By the same token as the symmetry under time shifts brings about, for a particular dynamical system, a particular classical Hamiltonian and its quantum counterpart, the symmetry under the N = 1 supersymmetry transformations (7.1) brings about, by Noether’s theorem, a particular classical supercharge and its quantum counterpart. The quantum Hamiltonian and the supercharge thus derived (we will see in many examples how this general procedure works in the third part of our book) must obey the same algebra as the generators (7.5) and (7.6). ˆ acting in the Hilbert space of our quantum system If the Hamiltonian H is positive definite (and, if the spectrum of the Hamiltonian is bounded from below, which we assume, one can always make it positive definite by adding ˆ an appropriate constant), the algebra (7.7) dictates that the spectrum of Q ˆ − eA/c) ˆ is Hermitian. This applies e.g. to the operator σ · (P is real, i.e. Q whose square gives the Pauli Hamiltonian (5.14). Now, we saw in Chap. 5 that the algebra (7.7) treated as a symmetry algebra of a dynamical system is too small and too poor to entail nontrivial dynamical consequencies. We are still discussing in detail the N = 1 case for two reasons: (1) It is the simplest possible superspace construction. Having understood this, it will be easy to understand everything that follows. (2) Though N = 1 dynamical systems not involving some extra symmetries are devoid of nontrivial physical content, N = 1 superfields represent a convenient technical tool that is useful to keep in theorist’s kit. N = 1, d = 1 superfields were used in [68] to describe one-dimensional supergravity. We will use this tool in Chap. 9 to describe in the supersymmetric language generic complex geometry and in Chap. 12 to write a generic bi-HKT SQM action1 and explore the conditions when it is reduced to the HKT one. 1A

reader should wait till Chap. 10 to learn what “bi-HKT” means.

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To derive in the superspace approach particular Lagrangians (and then the corresponding Hamiltonians and supercharges) as functions of dynamical variables, we have to consider supervariables, alias superfields. Definition 7.2. An N = 1 superfield is a function X (t, θ) = x(t) + iθΨ(t) .

(7.8)

The ordinary dynamical variable x(t) and the Grassmann dynamical variable Ψ(t) are its bosonic and fermionic components.2 The transformations (7.1) of the superspace coordinates induce the transformation ˆ δX = iQX (7.9) or in components: δx(t) = iΨ(t) , δΨ(t) = −x(t) ˙ .

(7.10)

Generically, x(t) and Ψ(t) may be complex, but one can also impose a constraint X = X¯ which makes them real.3 This constraint is compatible with the transformation law (7.10). One can say that the real superfield X (t, θ) represents an irreducible representation of the superalgebra (7.7).4 Theorem 7.2. Let XM be a set of real superfields that all vanish at t = ±∞. The functional 5 Z Z ∞ S = dθ dt F (XM ) (7.11) −∞

is invariant under the transformations (7.1). Proof. If XM are superfields, so is F (XM )—it also transforms as in (7.9). We derive   Z ∂F ∂F + iθ = 0. (7.12) δS =  dθdt ∂θ ∂t Indeed, the second term includes a time integral of a total time derivative and vanishes. The first term vanishes too due to the property (4.48). 2 We

keep using this physical terminology, though one cannot really talk about bosons and fermions in mechanical systems; one should go over to field theory for that. 3 That is why we have chosen the capital Ψ for the Grassmann component of X —to distinguish it from the complex fermion variables entering the expansion (7.26), etc. 4 Note that we are dealing here with infinite-dimensional representations. Finitedimensional representations of the superalgebras were discussed in Chap. 5: a doublet of denenerate states for N = 2 algebra, a quartet of denenerate states for N = 4 algebra, etc. R 5 The Berezin integral dθ entering (7.11) was defined in Eq. (4.46).

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Suppose that we have only one real superfield X at our disposal. Substituting (7.8) into (7.11) and integrating over dθ, we express the functional (7.11) in components: Z S = i F 0 [x(t)] Ψ(t) dt . (7.13) This expression involves no time derivatives and is too simple to play the role of an action functional of a dynamical system. More interesting expressions can be derived if we introduce the operator ∂ ∂ − iθ (7.14) D = ∂θ ∂t called the supersymmetric covariant derivative. It has a nice property: Theorem 7.3. If X is a real superfield, iDX is also a real superfield—it is real and transforms in the same way as X . Proof. Indeed, the operators (7.6) and (7.14) anticommute. The infinitesˆ . The variation imal variation of X under (7.1) is by definition δX = iQX of DX is then ˆ ) = iQ(DX ˆ δ(DX ) = D(δX ) = D(iQX )

(7.15)

— the same law as for X . The reality of iDX = −Ψ + θx˙ is manifest.6 The time derivative of X is also a superfield, of course. Algebraically, ˆ H] ˆ = 0 or else from the relation D2 = −i∂/∂t and that follows from [Q, from the theorem just proven. Thus, the functional7 Z S = dθdt F (XM , X˙M , DXM ) (7.17) is a supersymmetric invariant. The simplest example giving really a dynamical system is Z i S = dθdt X˙ DX . (7.18) 2 6 A distinguishing feature of iDX is, however, the Grassmann nature of its lowest θindependent component. In fact, besides the bosonic superfields (7.8), one can also consider the Grassmann superfields of the type

Φ(t, θ) = φ(t) + θF (t) ,

(7.16)

where φ(t) is a real fermionic and F (t) is a real bosonic variable. We will encounter such superfields in Chaps. 8,10. 7 We will not display the limits for the time integral anymore. Heretofore, speaking about the invariance of the Lagrangian, we will always mean invariant up to a total derivative and, speaking about the invariance of the action, we will always mean invariant up to a boundary term.

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Substituting there (7.8) and integrating by dθ, we obtain the system with the Lagrangian x˙ 2 i ˙ L = − ΨΨ . (7.19) 2 2 Its bosonic part describes free motion along the x axis. N =2

7.2

Consider now a superspace including t and a complex Grassmann variable θ. The supersymmetry transformations are now defined as θ → θ + , θ¯ → θ¯ + ¯ , t → t + i(θ¯ − θ¯ )

(7.20)

with complex Grassmann . An element of the supergroup is convenient to write as √ ˆ¯ , ˆ + ¯Q)]} ˆ + 2(Q (7.21) g(α, , ¯) = exp{i[αH where   ∂ ∂ ˆ = − √i Q + iθ¯ , ∂t 2 ∂θ   ∂ ∂ ˆ ¯ = − √i + iθ . (7.22) Q ∂t 2 ∂ θ¯ The supercharges (7.22) satisfy the algebra (5.1), which means that we are on the right track to describe in the superspace approach the supersymmetric models of the previous chapter. We also introduce the covariant derivatives ∂ ∂ − iθ¯ , D= ∂θ ∂t ∂ ∂ ¯ D = − ¯ + iθ . (7.23) ∂t ∂θ ˆ¯ ¯ are nilpotent and anticommute with both Q ˆ and Q. D and D A generic N = 2 superfield reads ¯ + θθF ¯ , Z = z + θψ + θχ

(7.24)

where z and F are complex bosonic and ψ and χ are complex fermionic variables. Note that the numbers of bosonic and fermionic variables in the expansion (7.24) are equal. Theorem 7.4. The latter property holds for all superfields.

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Proof. Consider an arbitrary superfield Φ(t, θα ) depending on t and several real Grassmann coordinates θα=1,...,n . It is clear that its θ expansion gives altogether 2n components (2n+1 real components if Φ is complex). A half of them multiply Grassmann-even monoms θα1 · · · θα2k and another half multiply Grassmann-odd monoms. As was the case for N = 1, the supertransformations (7.20) can be written in a compact form: √ ˆ¯ . ˆ + ¯Q)Z (7.25) δZ = i 2(Q However, (7.24) is not an irreducible representation of the superalgebra (5.1). Irreducible ones can be constructed by imposing certain constraints ¯ This gives a on Z. To begin with, we can impose the constraint Z = Z. real N = 2 superfield: ¯ X = x + θψ + ψ¯θ¯ + θθF

(7.26)

with real x, F . A real superfield stays real after a supertransformation (7.25). The superfield X has the components at the three levels: (i) the θindependent bosonic component x, (ii) the fermion components ψ and ψ¯ that appear in the linear terms of the θ expansion and (iii) the bosonic component F in the quadratic term. Using the convenient notation suggested in [69], we can denote this superfield (1, 2, 1), where the numbers count the real degrees of freedom in the θ-independent, linear and quadratic sectors.8 One can ask: how do we know that (7.26) is irreducible; how to prove that one cannot impose on X an additional constraint compatible with supersymmetry? It follows from the fact that any superfield representation of the N = 2 supersymmetry algebra must have at least 2 bosonic and 2 fermionic components. Indeed: take any bosonic component and act on it by two different real supercharges available in the algebra; you obtain two different fermionic components. And starting from a fermion component, we obtain two different bosonic components. The superfield (7.26) saturates this minimum, and the number of components cannot be reduced further. The same concerns the chiral superfields (7.29) and (7.30) discussed below. The supertransformations of the components of (7.26) induced by (7.25) 8 And

the N = 1 superfield (7.8) is, obviously, (1, 1) in these terms!

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read δx = ψ − ¯ψ¯ , δψ = ¯(F − ix) ˙ , δ ψ¯ = (F + ix) ˙ , ¯˙ δF = −i(ψ˙ + ¯ψ)

(7.27)

One can observe the pattern: the variation of a certain component of X ¯ and the involves a higher component (ψ and ψ¯ for δx, F for δψ and δ ψ) time derivative of a lower component. This pattern also holds for more complicated superfields to be considered later. In particular, the supersymmetric variation of the highest superfield component is always a total time derivative. For N = 1, the only irreducible superfield was the real superfield (7.8). But the case N = 2 is more rich, and besides the real superfield (7.26), there exist also two other irreducible representations of (5.1). Definition 7.3. A holomorphic chiral N = 2 superfield Z is a generic su¯ = 0. An antiholomorphic chiral perfield (7.24) satisfying the constraint DZ ¯ N = 2 superfield Z is a generic superfield (7.24) satisfying the constraint9 DZ¯ = 0. ˆ¯ = 0, one immediately sees that ¯ Q} ˆ = {D, ¯ Q} Bearing in mind that {D, the variation (7.25) is holomorphic chiral if Z is holomorphic chiral. The ¯ = 0 kills a half same is true for antiholomorphic fields. The condition DZ of the degrees of freedom. A generic solution of this constraint is10 √ (7.28) Z = z + 2 θψ − iθθ¯z˙ . We see that the term ∝ θ¯ is now absent and the term ∝ θθ¯ is not independent any more, but is given by the time derivative of the lowest term. A convenient technical trick is to introduce a “holomorphic time” tL = ¯ It transforms under supersymmetry as tL → tL − 2iθ¯ t − iθθ.  and does ¯ The covariant derivative D ¯ reduces in this not “feel” the presence of θ. 9 In the slang used by the physicists these fields are called “left chiral” and “right chiral”. This terminology came from field theories, where the words “left” and “right” have a physical meaning: the particle representing an elementary excitation of a chiral field is not symmetric under mirror reflection, like a left or right hand (the word chiral comes from the Greek root χε ´ρι meaning “hand”). But the notions of left and right make, of course, little sense for mechanical systems, and we will try not to use these words, allowing, however, for their traces in the notation tL and tR below. √ 10 The factor 2 is a convenient convention.

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¯ = 0 have no frame simply to ∂/∂ θ¯ and the solutions to the equation DZ ¯ dependence on θ: √ Z = Z(tL , θ) = z(tL ) + 2 θψ(tL ) . (7.29) Similarly, ¯ = z¯(tR ) − ¯ R , θ) Z¯ = Z(t

√

¯ R) 2 θ¯ψ(t

(7.30)

¯ with tR = tL = t + iθθ. Definition 7.4. The set of coordinates (tL , θ) describes the holomorphic ¯ describes the antiholomorphic chiral N = 2 superspace and the set (tR , θ) chiral N = 2 superspace. The lowest bosonic term in the expansion (7.29) or (7.30) includes two real degrees of freedom, the linear in θ or θ¯ term gives two real fermionic degrees of freedom, and there is no quadratic term. These are (2, 2, 0) superfields. The supertransformations of the components induced by (7.25) read √ √ δψ = −i 2 ¯z˙ δz = 2 ψ , √ √ δ¯ z = − 2 ¯ψ¯ , δ ψ¯ = i 2 z¯˙ . (7.31) √ Note that they coincide with (5.53) up to an irrelevant factor 2 ! Besides the real superfield (7.26) and the chiral superfields (7.29), (7.30), there are no other irreducible infinite-dimensional representations of the N = 2 supersymmetry algebra. This is a nontrivial statement. As we will see in Chap. 9 (see the discussion after Theorem 9.4), it is in fact a corollary of Theorem 2.6—the Newlander-Nirenberg theorem. To derive the Lagrangian (5.52) of the Landau problem in the superspace approach, we need to write a superfield action. According to a generalization of Theorem 7.2 for N = 2 (there are similar generalizations for higher N ), any full superspace integral of a generic superfield is a supersymmetric invariant. For the action to be real, this superfield should be real too.11 There are many ways to cook up a real superfield out of the chiral superfield (7.28) that we have at our disposal. One can simply multiply Z by Z¯ and the product Z Z¯ is obviously real. But to obtain a nontrivial ¯ out of our dynamics, we should also get the covariant derivatives D and D 11 We hasten to comment that there is an alternative: one can write the integral of a chiral superfield over the chiral superspace and add the complex conjugate, as we will do in Eq. (7.51) below. But one would not obtain the Landau Hamiltonian in this way.

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tool kit. As was also the case for N = 1, they make a superfield out of a superfield. But the nature of the superfield changes: due to the nilpotency ¯ DZ is antiholomorphic chiral and D ¯ Z¯ is holomorphic chiral of D and D, 12 for any Z. One can be convinced that  the action  Z 1¯¯ B ¯ ¯ S = dθdθdt DZ DZ − Z Z (7.32) 4 2 does the job and the component Lagrangian derived from it coincides up to a total derivative with (5.52). Equations (5.52) and (7.32) are two different ways of representing the same supersymmetric system. In the first case, the Lagrangian is expressed in components, and to show that it is supersymmetric, one has to check the invariance of the action under the transformations (5.53) explicitly. On the other hand, supersymmetry of the action (7.32) is a corollary of Theorem 7.2 (its generalization for higher N ). The law of transformations of components follows from the transformation law (7.20) of the superspace coordinates. One may but does not need to check anything. The superspace approach gives one, if you will, an “industrial” method to write supersymmetric Lagrangians without much effort—like the calculus deleloped by Newton and Leibniz allowed one to calculate the volume of a ball—the problem first solved by Archimedes by ingenous geometric construction—in two lines. Besides the component description and N = 2 description, Landau’s problem also admits a third intermediate way of its description. We pose Ψ1 + iΨ2 θ1 + iθ2 x1 + ix2 √ √ , ψ= , and θ = i √ (7.33) z= 2 2 2 to express (7.28) and its conjugate in the form 1 Z = √ (1 + iθ2 D1 )(X1 + iX2 ), 2 1 Z¯ = √ (1 − iθ2 D1 )(X1 − iX2 ), (7.34) 2 where X1,2 = x1,2 + iθ1 Ψ1,2 are N = 1 superfields and D1 is the N = 1 covariant derivative (7.14) in the superspace (t, θ1 ). ¯ by13 −idθ1 dθ2 , integrating it over dθ2 and Replacing in (7.32) dθdθ trading θ1 for θ, we Z may express the action asZ iB i dθdt X˙M DXM − εM N dθdt XM DXN . (7.35) S = 2 2 R ¯ X coincides with the highest component of the superthat the integral dθdθ field (7.26). This is how the construction of supersymmetric actions is described in the textbook [70]: construct a real superfield of the basic superfields in your kit and then pick up the highest component in√its θ expansion. √ 13 Note that, if θ = i(θ + iθ )/ 2, then we must define dθ = −i(dθ − idθ )/ 2 to 1 R 2 1 2 √ R harmonize the properties dθ θ = 1 and dθ1,2 θ1,2 = 1. Also, D = −i(D1 − iD2 )/ 2. 12 Note

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Going down to components, we obtain L =

i ˙ B 1 2 x˙ − Ψ εM N (x˙ M xN + iΨM ΨN ) , M ΨM + 2 M 2 2

(7.36)

which is equivalent to (5.52). In the expression (7.35), θ is real, and the transformation (7.10) of the N = 1 superfield components gives one of the complex supersymmetry √ transformations that appear in (7.31) : N =1 = 2 Im(N =2 ). And another √ transformation with the real Grassmann parameter η (the real part of 2 N =2 ) is realized as δXM = η εM N DXN .

(7.37)

One can be explicitly convinced that (7.37) is a symmetry using D2 = −i∂/∂t. Rather often such a mixed description, when a part of supersymmetries is realized by superspace transformations and another part as a transformation of superfields of low rank, turns out convenient. We will meet many such constructions later. We concentrated in the last couple of pages on the Landau problem ¯ in with a homogeneous magnetic field. But one could also replace BZ Z/2 ¯ (7.32) by any function F (Z, Z) to obtain a system with a non-homogeneous magnetic field B(z, z¯) = 2

∂2F . ∂z∂ z¯

(7.38)

The Hamiltonian of this system was written before in Eq. (6.63). To derive the Lagrangian (5.48) for the Witten model, one has to take the real N = 2 superfield (7.26) and write   Z 1¯ ¯ L = dθdθ DXDX − W (X) 2 1 2 = (x˙ + F 2 ) − iψ˙ ψ¯ − W 00 (x)ψ ψ¯ − F W 0 (x) . (7.39) 2 This resembles (5.48), but does not coincide with it completely: the Lagrangian L depends on an extra variable F , which was absent in (5.48). One can notice, however, that F is an auxiliary non-dynamical variable: it enters the Lagrangian without derivatives and the corresponding equation of motion is simply ∂L = F − W 0 (x) = 0 . ∂F

(7.40)

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It has a trivial solution F = W 0 (x). By substituting this solution in (7.39), we reproduce (5.48). When doing the same for the supertransformation (7.27), we reproduce (5.49). One can mention in parentheses at this point that F is auxiliary and can be algebraically excluded for the Lagrangian (7.39), but if one writes a more complicated action including extra time derivative, like Z ¯ ¯ X˙ , S = dθdθdt DX D (7.41) the component Lagrangian would depend on F˙ [49]. A final comment in this section concerns the supersymmetric oscillator. In the previous chapter, we described it first using the holomorphic bosonic variables a, a ¯ and then noted that the same system, formulated in the ordinary coordinate representation, has the Hamiltonian (5.29) and represents a particular case of Witten’s model. It is this second formulation which can also be derived in the superspace approach. We do not know how to directly derive in this way the Lagrangian (5.46) involving bosonic holomorphic variables. The problem is that the supersymmetric transformations (5.47) do not involve time derivatives, as is always the case for the transformation law of superfield components. 7.3

N =4

Consider the superspace including t and two complex Grassmann variables θj and their conjugates θ¯j . They transform under supersymmetry as θ j → θ j + j , θ¯j → θ¯j + ¯j , t → t + i(j θ¯j + ¯j θj ) The generators of these transformations   ∂ ∂ ˆ j = − √i + iθ¯j , Q ∂t 2 ∂θj   i ∂ ∂ ˆ ¯ √ Qj = − + iθj ∂t 2 ∂ θ¯j

(7.42)

(7.43)

satisfy the extended supersymmetry algebra ˆ ˆ¯ } = 0 , ˆj , Q ˆ k } = {Q ¯j , Q {Q k

ˆ¯ } = −iδ j ∂ , ˆj , Q {Q k k ∂t

(7.44)

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as in (5.36). The covariant derivatives ∂ ∂ − iθ¯j , Dj = ∂θj ∂t ∂ ∂ ¯j = − + iθj . D j ¯ ∂t ∂θ satisfy a similar algebra,

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(7.45)

¯ k } = 2iδ j ∂ , (7.46) {Dj , D k ∂t and anticommute with all the supercharges. A generic complex N = 4 superfield includes many terms in its θ expansion (up to ∼ θ4 ) and 16 + 16 real bosonic and fermionic components. This representation of the algebra (5.36) is far from being irreducible. In N = 1 superspace, we had only one superfield (7.8). In N = 2 superspace, there are irreducible superfields of two types: real and chiral. In N = 4 superspace, there are several types of irreducible superfields. ¯j, D ¯ k} = 0 , {Dj , Dk } = {D

7.3.1

(2, 4, 2)

The direct generalizations of the N = 2 chiral superfields discussed in the previous section are N = 4 chiral superfields. Definition 7.5. A holomorphic chiral N = 4 superfield is a generic N = 4 ¯ j Z = 0. An antiholomorphic chiral superfield Z satisfying the constraint D j field satisfies the constraint D Z = 0. If we introduce the holomorphic time tL = t − iθj θ¯j ,

(7.47)

¯ j Z = 0 becomes the constraint D ∂Z/∂ θ¯j fixed

tL

= 0.

In this frame, the holomorphic chiral superfield does not depend on θ¯j and has only 3 terms in its expansion:14 √ (7.48) Z(tL , θj ) = z(tL ) + 2 θj ψ j (tL ) + 2 θ1 θ2 F (tL ) . Now, z(tL ) gives two real bosonic components, there are also two real bosonic components in the quadratic term ∼ θ2 , and the linear in θ term 14 Recall that the indices are raised and lowered according to (5.37). The complex conjugate of ψj will be denoted as ψ¯j —in the same way as for θj . It follows that θ¯j = −θj and ψ¯j (tR ) = −ψ j (tL ).

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provides us with four real fermionic components. It is a superfield of the type (2, 4, 2). By the same token, an antiholomorphic chiral superfield can be represented as √ ¯ R , θ¯j ) = z¯(tR ) + 2 θ¯j ψ¯j (tR ) + 2 θ¯2 θ¯1 F¯ (tR ) , (7.49) Z(t where tR = tL = t + iθj θ¯j ,

(7.50)

The field (7.48) lives in the holomorphic chiral N = 4 superspace and the field (7.49) in the antiholomorphic chiral N = 4 superspace. Consider the system Z 1 ¯ R , θ) ¯ 2 θdt Z(t ¯ Z(tL , θ) d2 θd S= 4 Z Z 1 1 ¯ R W (Z) , d2 θdtL W (Z) + d2 θdt + (7.51) 2 2 with d2 θ = dθ1 dθ2 , d2 θ¯ = dθ¯2 dθ¯1 . (7.52) W (Z) is an arbitrary holomorphic function. The first term in (7.51) involves the integral over the whole superspace and the second and the third terms ¯ All these involve the integrals over the chiral superspaces (tL , θ) and (tR , θ). integrals are supersymmetric invariants. Substituting in this expression (7.48), (7.49), (7.47) and (7.50), integrating over d4 θ (doing so for the first term, one has to keep in mind ¯ depend on the different arguments and one should expand that Z and Z z(tL ) = z − iθj θ¯j z˙ − (θj θ¯j )2 z¨/2, and similarly for other variables), and omitting some total time derivatives, we obtain i L = z˙ z¯˙ + F F¯ − (ψ˙ j ψ¯j − ψj ψ¯˙ j ) 2 (7.53) − F W 0 (z) − F¯ W 0 (z) + W 00 (z)ψ1 ψ2 + W 00 (z) ψ¯2 ψ¯1 . ¯ We see that the variables F and F are not dynamical and one can exclude them by substituting the solutions F = W 0 (z), F¯ = W 0 (z) of the corresponding (purely algebraic) equations of motion in the Lagrangian. Doing so and going over to the classical and then quantum Hamiltonian, we derive ˆ = π H ˆπ ˆ † + W 0 (z)W 0 (z) − W 00 (z)ψ1 ψ2 − W 00 (z) ψˆ¯2 ψˆ¯1 . (7.54) When the superpotential W (Z) = Z3 /3 is chosen, this Hamiltonian coincides with the Hamiltonian (5.39) (our first example of an extended supersymmetric system). In fact, the system just considered is none other

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than the SQM version of the Wess-Zumino supersymmetric field theory system [71]. To define the chiral N = 4 superfields, we imposed the constraints ¯ j Z = 0 or Dj Z = 0. But one could also impose a “mixed” constraint like D ¯ 1Z = 0 . D2 Z = D

(7.55)

A superfield satisfying the constraint (7.55) can be called a “twisted” or “mirror” chiral superfield. If the action of the model includes only such mirror fields and their complex congugates, it does not really matter— just redefine θ2 ↔ θ¯2 and Z becomes a conventional holomorphic chiral superfield. Nontrivial models appear when the action includes both ordinary and mirror field. Such models were first constructed in the field theory context in [72]. Different SQM models including both ordinary and mirror (2, 4, 2) superfields were studied in [73]. We will discuss them in Chap. 10. 7.3.2

(3, 4, 1) and (4, 4, 0)

The number of real fermionic components in an irreducible N = 4 superfield is equal to 4. The same concerns the bosonic components, but not all of them are always dynamical. In the example that we have just considered, there were 2 dynamical and 2 auxiliary bosonic degrees of freedom. Superfields with a different dynamic-auxiliary decomposition can also be constructed. We will discuss two such superfields: the superfield (3, 4, 1) and the superfield (4, 4, 0). Take a generic N = 4 superfield V and impose the condition of reality, V = V¯ . We are left at this stage with 8 bosonic and 8 fermionic real components. Consider now the superfield ¯ k + Dk D ¯ j )V . Φjk = i(Dj D

(7.56)

It satisfies the pseudoreality condition, Φjk = εjl εkp Φlp ,

(7.57)

Dp Φjk + Dj Φkp + Dk Φpj = 0 .

(7.58)

and also the constraint

Inversely, one can show that any symmetric in j ↔ k complex N = 4 superfield Φjk satisfying the constraints (7.57) and (7.58) can be represented in the form (7.56). A remarkable fact is that Φjk does not change under the transformation ¯, V → V +Z+Z

(7.59)

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where Z is a chiral (2, 4, 2) superfield of the preceding subsection. This ¯ = 0 and {Dj , D ¯ j Z = Dj Z ¯ k } = 2iεjk ∂/∂t. The chiral follows from D superfield involves, as we know, 4 bosonic and 4 fermionic components. To count the independent components in Φjk , one has to subtract these numbers from 8+8 components in V . One is thus left with 4+4 components meaning that Φjk is an irreducible superfield. In fact, the transformation (7.59) is nothing but a supergauge symmetry. A reader physicist who already has some notions about supersymmetry, encountered this transformation (its field theory version) e.g. in Chap. 5 of the standard textbook in supersymmetry [70]. It is thus not accidental that the first example of an SQM system based on the superfield Φjk was found (in the component form) as an effective Hamiltonian of a 4-dimensional supersymmetric gauge theory put in a small spatial box [74]. Its superfield description was worked out in [75]. To mark out the independent components explicitly, one can use the freedom (7.59) and bring V in the form (the so-called Wess-Zumino gauge): 1 ¯ M θ + (ψθ + ψ¯θ) ¯ θθ ¯ + F θ1 θ2 θ¯2 θ¯1 V = − AM θσ 2

(7.60)

with real AM and F and complex ψj . The shorthands in the equation above are spelled out as ¯ M θ ≡ θ¯j (σM ) k θk , θσ j

¯ ≡ θ¯j θj , θθ

ψθ ≡ ψ j θj ,

ψ¯θ¯ ≡ ψ¯j θ¯j . (7.61)

It is now convenient to trade the pseudoreal superfields Φjk for the triplet of real superfields ΦM =

i (σM )jk Φjk , 2

Φjk = i(σM )jk ΦM

(7.62)

with (σM )jk and (σM )jk given in (3.17). In these terms, the constraint (7.58) acquires the form Ds ΦM =

i εM N P (σP )s j Dj ΦN . 2

(7.63)

The solution to (7.63) is ¯q Ds ΦM = (σM )sq Ξ

(7.64)

¯ q . In other words, the constraint (7.58) eliminates the spin with arbitrary Ξ 3/2 part in Ds ΦM . ¯ q indicates that we are dealing with a pair of antiholoThe bar in Ξ morphic chiral superfields. Antiholomorphicity follows not from (7.63), but

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from the definitions (7.56) and (7.62). It is not so much an issue, however. The identity ¯ s ΦM = (σM )sq Ξq D

(7.65)

with holomorphic Ξq also holds. ΦM kind of straddles the middle stretch ¯q. of the seesaw [76] between Ξq and Ξ The component expansion of ΦM reads ¯ M ψ¯ + εM N P A˙ N θσ ¯ P θ + F θσ ¯ Mθ ΦM = AM + ψσM θ − θσ ¯ M θ)(ψθ ˙ − ψ¯˙ θ) ¯ + A¨M θ1 θ2 θ¯2 θ¯1 , + i(θσ

(7.66)

¯ M ψ¯ = θ¯j (σM ) k ψ¯k . The simplest action where ψσM θ = ψ j (σM )jk θk and θσ j Z 1 ¯ 2 θdt (ΦM )2 d2 θd (7.67) S = − 12 gives the component Lagrangian 2 1 ˙ 2 i ¯˙ ¯˙ + F . (AM ) + (ψ ψ − ψψ) (7.68) 2 2 2 We observe the presence of the complex fermion doublet, of three dynamical variables AM (and the bosonic part of the Lagrangian describes free 3dimensional motion) and of the single auxiliary variable F , which can be excluded thanks to the equation of motion F = 0. We postpone the discussion of more complicated models based on the superfield ΦM to Chap. 10.

L =

To define the irreducible superfield15 (4, 4, 0), we take a couple of holomorphic chiral N = 4 superfields ZA and impose on top of the constraints ¯ j ZA = 0 the constraint D ¯B = 0 . ¯jZ Dj ZA − εAB D

(7.69)

This constraint kills a half of the degrees of freedom, and we are left once again with 4 bosonic and 4 fermionic variables. The solution of (7.69) may be presented as ¯ 1 Z¯ B − iθ2 θ¯2 Z˙ A , ZA = Z A + θ2 εAB D ¯ A = Z¯ A − θ¯2 εAB D1 Z B + iθ2 θ¯2 Z¯˙ A , Z

(7.70)

where Z A are the holomorphic chiral N = 2 superfields (7.28) living in the subspace (t; θ1 , θ¯1 ). At the level of components, we have two complex bosonic variables z A and two complex fermionic variables ψ A . 15 It

was first introduced in [77]. In our description, we follow the approach of [73].

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The simplest invariant action reads Z 1 ¯A . ¯ 2 θdt ZA Z d2 θd (7.71) S = 8 The corresponding component Lagrangian is very simple: i L = z˙ A z˙ A + (ψ¯A ψ˙ A − ψ¯˙ A ψ A ) . (7.72) 2 It does not involve any auxiliary variables.16 Similar to what we had for (2, 4, 2), one can also define a mirror (4, 4, 0) superfield by interchanging θ2 ↔ θ¯2 in the constraints. 7.4

Harmonic Superspace

The goal of this book is to describe geometric structures of the smooth manifolds using supersymmetric methods. And the methods discussed so far have been sufficient to describe (i) generic manifolds with their de Rham complex, (ii) complex manifolds with their Dolbeaux complex, (iii) K¨ahler manifolds and (iv) Obata-flat HKT manifolds. In addition, they will allow us to describe new geometrical structures not noticed by mathematicians heretofore (see Chaps. 10,11). But these methods turn out to be not sufficient to describe in a complete way generic HKT manifolds and hyper-K¨ ahler manifolds. To do so, we need to learn an elaboration of the superspace technique developed in Dubna — the harmonic superspace technique [78]. We saw in the preceding section that the higher is N in the supersymmetry algebra (5.36), the more intricate is its superfield description. The problem is to find a set of constraints that the superfield should obey to be an irreducible representation of (5.36). We managed to do so for N = 1, N = 2 and N = 4, but this direct approach does not work for N = 8. An irreducible N = 8 superfield must include 8 ordinary and 8 Grassmann real components, but a generic N = 8 superfield includes 256+256 components, and a set of constraints in the ordinary N = 8 superspace reducing it to 8+8 is not known. In addition, already at the N = 4 level, we will see in Chaps. 10,12 that the constraints (7.69) that we used to define a (4, 4, 0) superfield are too rigid. They are not general enough to describe interesting geometry. Harmonic superspace was originally invented to describe extended supersymmetric field theories. The adaptation of this technique to SQM 16 The extra variables that were present in the (2, 4, 2) superfields ZA have been eliminated by the constraints (7.69).

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systems was worked out in Ref. [79], and we will mostly follow their conventions. Let us harmonize first the N = 4 superspace and leave the N = 8 superspace for the dessert—the complications that will show up in that case are not so principal. The harmonic superspace technique capitalizes on the presence of the R-symmetry that rotates complex supercharges and the associated fermion variables. [We discussed it above simultaneously with introducing the extended supersymmetry algebra (5.36).] In our case, the group is U (2). Its U (1) subgroup is present also in the case of the mininal N = 2 supersymmetry. The new relevant part is SU (2). The main idea is to extend the N = 4 superspace (t; θj , θ¯j ) by adding extra commuting coordinates parameterizing this SU (2) group. For our purposes, the most convenient way of parameterization is introducing a complex doublet u+(j=1,2) of unit length, def

u+j u+j = u+j u− j = 1. u+ j

(7.73) +k

One may write an explicit expression of = εjk u via the three Euler angles:   cos(θ/2) eiφ/2 + iψ/2 uj = e , (7.74) i sin(θ/2) e−iφ/2 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π, 0 ≤ ψ ≤ 4π. Then   i sin(θ/2) eiφ/2 −iψ/2 u− = e . (7.75) j cos(θ/2) e−iφ/2 The matrix  + − u1 u1 (7.76) U = − u+ 2 u2 belongs to SU (2) so that the property − − + u+ j uk − uj uk = εjk

(7.77)

holds. Introduce now the odd coordinates θ± = θj u± θ¯± = θ¯j u± j , j . They are ± SU (2) invariants. Note that θ¯ are not conjugates of θ± . We have instead θ+ = −θ¯− , θ− = θ¯+ . (7.78) We will also use in the following a special pseudoconjugation operation— a combination of the ordinary complex conjugation and the following trans− − + formation of the harmonics:17 u+ j → uj , uj → −uj . This transformation 17 It may be interpreted as the antipodal reflection on the 2-sphere SU (2)/U (1): θ → π − θ, φ → φ + π, supplemented by the U (1) reflection ψ → −ψ, in the parameterization (7.74).

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acts as ± =θ ¯± , θf

± = −θ ± , θ¯f

± ±j uf . j =u

(7.79)

Definition 7.6. The N = 4 one-dimensional harmonic superspace is a superspace with the coordinates t; θ+ , θ¯+ , θ− , θ¯− ; u± j .

(7.80)

The supertransformations of coordinates (7.42) induce the transformations t → t + i(− θ¯+ − + θ¯− + ¯+ θ− − ¯− θ+ ) ,

±

θ ± → θ ± + ± , θ¯± → θ¯± + ¯±

(7.81)

¯j u± j .

A generic harmonic superfield is a function

j u± j

±

with  = and ¯ = in this superspace:

Φ = φ(t, u) + . . . + D(t, u) θ¯+ θ+ θ¯− θ− .

(7.82)

The supercoordinate transformations (7.81) generate the transformations of the components of (7.82). As in the ordinary superspace, one defines the covariant derivatives. In this case, there are covariant derivatives of two types. Firstly, we have Grassmann covariant derivatives, ∂ ∂ ∂ ∂ − iθ¯+ , D− = − + − iθ¯− , − ∂θ ∂t ∂θ ∂t ∂ ∂ + + ∂ − − ∂ ¯ ¯ D = − ¯− − iθ , D = ¯+ − iθ . ∂t ∂t ∂θ ∂θ Secondly, there are harmonic covariant derivatives D+ =

∂ ∂ ∂ ∂ + θ¯+ ¯+ − θ− − − θ¯− ¯− , ∂θ+ ∂θ ∂θ ∂θ ∂ ∂ = ∂ ++ + θ+ − + θ¯+ ¯− , ∂θ ∂θ −− − ∂ − ∂ ¯ =∂ +θ + θ ¯+ , ∂θ+ ∂θ

(7.83)

D0 = ∂ 0 + θ+ D++ D−−

(7.84)

where ∂ 0 = u+ j

∂ − ∂ , + − uj ∂uj ∂u− j

∂ ++ = u+ j

∂ , ∂u− j

∂ −− = u− j

∂ . ∂u+ j

(7.85)

They satisfy the su(2) algebra [D++ , D−− ] = D0 ,

[D0 , D±± ] = ±2D±± ,

(7.86)

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the same as the “short” derivatives (7.85). The other nonzero (anti)commutators involving the covariant derivatives are ¯ ∓] = D ¯± [D±± , D ¯ + } = −{D+ , D ¯ − } = 2i ∂ . {D− , D (7.87) ∂t Note that all the Grassmann covariant derivatives (7.83) anticommute and the harmonic covariant derivatives (7.84) commute with the supercharges—generators of the transformations (7.81): [D±± , D∓ ] = D± ,

Q+ =

∂ ∂ + iθ¯+ , ∂θ− ∂t

etc.

Corollary. If Φ is a harmonic superfield (7.82), then DΦ is also a harmonic superfield with the same transformation law for the components, where D is any covariant derivative in (7.83) and (7.84) (cf. Theorem 7.3). The operator D0 is called the harmonic charge. If the parameterization 0 (7.74) for u+ j is chosen, its short version is expressed as ∂ = −2i∂/(∂ψ). + The variables uj as well as θ+ and θ¯+ have the harmonic charge +1. And − ¯− the variables u− carry charge −1. All the harmonic superfields j ,θ ,θ that we are going to deal with have a definite integer value of this charge. The operator D++ acting on a superfield of harmonic charge q produces a superfield of charge q + 2. The operator D−− acting on a superfield of charge q produces a superfield of charge q − 2. For any expression X q carrying a definite harmonic charge q, the property f fq = (−1)q X q X

(7.88)

holds. R The integral dt d2 θ+ d2 θ− Φ is invariant under the transformations (7.81). It still depends on u± , however, and to get rid of this dependence and to write down an invariant action, one has to integrate over the harmonics, in other words—overRthe group SU (2) that they parameterize. We (q) normalize the group volume du to 1. For any function R q of harmonics F having a nonzero Rharmonic charge the integral F (u) du vanishes. And R q, − + + − − the integrals like u+ u du or u u j k j k ul up du must be expressed via the SU (2) invariant tensor εjk . Bearing in mind (7.73) and (7.77), this gives Z 1 − εjk , u+ j uk du = 2 Z 1 + − − u+ (εjl εkp + εjp εkl ) , etc. (7.89) j uk ul up du = 6

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But why is it convenient to extend the N = 4 superspace in such a way? The harmonic superspace described above seems to be a rather complicated construction. Why not stay with the standard N = 4 superspace of the previous section? The point is that the harmonic superspace approach allows one to find simple constraints to be imposed on (7.82) that single out irreducible representations of the superalgebra. These constrained irreducible superfields can be handled with more ease than generic superfields (7.82) and also are much more useful: we will use them in the third part of this book to describe nontrivial geometric structures on the manifolds. To begin with, we impose the constraints ¯ +Φ = 0 . D+ Φ = D (7.90) The superfields Φ satisfying this constraint are called the Grassmannanalytic or G-analytic superfields. They have many common features with the chiral N = 2 superfields (7.29) and (7.30). As we saw, a holomorphic chiral N = 2 superfield lives in fact in the reduced holomorphic chiral ¯ but only θ and the holomorphic time superspace that does not include θ, ¯ tL = t − iθθ. Similarly, a G-analytic superfield lives in fact in the reduced analytic superspace (tA ; θ+ , θ¯+ ; u) ≡ (ζ, u) (7.91) including the harmonics, the analytic time tA = t + i(θ+ θ¯− + θ− θ¯+ )

(7.92) + − − ¯ ¯ and the odd coordinates θ and θ , but not θ and θ . As a result, the θexpansion for the G-analytic superfields is much shorter than for the generic ones. Indeed, in the analytic basis, the Grassmann covariant derivatives acquire the form ∂ ∂ ∂ D− = − + − 2iθ¯− , D+ = − , ∂θ ∂θ ∂tA ¯ − = ∂ − 2iθ− ∂ . ¯+ = − ∂ , D (7.93) D ∂tA ∂ θ¯− ∂ θ¯+ And we see that the constraints (7.90) boil down in the analytic basis to the condition that Φ does not depend on θ− and θ¯− The analytic space coordinates transform as tA → tA + 2i(− θ¯+ − ¯ − θ+ ) , +

θ + → θ + + + , θ¯+ → θ¯+ + ¯+ .

(7.94)

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These transformations do not include θ− or θ¯− , and hence the analytic superspace (7.91) represents an invariant subspace of the full harmonic superspace (7.80). It is noteworthy that the analytic time is invariant under the pseudoconjugation defined in (7.79) (but not under the ordinary complex conju+ = θ ¯+ and θf ¯+ = −θ+ gation). This means together with the fact that θf that the pseudoconjugation leaves the whole analytic superspace intact. The most important for geometric applications object is the G-analytic superfield q + (ζ, u) of harmonic charge +1. Its θ-expansion is short, as promised (cf. the short expansions (7.29), (7.30) for the chiral superfields): q + (ζ, u) = f + (tA , u) + θ+ χ(tA , u) + θ¯+ κ(tA , u) + θ+ θ¯+ A− (tA , u), (7.95) where f + (tA , u) and A− (tA , u) are Grassmann-even variables of harmonic charge +1 and −1, respectively, and χ(tA , u) and κ(tA , u) are Grassmannodd variables of zero harmonic charge. We have decreased the number of components in the θ-expansion, compared to the ordinary (non-harmonic) superspace description, but had to pay, as it seems, a heavy price for that: f + (tA , u), etc. depend not only on time, but represent infinite series in u+ and u− . It is convenient to expand over the symmetrized products of the harmonics. For example, + − (ijk) f + (tA , u) = f j (tA ) u+ (tA ) u+ j +A i uj uk + . . .

(7.96)

Then the different terms of this expansion are orthogonal to each other with respect to the measure (7.89). It seems that we have to deal with an infinite number of ordinary and Grassmann time-dependent variables that multiply an infinite number of monoms in these series. A remarkable fact, however, is that one can eliminate almost all of them by imposing a constraint D++ q + = 0 .

(7.97)

Indeed, the harmonic derivatives in the analytical basis read ∂ ∂ ∂ + θ+ − + θ¯+ ¯− , D++ = ∂ ++ + 2iθ+ θ¯+ ∂tA ∂θ ∂θ ∂ ∂ ∂ D−− = ∂ −− + 2iθ− θ¯− + θ− + + θ¯− ¯+ . (7.98) ∂tA ∂θ ∂θ Expanding the L.H.S. of (7.97) in θ and u, one obtains an infinite number of conditions. And it turns out that this infinity almost matches the infinity of the terms in the expansion of q + (ζ, u). A general solution to the constraint (7.97) is q + (ζ, u) = f j (tA )u+ + θ+ χ(tA ) + θ¯+ κ(tA ) − 2iθ+ θ¯+ f˙j (tA )u− . (7.99) j

j

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We are left with only two complex bosonic variables f j (tA ) and two complex fermionic variables χ(tA ) and κ(tA ). This set of components matches the set of components of the irreducible N = 4 superfield (4, 4, 0) discussed above! A legitimate question is why have we put so much attention to a rather nontrivial and complicated way to rederive the (4, 4, 0) superfield, which we have already derived earlier without breaking so much sweat. The answer is the following. Yes, for the linear multiplet q + satisfying the constraint (7.97), the harmonic description is not at all necessary. All the results coincide with the results that can be derived in a traditional way, which is simpler. But only the harmonic description allows one to treat the so-called nonlinear multiplets satisfying the nonlinear constraints [80] D++ q +a = L+3a (q +b , u) ,

(7.100)

where L+3a is a set of 2n arbitrary functions carrying harmonic charge +3 of 2n G-analytic superfields q +b and harmonics.18 This is the way a SQM model describing the geometry of a generic HKT manifold can be constructed. In addition, only the harmonic superspace description allows one to incorporate in the construction external gauge fields. We will continue this discussion in Chap. 12. Another interesting N = 4 harmonic superfield is the G-analytic superfield V ++ (ζ, u) of harmonic charge +2. To eliminate higher-order terms in the harmonic expansion, we impose the constraint D++ V ++ = 0 . To make V require

++

(7.103)

an irreducible representation of the superalgebra, we also ++ . V ++ = V] +

++

(7.104) + ++ ¯ =D V =0

A general solution to the analyticity constraints D V and the constraints (7.103), (7.104) reads [79] V ++ = v (jk) (tA )u+ u+ + iθ+ ψ j (tA )u+ + iθ¯+ ψ¯j (tA )u+ j

k

j

j

− + iθ+ θ¯+ [F (tA ) − 2v˙ (jk) (tA )u+ j uk ] , 18 The

(7.105)

variables q +a satisfy the additional pseudoreality constraint +a = −Ω q +b ≡ −q + , qg ab a

(7.101) Ω2

where Ωab is the symplectic matrix (satisfying Ωab = −Ωba , = −1) and e is the pseudoconjugation. This constraint is equivalent to the constraint (3.15): q ja = εjk Ωab q kb .

(7.102)

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where F is a real function of tA and vjk = vkj is a set of functions satisfying the pseudoreality constraint like in (7.57). As we have seen, such a set can be described by three real independent functions. We see that the physical component content of the superfield (7.105) is the same as in (7.66). In other words, V ++ gives a harmonic description of the (3, 4, 1) multiplet. In the harmonic approach, there are two ways to write down invariant actions. Either we integrate a generic superfield Φ over the full superspace, Z Z S = dt d2 θ+ d2 θ− du Φ = dt dθ¯+ dθ+ dθ¯− dθ− du Φ , (7.106) or else we can integrate a G-analytic superfield Φ++ of double harmonic G charge over the analytic superspace,19 Z S = i dtA dθ¯+ dθ+ du Φ++ (7.107) G . Remark. If we want the action to be real, we have to require Φ to be real and Φ++ G to be pseudoreal—invariant under the pseudoconjugation [this immediately follows from (7.78) and (7.79)]. The simplest nontrivial invariant action involving V ++ reads Z S = − dt d2 θ+ d2 θ− du V ++ V −− , (7.108) where V −− = (D−− )2 V ++ . Going down to components, we reproduce up to irrelevant coefficients the free Lagrangian (7.68). We go over now to N = 8 harmonic superspace. An ordinary N = 8 superspace includes four complex odd coordinates, which we denote θjα , with both j and α taking two values. Both indices are lifted under complex conjugation: θjα → θ¯jα . To harmonize it, we consider the projections ¯±α = θ¯jα u± . We then introduce the analytic time tA given θα± = θαj u± j and θ j by the same formula (7.92) as for N = 4, but with an extra summation over α. The N = 8 analytic superspace includes the variables tA , θα+ , θ¯+α and u. Consider now a G-analytic superfield Q+ (ζ, u) = F + (tA , u) + θα+ χα (tA , u) + . . . + (θα+ θ¯+α )2 P −3 (tA , u). (7.109) Take 2n such superfields Q+a and impose on them the pseudoreality constraint (7.101)—the only constraint that one has to impose in this case. 19 The

symbols dθ¯+ , dθ+ carry harmonic charge −1, cf. Eq. (7.83).

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Consider then the superfield action (see Chap. 5 of the book [78]) 20   Z 1 2 ¯+ 2 + 1 + ++ +a +4 +b S = dtA du d θ d θ Q D Q + L (Q , u) (7.110) 4 2 a with ∂ D++ = ∂ ++ + 2iθα+ θ¯+α . ∂tA

(7.111)

Here L+4 is an arbitrary real function of Q+a and u± that carries harmonic charge +4. The superfields Q+a (ζ, u) involve in their harmonic expansion an infinity of Grassmann-even and Grassmann-odd variables. How to deal with them and what dynamics does the action (7.110) describe? It turns out that almost all these variables have a non-dynamical auxiliary nature. They are algebraically determined by the equations of motion following from (7.110). The latter can be compactly written in the superfield form: D++ Q+a = Ωab

∂L+4 ∂Q+b

(7.112)

with Ωab = −Ωab . By going down to components and resolving a set of equations following from (7.112),21 we are left with 4n real bosonic and 8n real fermionic dynamical variables. Plugging in (7.110) the solutions for the auxiliary variables, we derive a dynamical SQM model describing the motion along a 4n-dimensional hyper-K¨ahler manifold. We will discuss in detail the hyper-K¨ahler and HKT models in Chap. 12.

20 The factor 1/4 in the action comes from our convention (7.52) for the Grassmann measure. The convention in [78] was d2 θ+ = 21 dθ1+ dθ2+ , d2 θ¯+ = 12 dθ¯+2 dθ¯+1 . 21 These equations are rather intricate and can be analytically resolved only in some special cases.

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PART 3

SYNTHESIS

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b2530   International Strategic Relations and China’s National Security: World at the Crossroads

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Chapter 8

Supersymmetric Description of the de Rham Complex

8.1

Basic Structures

In Chaps. 5–7 we presented the simplest SQM models. Now we begin the discussion of more complicated models describing the motion along curved manifolds. By some historical reason, physicists call them the supersymmetric sigma models. The study of these models will allow us to reproduce and elaborate the geometric constructions of the first part. Definition 8.1. An ordinary mechanical1 sigma model is a dynamical system including D dynamical variables xM (t) with the Lagrangian L =

1 gM N (x) x˙ M x˙ N . 2

(8.2)

Here xM have the meaning of the coordinates of a D-dimensional manifold and gM N (x) is its metric tensor. The equations of motion derived from (8.2) coincide with (4.1) and describe the motion along the geodesic lines. We will consider in this and the subsequent chapters many different supersymmetric sigma models including, besides xM (t), Grassmann dynamical variables. 1 Physicists discuss also sigma models involving not a finite number of dynamical variables xM (t), but the fields xM (t, ξ), where the components of ξ are the flat spatial coordinates that have nothing to do with xM . Such models are characterized by the actions like Z 1 S = dtdξ gM N (x) ∂µ xM ∂ µ xN (8.1) 2

with ∂µ = (∂t , ∂ξ ). We had to mention that to give the reader a proper perspective, but we will not consider field-theory models further. 161

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For example, one can take a set of D N = 1 superfields (7.8) and generalize (7.18) by writing Z i dθ gM N (X ) X˙ M DX N . (8.3) L = 2 In components, this gives L =

1 gM N (x)(x˙ M x˙ N + iΨM ∇ΨN ) , 2

(8.4)

˙ M + ΓM ∇ΨM = Ψ ˙ P ΨS PSx

(8.5)

where

ΓM PS

and are the Christoffel symbols (1.17). (The operator ∇ is a covariant time derivative: if xM and ΨM are vectors, ∇ΨM is also a vector). By construction, the Lagrangian (8.4) is invariant under the supertransformations (7.10) for every pair (xM , ΨM ). However, as we mentioned before, N = 1 supersymmetry is in fact not a real symmetry—it does not entail dynamical consequencies, like the double degeneracy of the spectrum. We will see later that for complex manifolds the Lagrangian (8.4) enjoys an additional “hidden” supersymmetry [cf. Eq. (7.37)]. But in a general case, the study of the system (8.4) will not help us to elucidate geometric properties of the manifold, and we have to invent something else. Well, if we want to have N = 2 supersymmetry, the best way to proceed is to express the Lagrangian in terms of N = 2 superfields. For a generic real manifold, the only option is to take D real superfields (7.26), with the dynamical bosonic components coinciding with the coordinates xM . To supersymmetrize (8.2), we write the Lagrangian [81] Z 1 ¯ gM N (X)DX ¯ M DX N dθdθ (8.6) L= 2 with N = 2 covariant derivatives (7.23). In components, this gives 
 1 L = gM N (x) x˙ M x˙ N + F M F N + i ψ¯M ∇ψ N − ∇ψ¯M ψ N 2 1 + ΓM,P Q F M ψ P ψ¯Q + (∂P ∂Q gM N ) ψ¯P ψ¯M ψ Q ψ N . 2

(8.7)

The auxiliary variables F M enter the Lagrangian without derivatives and the corresponding equations of motion ∂L/∂F M = 0 are purely algebraic and can be resolved for F M . The solution reads P ¯Q F M = −ΓM P Qψ ψ .

(8.8)

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Supersymmetric Description of the de Rham Complex

Plugging this into (8.7), we derive 
 1 L = gM N x˙ M x˙ N + i ψ¯M ∇ψ N − ∇ψ¯M ψ N 2 1 − RP M QN ψ¯P ψ¯M ψ Q ψ N , 4 where RP M QN is the Riemann tensor.

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(8.9)

Using the N = 2 superfields was the best option, but one also could express the Lagrangian of the model in terms of N = 1 superfields [82]. Consider along with the set of bosonic real N = 1 superfields2 X M = xM +iΘΨM a set of D Grassmann real superfields (7.16): ΦM = φM +ΘF M . Then the Lagrangian Z n o 1 P S L= (X )DX Φ ] , (8.10) dΘ gM N (X ) iX˙ M DX N + ΦM [DΦN + ΓN PS 2 being expressed in components, gives √ the same expression (8.7) after the identification ψ M = (ΨM − iφM )/ 2. Besides the explicit N = 1 supersymmetry mixing the components of X M and ΦM , the Lagrangian (8.10) enjoys the extra “hidden” N = 1 symmetry under the transformations δX M = −iηΦM , δΦM = η X˙ M

(8.11)

with real Grassmann η. These √ transformations correspond to the transformations (7.27) with  = ¯ = η/ 2. Note the rule of thumb: the farther in the θ expansion is placed a superfield component—the less is the degree of the time derivative with which it enters the component Lagrangian. The lowest component of X M ˙ and gives the term ∝ x˙ 2 , the middle component gives the structure ∝ ψ, the auxiliary field F enters without derivatives. This is quite natural. For the highest component of X M , it is only the terms ∝ ∂/∂θ and ∂/∂ θ¯ in D ¯ ¯ that work, while for the lower components, the terms ∝ θ∂/∂t and D and ∝ θ∂/∂t in the covariant derivatives contribute—for lower components we need to increase the number of the factors θ, rather than diminish them. This brings about time derivatives in the component Lagrangian. It is also possible to write the structures with extra derivatives like Z ¯ κM N (X)DX ¯ M d DX N dθdθ dt in the superfield Lagrangian, then the component F would enter with a time derivative. But in that case, the lowest component x would enter with 2 We

write here capital Θ in order not to confuse the real Θ with the complex θ in (8.6).

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still higher derivatives ∝ x¨ ˙ x. We will not consider the systems with higher derivatives in our book. Our next task is to construct the supercharges in this model. A general method to derive conserved supercharges in any Lagrangian system is provided by Noether’s theorem. It follows from (7.27) and (8.8) that the Lagrangian (8.9) is invariant under the following nonlinear supersymmetry transformations δxM = ψ M + ψ¯M ¯,
P ¯Q δψ M = −¯  ix˙ M + ΓM , P Qψ ψ


P ¯Q δ ψ¯M =  ix˙ M − ΓM .(8.12) P Qψ ψ

From this and using the general Theorem 4.6, we derive     i 1 i 1 Q = √ ψ M ΠM + ΓM,N P ψ N ψ¯P = √ ψ M ΠM − ∂M gN P ψ N ψ¯P , 2 2 2 2     ¯ = √1 ψ¯M ΠM − i ΓM,N P ψ N ψ¯P = √1 ψ¯M ΠM + i ∂M gN P ψ N ψ¯P , Q 2 2 2 2 (8.13) where i (∂Q gP M − ∂P gQM ) ψ P ψ¯Q 2

(8.14)

gM N 1 PM PN + RP M QN ψ¯P ψ¯M ψ Q ψ N , 2 4

(8.15)

ΠM = gM N x˙ N +

√ is the canonical momentum of xM , ΠM = ∂L/∂ x˙ M , and the factor 1/ 2 is introduced for convenience. The classical Hamiltonian is obtained by the Legendre transformation of the Lagrangian (8.9). It reads H = where i PM = ΠM + (∂P gQM − ∂Q gP M )ψ P ψ¯Q . 2 For some purposes (in particular, for quantization, which is our next task), it is more convenient to express the classical supercharges and the Hamiltonian in terms of the tangent space fermion variables ψA = eAM ψ M . Then the supercharges acquire the following form [83]: 1 ¯ Q = √ ψC eM C (PM − iωAB,M ψA ψB ) , 2 ¯ ¯ = √1 ψ¯C eM Q C (PM − iωAB,M ψA ψB ) , 2

(8.16)

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where ωAB,M is the spin connection. When comparing (8.13) and (8.16), it is necessary to have in mind that the canonical momenta ΠM and PM do not coincide. ΠM is given by the variation of the Lagrangian with respect to x˙ M when ψ M are kept fixed, while PM is this variation taken with fixed ψA . The difference stems from the terms including the time derivatives of the fermion variables [84, 85] : ∂ ψ¯˙ N ∂L ∂ ψ˙ N ∂L + PM = Π M + ∂ x˙ M ∂ ψ˙ N ∂ x˙ M ∂ ψ¯˙ N i N ¯ = ΠM + [(∂M eN (8.17) B ) eAN − (∂M eA ) eBN ]ψA ψB . 2 M ¯ The only nonzero Poisson brackets in the basis (x , PM ; ψA , ψA ) are   M PN , xM P = δN , ψA , ψ¯B P = iδAB , (8.18) whereas in the basis (xM , ΠM ; ψ M , ψ¯M ), the Poisson brackets are more complicated:  M ΠN , xM P = δN , {ψ M , ψ¯N }P = ig M N , 1 1 {ΠM , ψ N }P = ψQ ∂M g N Q , {ΠM , ψ¯N }P = ψ¯Q ∂M g N Q . (8.19) 2 2 Thus, the basis (xM , PM ; ψA , ψ¯A ) is orthogonal with respect to the symplectic Poisson bracket structure, in contrast to the basis (xM , ΠM ; ψ M , ψ¯M ). In the former case, the Poisson bracket of arbitrary phase space functions has a canonical form (4.59). ¯ P gives the classical Hamiltonian The Poisson bracket −i{Q, Q} 1 MN (PM − iωAB,M ψA ψ¯B )(PN − iωCD,N ψC ψ¯D ) H = g 2 1 + RABCD ψ¯A ψ¯B ψC ψD , (8.20) 4 which coincides with (8.15). To quantize this system, we take the classical supercharges (8.16) and apply the recipe suggested in Ref. [57] and discussed earlier on p. 112. Namely, we order the classical supercharges using the symmetric (or antisymmetric for the products of Grassmann-odd variables) Weyl prescription and then replace PM → PˆM = −i∂/∂xM and ψ¯A → ψˆ¯A = ∂/∂ψA . In particular, we have to replace i 1 Mˆ M ˆ M ˆ M eM C PM → (eC PM + PM eC ) = eC PM − (∂M eC ) , 2  2  1 ∂ ∂ ∂ ∂ ¯ 2ψC ψA +2 ψC ψA + ψA ψC − ψC ψA ψC ψA ψB → 6 ∂ψB ∂ψB ∂ψB ∂ψB ∂ 1 = ψC ψA + (δBC ψA − δBA ψC ) . (8.21) ∂ψB 2

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After some calculation using the expression (1.53) for the spin connec1 N N tion (with GN M P → ΓM P ) and the identity ΓN M = 2 ∂M ln g, where g = det(gM N ), we obtain   ∂ ∂ 1 ˆ = − √i ψ M + ω ψ Q − ∂ ln g , AB,M A M ∂xM ∂ψB 4 2   ∂ ∂ 1 ˆ ¯ = − √i eM ∂ + ω ψ Q − ∂ ln g . (8.22) AB,M A M C ∂ψC ∂xM ∂ψB 4 2 These supercharges are Hermitially conjugate to each other in the Hilbert space involving the wave functions Ψ(xM , ψA ) = Ψ0 (xM ) + ΨA (xM )ψA + . . .

(8.23)

with the complex-valued coefficients and endowed with the inner product with the flat measure [see Eq. (6.24)] Y Y ¯ dµflat ∼ dxM dψA dψ¯A e−ψA ψA . (8.24) M

A

If we want to make a contact with geometry, we have to derive the expressions for the covariant operators acting in the Hilbert space with the √ measure involving the extra factor g. This is achieved by a similarity transformation ˆ ˆ¯ flat )g 1/4 . ˆ cov , Q ¯ cov ) = g −1/4 ( Q ˆ flat , Q (Q (8.25) This transformation kills the last terms in (8.22) and we finally derive   i M ∂ ∂ cov ˆ Q = −√ ψ + ωAB,M ψA , ∂xM ∂ψB 2   ∂ ∂ i M ∂ cov ˆ ¯ + ωAB,M ψA , (8.26) Q = − √ eC ∂ψC ∂xM ∂ψB 2 where the operator ∂ ∂ DM = + ωAB,M ψA (8.27) M ∂x ∂ψB acts on the coefficients in the expansion (8.23) as the spinor part in the covariant derivative (1.50). ˆ¯ cov } ˆ cov , Q The operators (8.26) are nilpotent. The anticommutator {Q 3 gives the covariant quantum Hamiltonian, which has the form 1 1 √ ˆ cov = − √ DM g g M N DN − RABCD ψˆ¯A ψB ψˆ¯C ψD . (8.28) H 2 g 2 3 The different coefficients and signs before the last term in (8.20) and (8.28) are not a mistake. In the classical limit, (8.28) is reduced to (8.20), bearing in mind the cyclic property of the Riemann tensor [the second line in (1.21)].

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In the first term in (8.28) with suppressed Grassmann variables, the reader will recognize the Laplace-Beltrami operator (1.33). Note that the quantum Hamiltonian is not obtained by the procedure outlined above—Weyl ordering supplemented by the similarity transformation. If you want to preserve a symmetry (in our case, supersymmetry) at the quantum level, it is the symmetry generators (the quantum supercharges in our case) that should be derived this way. We are ready now to prove the key theorem ˆ¯ cov on the wave ˆ cov and Q Theorem 8.1. The action of the supercharges Q M functions Ψ(x , ψA ) is isomorphic to the action of the exterior derivative operator d and its conjugate d† in the de Rham complex. Proof. Consider a wave function of fermion charge p (i.e. a function involving p Grassmann factors). Represent it as Ψp = AM1 ...Mp (x) ψ M1 · · · ψ Mp , (8.29) M M where ψ = eA ψA and AM1 ...Mp (x) is totally antisymmetric. There are ˆ cov . The action of the derivative term is two terms in Q i i M ∂ Ψp = − √ (∂M AM1 ...Mp ) ψ M ψ M1 · · · ψ Mp −√ ψ M ∂x 2 2 ip M N M2 1 − √ AM1 ...Mp (∂M eM · · · ψ Mp . (8.30) A )eAN ψ ψ ψ 2 The action of the term ∝ ωAB,M gives ip M N M2 1 · · · ψ Mp . − √ ωAB,M eAN eM B AM1 ...Mp ψ ψ ψ 2 Adding this to the second term in (8.30), we obtain an expression involving the factor M1 M N 1 eAN (∂M eM A + ωAB,M eB )ψ ψ . Using (1.50) and (1.52), we represent it as P M N M N 1 1 −eAN ΓM = −ΓM = 0. M P eA ψ ψ MN ψ ψ We are left with the result ˆ cov Ψp = − √i (∂M AM ...M ) ψ M ψ M1 · · · ψ Mp , Q (8.31) 1 p 2 √ which has the same form, up to the factor −i/ 2, as Eq. (1.28) defining the action of d. One has only to map ψ M ↔ dxM . The operator d† is adjoint to d with the measure (1.27) including the √ ˆ¯ cov is adjoint to Q ˆ cov with a similar covariant factor g. And the operator Q ˆ¯ cov must be isomorphic to ˆ cov is isomorphic to d, Q measure. Thus, if Q d† .

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Euler Characteristic

We have succeeded in translating the content of Chap. 1.2 into supersymmetric language by mapping the de Rham complex of differential forms to the Hilbert space of a certain SQM system. A legitimate question is: what have we gained by that? Was the traditional language of differential forms not good enough for working with this system? There are two answers to this question. Firstly, one can also derive in this way certain new mathematical results; we will display them later. And secondly, many known mathematical results can be derived in a simpler way when using supersymmetry. As an example, we will prove here the important theorem. Theorem 8.2. [Hodge] Any differential form α can be uniquely presented as a sum of an exact, coexact and a harmonic form, α = dβ + d† γ + h .

(8.32)

Proof. This immediately follows from the general properties of the spectrum of a SQM system established in Chap. 5: it consists of degenerate ˆ¯ ≡ Q ˆ and Q ˆ † and of some unpaired doublets interrelated by the action of Q zero-energy states, as in Fig. 5.2. Exact forms map to the right tower of states in this figure, coexact forms map to the left tower and harmonic forms to the zero-energy states at the bottom. A related classical mathematical notion that shines differently, when looking through the supersymmetric glasses, is the de Rham cohomology. We already mentioned it on p. 22. It is the space of all closed forms, i.e. the forms satisfying the condition dαp = 0, factorized over the space of exact forms that can be represented as αp = dβp−1 . For a given degree p, the p coset closed/exact represents a vector space HdR (M). p Definition 8.2. The Betti number bp is the dimension of HdR .

Definition 8.3. The Euler characteristic of the manifold is the alternating sum X χ(M) = (−1)p bp (M) . (8.33) p

I give here two elementary examples. (1) For S 2 , b0 = 1 (a closed but not exact 0-form is a real constant C) 0 and HdR (S 2 ) = R. Similarly, b2 = 1 : a closed but not exact 2-form is

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a real constant multiplied by dx1 ∧ dx2 . There are no closed but not exact 1-forms and b1 = 0. The Euler characteristic is χ(S 2 ) = 1 − 0 + 1 = 2 . 2

(8.34)

4

(2) For T , b0 = b2 = 1, as for the sphere, but there exist now closed 2 but not exact 1-forms,5 α1 = Adx1 + Bdx2 . Then HdR (T 2 ) = R2 and b1 = 2. The Euler characteristic is χ(T 2 ) = 1 − 2 + 1 = 0 .

(8.35)

Remark. It is a proper place to say that, when we are discussing forms, the de Rham and other complexes, we always have in mind an important restriction. We always assume that our space of forms is endowed by a well-defined and finite inner product (1.27). Mathematicians also do that, but for us it is especially necessary. If the inner product is not defined, we cannot map the complex to the Hilbert space of an SQM system, and the whole ideology of this book fails. The evaluation of the Euler characteristic above also implies the existence of the inner product. If this restriction is lifted, the evaluation is not correct. Indeed, consider the form dφ on S 2 . It seems to have the same properties as the form dx1 on T 2 ,R being closed but not exact. But the norm (1.27) for this form diverges ∼ dθ/ sin θ, and it is not admissible for this reason. (On the other hand, the form dθ is closed, has a finite norm, but it is also exact (θ is a legitimate 0-form on S 2 ) and hence does not contribute in b1 .) The Betti numbers and hence the Euler characteristic are topological invariants. We will show now how the latter can be calculated in the SQM framework. The key observation is Theorem 8.3. The Euler characteristic of a smooth manifold coincides with the Witten index of the supersymmetric sigma model (8.3). Proof. A look at Fig. 5.2 makes it evident that any eigenstate Ψ of an ˆ but is not representable as SQM Hamiltonian that is annihilated by Q, 0 ˆ , does not belong to a supersymmetric doublet of states carrying Ψ = QΨ nonzero energy. Thus, bp counts the number of zero-energy states of the Hamiltonian (8.28) having fermion charge p. If p is even, these are bosonic states and, if p is odd, these are fermionic states. Then the sum (8.33) coincides with the definition (6.53) of the Witten index for this system. 4 Actually,

for any manifold M of dimension D, b0 = bD = 1. form dx1 is not exact because x1 is not an admissible 0-form —it is not uniquely defined on the torus. 5 The

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In Chap. 6, we explained how one can calculate the Witten index of an SQM system as a phase-space integral (6.54) to which the functional integral (6.48) is reduced in the small β limit.6 Substituting there the Hamiltonian (8.20) and integrating over the momenta, we obtain the integral representation of the Euler characteristic: χ = 1 (2πβ)D/2

Z

√ dD x g

Z Y D A=1

  β dψA dψ¯A exp − RABCD ψ¯A ψ¯B ψC ψD . (8.36) 4

The first immediate observation is that the fermion integral vanishes when D is odd. And, indeed, the Euler characteristic of any odd-dimensional manifold vanishes, as follows from the definition (8.33) and from the identity bp = bD−p , which stems from the fact that p-forms and (D − p)-forms are interrelated by the duality transformation ?. If D is even and RABCD 6= 0, the fermion integral does not generally vanish. A nonzero contribution stems from the D/2-th term in the expansion of the exponential. The factor β D/2 cancels the factor β −D/2 in front of the integral in (8.36) that came from the momentum integrations. In the simplest 2-dimensional case, we obtain Z 1 √ d2 x g R , (8.37) χ2 = 2π where R = R1212 is the Gaussian curvature. This is none other than the very well known Gauss-Bonnet formula. For an arbitrary D = 2n we derive the generalized Gauss-Bonnet formula, Z n Y 1 D √ χ2n = d x g εA1 B1 ...An Bn εC1 D1 ...Cn Dn RAk Bk Ck Dk . (8.38) (8π)n n! k=1

In four dimensions, this can be presented as Z 1 √ ˜ ABCD , d4 x g RABCD R χ4 = 32π 2

(8.39)

where ˜ ABCD = 1 εABEF εCDGH REF GH (8.40) R 4 is the dual Riemann tensor. The formula (8.38) is the second example of an index theorem—an integral representation for a topological invariant—that we consider in our 6 We already noted in Chap. 6 and we will get convinced in Chap. 14 that this recipe works in many, but not in all cases. But in this case it works.

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book [the first such example was the magnetic flux (6.65)]. More examples will be given in Chaps. 13,14. The generalized Gauss-Bonnet formula was first derived by Chern back in 1942 [86]. To prove it, Chern used traditional mathematical methods. Its supersymmetric proof was given in 1983–1984 in Refs. [60]. As you have seen, this proof is very simple.7 8.3

Deformations



Morse Theory



Quasitorsions

The Lagrangian (8.6) is not the most general supersymmetric Lagrangian ¯ that one can construct with the N = 2 (1, 2, 1) superfields X M (t, θ, θ). By the same token as in (7.39), one can add the superpotential term Z Lpot = −

¯ W (X M ) . dθdθ

(8.41)

Adding this to (8.6) and performing the same standard manipulations as for the undeformed Lagrangian (8.6), which we will not describe in detail anymore, we arrive at the following form of the quantum supercharges: ˆ 0 eW , ˆW = Q ˆ 0 − √i ∂M W ψ M = e−W Q Q 2 ∂ ˆ ˆ ˆ¯ e−W , ¯W = Q ¯ 0 + √i eM Q = eW Q (8.42) 0 A (∂M W ) ∂ψA 2 ˆ ˆ and Q ¯ are the undeformed covariant quantum supercharges where Q 0

0

(8.26). In the language of the differential forms, this is translated as replacing the operators d and d† by dW and d†W defined as dW α = dα + dW ∧ α , d†W α = d† α − hdW, αi ,

(8.43)

where hdW, αi stands for the interior product, hdW, αi = p(∂M1 W )αMM12 ...Mp dxM2 ∧ · · · ∧ dxMp .

(8.44)

The supercharges (8.42) are nilpotent, as the supercharges (8.26) are. The ˆ ˆW , Q ¯ W } gives the quantum Hamiltonian: anticommutator {Q ˆW = H ˆ 0 + 1 ∂M W ∂ M W + 1 ∂M ∂N W (ψ M ψˆ¯N − ψˆ¯N ψ M ) , (8.45) H 2 2 7 Of course, to appreciate its simplicity, one had to first learn the basics of supersymmetric dynamics outlined in Chaps. 5–7. Any new method requires some time and effort to learn. But having done this, a scholar is rewarded by the rich harvest of interesting and nontrivial results.

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ˆ 0 was written in (8.28). where H Similarly, the Poisson bracket of the classical supercharges gives the classical Hamiltonian: 1 (8.46) HW = H0 + ∂M W ∂ M W + ψ M ψ¯N ∂M ∂N W , 2 What is the Witten index of the new system? The point is that it is exactly the same as of the old one! Indeed, a smooth deformation cannot change the value of the index. It is possible that a zero-energy eigenstate of ˆ 0 acquires a nonzero energy after deformation or other the Hamiltonian H way round. However, it cannot do so alone, but only together with its superpartner—all the states with nonzero energies are grouped into supersymmetric doublets. And this does not change the difference (6.53) of the bosonic and fermionic zero-energy states. ˆ W substituted for The index is still given by the integral (6.54) with H ˆ H0 . And this integral, in contrast to the integral (8.38), can be evaluated in a simple algebraic way. As is seen from Eq. (8.46), the Hamiltonian acquires the potential term ∝ (∂M W )2 after deformation. This potential turns to zero at the critical points of the function W . For a compact manifold and for a smooth function W , such critical points are always there.8 With our smooth deformation, we can scale up W so that the integrand in (6.54) would be exponentially suppressed everywhere except the neighbourhood of the critical points. We can thus evaluate the contribution of each critical point separately and then add them up. Substituting (8.46) in (6.54) and integrating over the momenta, we obtain Z Y 1 D √ d x g dψA dψ¯A IW = (2πβ)D/2 A   1 exp −β ∂M W ∂N W g M N + ψ M ψ¯N ∂M ∂N W 2  1 + RABCD ψ¯A ψ¯B ψC ψD . (8.47) 4 At the vicinity of each critical point, one can expand (P )

W (x) = W0

+

1 (P ) M N b x x + ... 2 MN

(8.48)

(P )

If bM N are chosen to be large enough, one can neglect in this expansion the higher-order terms and neglect also the term ∝ RABCD in the exponential. 8 To

give a Greek-style proof of this assertion, we invite the reader to look at Fig. 8.1.

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z

173

maxima

saddle points

minima 2

2

a) S

b) T

Fig. 8.1 Critical points of the Morse function z(M). The Morse indices are fP = 0 at the minima, fP = 2 at the maxima, and fP = 1 at the saddle points.

The fermion integral can be done using (4.50), and we can write    D/2 Z β (P ) (P ) Q R M N β (P ) D √ M N (P ) d x g det(eA eB bM N ) exp − bM Q bN R x x g IW = 2π 2 =

det(b(P ) ) = (−1)fP , | det(b(P ) )|

(8.49) (P )

where fP is the number of negative eigenvalues in the matrix bM N . We have derived the formula representing the Euler characteristic of a manifold as W a sum over the critical points of a generic smooth function9 M → R: X χ(M) = (−1)fP . (8.50) P

This result is illustrated in Fig. 8.1. We embed S 2 and T 2 in R3 . Let W be the coordinate z of a manifold point. For the sphere, W has two critical points—a maximum and a minimum. The contribution of each point in the sum (8.50) is +1 and we derive χ(S 2 ) = 2. For the torus, W has 4 critical points: a maximum, a minimum and two saddle points. The contribution of each extremum is +1 and the contribution of each saddle point is −1. We derive that χ(T 2 ) = 0 is in agreement with (8.35). 9 Mathematicians call it the Morse function and the integer f is called the Morse index P of the critical point P .

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The Morse index fP also has a physical interpretation. Choose locally flat coordinates that diagonalize the matrix b in the vicinity of the critical point P and are rescaled so that gM N (P ) = δM N . The dominant contribution to the quantum Hamiltonian is i Xh 2 2 ˆ (P ) = 1 PˆM + ωM (xM )2 + ωM (ψ M ψˆ¯M − ψˆ¯M ψ M ) , (8.51) H 2 M

where ωM are the eigenvalues of bM N . This is the sum of D Hamiltonians (5.29) describing supersymmetric oscillators. We have seen that the ground state of each term in the sum (5.29) has zero energy and is bosonic if ωM > 0 and fermionic if ωM < 0. If the matrix b has fP negative eigenvalues, the ground state involves fP Grassmann factors and has fermion charge fP . We have constructed, however, only approximate ground states for the original Hamiltonian (8.45). The properties of the true ground states may not be the same. For example, for S 2 , two supersymmetric bosonic states of (8.51) go over to two supersymmetric ground states of the full Hamiltonian because IW = 2 in this case. On the other hand, IW = 0 for the torus and the exact Hamiltonian has no supersymmetric ground states. Four ˆ (P ) are shifted from zero if one takes into zero-energy ground states of H account the neglected terms in the Hamiltonian. Instead of four unpaired zero-energy ground states, one obtains two supersymmetric doublets with small (if the torus is large), but nonzero energies. The supersymmetric derivation of (8.50) and of other results of Morse theory is due to Witten [1]. Introducing the potential term (8.41) is not the only possible supersymmetric deformation of (8.6). We can also add the term [87] Z 1 ¯ BM N (X P ) DX M DX N + complex conjugate (8.52) dθdθ 2 with an antisymmetric BM N (X P ). Substituting the component expansion ¯ of X M and integrating over dθdθ, we derive [84, 87, 88] 1 LB = ∂M AN P Q ψ N ψ P ψ Q ψ¯M + ∂M A¯N P Q ψ¯N ψ¯P ψ¯Q ψ M 2  − 3ix˙ M (AM N P ψ N ψ P + A¯M N P ψ¯N ψ¯P ) + F M DM , (8.53) LB =

where AM N P =

1 (∂M BN P + ∂N BP M + ∂P BM N ) 3

(8.54)

and DM = 3(AM N P ψ N ψ P − A¯M N P ψ¯N ψ¯P ) .

(8.55)

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Adding this to (8.7) and excluding F M , we obtain the following component Lagrangian: L =


 1 1 gM N x˙ M x˙ N + i ψ¯M ∇ψ N − ∇ψ¯M ψ N − RP M QN ψ¯P ψ¯M ψ Q ψ N 2  4 1 1 + −3ix˙ M (AM N P ψ N ψ P + A¯M N P ψ¯N ψ¯P ) − DM DM 2 4  N P Q ¯M N ¯P ¯Q M ¯ ¯ + (∇M AN P Q ) ψ ψ ψ ψ + (∇M AN P Q ) ψ ψ ψ ψ , (8.56)

where ∇M AN P Q and ∇M A¯N P Q are the Levi-Civita covariant derivatives of the corresponding tensors (do not confuse ∇M with the covariant time derivative ∇ in (8.5)). The contribution (8.53) depends on an antisymmetric rank-3 tensor (8.54), which makes one to think about torsion, but it is not quite a torsion: the covariant derivative ∇M keeps its torsionless form. In addition, ordinary torsions are all real, whereas AM N P can be complex. We will call the latter quasitorsions. We see that, besides the terms ∝ ψ ψ¯ and ¯ 2 , the Lagrangian (and the corresponding Hamiltonian) contain the ∝ (ψ ψ) terms ∝ ψ 3 ψ¯ and ∝ ψ¯3 ψ. And that means that the fermion charge is not conserved anymore. The deformed covariant quantum supercharges can be obtained by our recipe consisting, as we recall, of three steps: (i) find classical supercharges by Noether’s theorem; (ii) write the corresponding quantum operators using the Weyl ordering prescription; (iii) perform the similarity transformation (8.25). The deformed supercharges differ from the undeformed ones by the ¯3 : extra terms ∝ ψ 3 and ∝ ψˆ ˆB = Q ˆ 0 + √i AM N P ψ M ψ N ψ P , Q 2 i ˆ =Q ¯ ¯ˆ 0 + √ A¯M N P ψˆ¯M ψˆ¯N ψˆ¯P . Q B 2

(8.57)

By construction, these supercharges satisfy the N = 2 supersymmetry algebra and hence are nilpotent. A dedicated reader who would like to check it explicity should take into account the action of the derivatives in ˆ ˆ 0 and Q ¯ 0 not only on AM N P , but also on the vielbeins in ψ M = eM ψA . Q A The latter contributions exactly cancel the contributions coming from the ˆ¯ with ˆ 0 and Q anticommutators of the structures ∝ ωψ 2 ψˆ¯ and ∝ ω ψˆ¯2 ψ in Q 0 the structures ∝ Aψ 3 and A¯ψˆ¯3 . The supercharges (8.57) can be represented in such a way that their

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nilpotency becomes obvious: ˆ B = eBM N ψM ψN Q ˆ 0 e−BM N ψM ψN , Q ˆ = eBM N ψˆ¯M ψˆ¯N Q ˆ¯ e−BM N ψˆ¯M ψˆ¯N . ¯ Q B 0

(8.58)

This representation can be translated into the language of differential forms: dB α = dα − dB ∧ α , ¯ αi , d†B α = d† α − hdB, M

(8.59)

N

where B = BM N dx ∧ dx . One can also make a more complicated deformation by adding to the Lagrangian the term Z 1 ¯ EM N P Q DX M DX N DX P DX Q + c.c. dθdθ (8.60) LE = 2 This gives the deformation like (8.59) with the 2-form B replaced by the 4form E. One can also write a term with 6 covariant derivatives, etc. But for a manifold of given dimension D, the number of different such deformations is finite. For 2-forms B, the deformation (8.59) is non-trivial when D ≥ 3, for 4-forms E, the deformation (8.59) is non-trivial when D ≥ 5, etc. The Witten index of all the deformed models described above coincides with the Witten index of the undeformed model and is equal to the Euler characteristic of the manifold. In this chapter, we only discussed the de Rham complex and its deformations that are known to mathematicians and have been studied by them previously. But one can also consider many other deformations that have not attracted the attention of mathematicians yet. These deformations and, in particular, the so-called quasicomplex deformation will be discussed in Chaps. 10,11.

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Chapter 9

Supersymmetric Description of the Dolbeault Complex

As we know from Chap. 2, there are two ways to describe complex manifolds. One way is to use complex coordinates. Another way is to use real coordinates, but take into account the presence of the complex structure tensor satisfying the integrability conditions (Theorem 2.6). Similarly, there are two possible supersymmetric descriptions. 9.1

N = 2 Superfield Description

¯ m = 0) and Consider a set of d holomorphic chiral (2, 2, 0) fields Z m (DZ m ¯ ¯ their antiholomorphic chiral conjugates Z . We generalize the superfield action (7.32) by writing Z S =

¯ dθdθdt



1 ¯ D ¯ Z¯ n¯ DZ m + W (Z, Z) ¯ hm¯n (Z, Z) 4

 (9.1)

with Hermitian hm¯n and real W . We will see soon that the term ∝ W describes a gauge field, like it does in the Landau system (7.32). But consider first the systems with W = 0. The component Lagrangian takes the form   i  m ¯˙ n¯ m˙n ¯ m ¯n ¯ ˙ ψ ψ −ψ ψ L = hm¯n z˙ z¯ + 2 i ih ¯ (2∂m hk¯n − ∂k hm¯n ) z˙ k − (2∂n¯ hmk¯ − ∂k¯ hm¯n ) z¯˙ k ψ m ψ¯ n¯ − 2 + (∂k ∂q¯hm¯n ) ψ k ψ m ψ¯ q¯ψ¯ n¯ . (9.2) We see that hm¯n is the complex Hermitian metric. By going to the quasireal notation (2.14) [with also PM = (pm , pm ¯ ) and ¯ ψ M = (ψ m , ψ¯m )], the Lagrangian (9.2) can be brought to the following nice 177

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form   1 1 M N M N P K L M gM N x˙ x˙ + igM N ψ ∇ψ − ∂P CKLM ψ ψ ψ ψ . (9.3) L= 2 6 The totally antisymmetric tensor CKLM has the meaning of torsion. It has the following non-vanishing components: Cklm ¯ = (∂l hkm ¯ − ∂k hlm ¯) ,

Ck¯¯lm = Cklm ¯ − ∂k ¯ hm¯ ¯ = (∂¯ l hmk l ) . (9.4)

This tensor was already written1 in Chap. 2 [Eq. (2.18)]. It was mentioned there that the torsion (9.4) enters the Bismut affine connection (B)

GM,N P = ΓM,N P +

1 CM N P . 2

(9.5)

The covariant derivative ∇ψ M = ψ˙ M + GM ˙ N ψP NP x entering (9.3) involves this connection. Note that for the K¨ahler manifolds the torsion (9.4) vanishes. The classical supercharges can be calculated using Noether’s theorem in a standard way, similar to what we did in the preceding chapter for real supersymmetric sigma models. They read     i i m n ¯p¯ m n ¯p¯ Q = ψ πm + Γm,np¯ ψ ψ = ψ πm − ∂m hnp¯ ψ ψ , 2 2     i i n ¯p¯ n ¯p¯ m ¯ m ¯ ¯ ¯ Q = ψ πm = ψ πm , (9.6) ¯ p¯ ψ ψ ¯ hnp¯ ψ ψ ¯ − Γm,n ¯ + ∂m 2 2 where the canonical momenta πm and πm ¯ are obtained by differentiating ¯ ¯ the Lagrangian over z˙ m , z¯˙ m , while keeping ψ m , ψ¯m fixed. Introducing the tangent space holomorphic and antiholomorphic Grass¯ ¯m ¯ mann variables2 ψ c = ecm ψ m and ψ¯c¯ = ecm and canonical momenta ¯ψ pm and pm ¯ given by the derivatives of the Lagrangian over the canonical velocities with fixed ψ c and ψ¯c¯, we derive i h c a ¯¯ b Q = em ψ p − iω ψ ψ , ¯ m ab,m c h i ¯ ¯c a ¯¯ b ¯ = em Q pm , (9.7) ¯ − iωa¯ b,m ¯ ψ ψ c¯ ψ 1 It is like a gun which an experienced playwright hangs on the wall in the study of the main character in the first act and which fires in the middle of the play. There are many such guns in our book. 2 We remind that, in contrast to the real case where we did not have to care about the position of the flat tangent space indices, we should do so for complex geometry—see the footnote on p. 33.

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where ωa¯b,m , ωa¯b,m ¯ are the spin connections corresponding to the standard Levi-Civita affine connections ΓM N P . The property ωa¯b,m ¯b,m = −ωb¯ a,m ¯ = ωa holds. Note, however, that the spin connections ωab,m ¯ and ωa ¯¯ b,m , which do not vanish for a generic complex manifold (see the discussion on p. 34) do not enter the expressions (9.7) for the supercharges. The canonical classical Hamiltonian can be represented in the following compact form: ¯ H = hn¯ m Pm P¯n¯ − (∂k ∂¯l hm¯n ) ψ k ψ m ψ¯l ψ¯n¯ ,

(9.8)

where (B)

¯

Pm = pm − iωa¯b,m ψ a ψ b ,

¯ (B) P¯n¯ = p¯n¯ − iωa¯b,¯n ψ a ψ b .

(9.9)

In contrast to the supercharges (9.7), the Hamiltonian (9.8) involves the Bismut spin connection 1 N K e e CN M K . (9.10) 2 A B Let us now turn to quantum theory. As in the previous chapter, we resolve the ordering ambiguities as prescribed in [57], i.e. we use the symmetric Weyl ordering for the supercharges supplemented by a similarity transformation (8.25). We obtain the following expressions for the covariant quantum supercharges:3   i ¯ aˆ b c ¯ ˆ = em ∂ (ln det e) − i ω ψ ψ , Q ψ p ˆ − ¯ m m ab,m c 2   i ¯ aˆ b ¯ ˆ ˆ ¯c¯ pˆm ¯ ¯ = em ∂ (ln det e) − i ω ψ , (9.11) Q ψ ψ − ¯ ¯ m ¯ ab,m ¯ c¯ 2 (B)

N K ωAB,M = eAN (∂M eN B + GM K eB ) = ωAB,M +

ˆ m ¯ ¯a¯ where pˆm , pˆm ¯m and ∂/∂ψ a . ¯ and ψ are the operators −i∂/∂z , −i∂/∂ z These expressions almost coincide in form with the classical expressions (9.7), but note the presence of the extra terms ∝ ∂M ln det e in the quantum supercharges. To understand their meaning, we turn our attention to the second term in the action (9.1). The component Lagrangian acquires the extra term ¯ m m ¯n ¯ ∆LW = i(z¯˙ m ∂m ¯ W − z˙ ∂m W ) + 2 ∂m ∂n ¯W ψ ψ .

(9.12)

3 To avoid unnecessary complications, we assumed that the complex vielbeins ea are m √ chosen so that their determinant is real: det e = det e¯ = det h.

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This brings about the extra terms in the supercharges: ¯ ˆ¯m ˆ¯ ˆ W = iψ m ∂m W , ∆Q ∆Q W = −iψ ∂m ¯W .

(9.13)

The shift (9.13) amounts to a similarity transformation, ˆ W = eW Qe ˆ −W . Q

(9.14)

This resembles the supercharge (8.42) of the previous chapter that described the de Rham complex deformed by the extra scalar potential,4 but the dynamical meaning of W is now different. Adding (9.13) to (9.11), we see that the momenta pˆm and pˆm ˆm + i∂m W ¯ are substituted by the structures p and pˆm ¯ − i∂m ¯ W . These structures also enter the Hamiltonian (9.17). This tells us that the vector AM = (i∂m W, −i∂m ¯ W)

(9.15)

has the meaning of the gauge vector potential [cf. the expression (5.14) for the Pauli Hamiltonian where one should replace e/c → −1]. The corresponding field strength tensor FM N = ∂M AN − ∂N AM has the components Fn¯ m = −Fm¯n = 2i∂m ∂n¯ W ,

Fmn = Fm¯ ¯n = 0.

(9.16)

The Hamiltonian acquires an extra bifermionic contribution proportional to Fn¯ m . The classical Hamiltonian of a generic model (9.1) (we will need it later) reads
HW = hn¯ m (Pm + i∂m W ) P¯n¯ − i∂n¯ W ¯

− (∂k ∂¯l hm¯n ) ψ k ψ m ψ¯l ψ¯n¯ − 2 ∂m ∂n¯ W ψ m ψ¯n¯ .

(9.17) N eM A eB FM N

Remark. The form (9.16) for FM N means that FAB = is not generic, but belongs to the algebra u(2n) ⊂ so(4n), in the same (B) way as the Bismut spin connection ωAB,M does. And that means that FAB commutes with the complex structure IAB . The inverse is also true: [F, I] = 0 ⇒ the holomorphic and antiholomorphic components of FM N vanish. Both statements follow rather straightforwardly from (9.16) and (2.52). We now note that the contribution (9.13) has a similar form to the extra term ∝ ∂M ln det e in the quantum supercharges. More precisely, by ˆ while choosing W = (1/4) ln det h, one can get rid of the extra term in Q, ˆ¯ choosing W = −(1/4) ln det h, one can get rid of the extra term in Q. Definition 9.1. The Abelian gauge field (9.15) derived from the prepotential 1 ln det h (9.18) W = 2 4 More

examples of similarity transformations will be given and discussed in Chap. 11.1.

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is called by mathematicians the connection of the canonical line bundle. The gauge field derived from W = (1/4) ln det h (which interests us more) is called the connection of the square root of the canonical line bundle. And the field derived from W = −(1/4) ln det h is the connection of the square root of the anticanonical line bundle. I gave these translations for the benefit of our readers who are mathematicians, but in the following we will stick to the physical terminology. We are now ready to prove the central theorem, the complex analog of Theorem 8.1 of the preceding chapter. Theorem 9.1. Consider the supersymmetric system (9.1) with W = (1/4) ln det h. The action of its quantum supercharges   ∂ a ∂ c ˆ W = −iem + ω ψ , Q ψ a¯ b,m c ∂z m ∂ψ b   ∂ 1 a ∂ ¯ ∂ ˆ ¯ W = −iem ∂ (ln det h) + ω ψ Q + , (9.19) m ¯ a¯ b,m ¯ c¯ ¯ ∂ψ c ∂ z¯m 2 ∂ψ b ¯ on the wave functions Ψ(z m , z¯m , ψ a ) is isomorphic to the action of the holomorphic exterior derivative operator ∂ and its conjugate ∂ † in the Dolbeault complex.

Proof. The proof is exactly the same as in the de Rham case. Consider a wave function of fermion charge p presented as Ψp = Am1 ...mp (x) ψ m1 · · · ψ mp

(9.20)

ˆ W Ψp . The action of the derivative term is and consider Q −iψ m

∂ Ψp = −i (∂m Am1 ...mp ) ψ m ψ m1 · · · ψ mp ∂z m a m n m2 1 −ip Am1 ...mp (∂m em · · · ψ mp . a )en ψ ψ ψ

(9.21)

The action of the term ∼ ωa¯b,m gives m n m2 1 −ip ωa¯b,m ean em · · · ψ mp . b Am1 ...mp ψ ψ ψ

Adding this to the second term in (9.21), we obtain an expression involving the factor m1 m n 1 X m1 = ean (∂m em b,m eb )ψ ψ . a + ωa¯

(9.22)

Look now at the “full” covariant derivative (1.50) with the Levi-Civita affine connections ΓM N P and the Levi-Civita spin connections. It vanishes

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according to (1.52). Choose in the L.H.S. of (1.52) all the free indices holomorphic, M → m1 , N → m, A → a. We derive m1 P m1 1 ∂m em = 0. (9.23) a + ΓmP ea + ωaB,m eB p¯ Note now that the index P is necessarily holomorphic, because ea does not 1 exist. Also the second term in (9.23) can be written as5 ωa¯b,m em b . Bearing all this in mind, we derive m n 1 X m1 = −Γm (9.24) mn ψ ψ = 0 . We are left with the result ˆ W Ψp = −i(∂m Am ...m ) ψ m ψ m1 · · · ψ mp , (9.25) Q 1 p which has the same form, up to the factor −i, as Eq. (2.41) defining the action of the exterior holomorphic derivative ∂ on the holomorphic (p, 0) forms. One has only to map ψ m −→ dz m . The operator ∂ † is adjoint to ∂ with the measure (2.39) including the √ ˆ¯ is adjoint to Q ˆ W with the factor det h = det g. And the operator Q W ˆ¯ must be ˆ same covariant measure. Thus, if QW is isomorphic to ∂, Q W † isomorphic to ∂ . Remark. This proof “explains” the “experimental fact” that the quantum supercharges of this model involve the conventional spin connection ωa¯b,m rather than the Bismut spin connection (9.10). In the latter case, we m1 m1 1 would have the torsionful Gm mn instead of Γmn in Eq. (9.24), the factor X ˆ W would not have a nice geometric interpretation. would not vanish and Q Consider now the gauge field derived from W = −(1/4) ln det h and consider the action of the corresponding quantum supercharges on the dual wave functions Ψ(z m , z¯m ; ψ¯a ) rather than on Ψ(z m , z¯m ; ψ a ) (it is like going from the coordinate to momentum representation in ordinary quantum mechanics). One can prove in exactly the same way as above that the action ˆ ¯ W is isomorphic in this case to the action of the operator ∂¯ on the of Q ˆ W is isomorphic to ∂¯† . forms (2.43) of the anti-Dolbeault complex. And Q For an arbitrary W , we are dealing with the twisted Dolbeault complex. The following fact6 known to mathematicians is a rather straightforward corollary of (9.11) and (9.13). ˆ W on the wave functions Theorem 9.2. The action of a generic Q m m a Ψ(z , z¯ ; ψ ) maps to the action of the nilpotent operator   1 (9.26) ∂W = ∂ − ∂ W − ln det h ∧ 4 5 Recall 6 See

that YA ZA ≡ Ya Za¯ + Ya¯ Za . e.g. Propositions 1.4.23 and 1.4.25 in [89].

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ˆ¯ on the wave functions in the Dolbeault complex. The action of a generic Q W Ψ(z m , z¯m ; ψ¯a ) maps to the action of the operator   1 ∂¯W = ∂¯ + ∂¯ W + ln det h ∧ (9.27) 4 in the anti-Dolbeault complex. ˆ¯ is isomorphic to the Dirac operator on the ˆW + Q Finally, the sum Q W manifold with the gauge field (9.15). But we postpone the discussion of this issue till Chap. 13. 9.1.1

Holomorphic torsions

In contrast to the superpotential term (8.41) for the de Rham supersymmetR ¯ W (Z, Z) ¯ canric sigma model, the gauge field term in (9.1) involving dθdθ ¯ not be considered as a deformation. We will see in Chap. 13 that W (Z, Z) cannot be infinitesimal—certain topological invariants expressed through the integrals involving7 W are equal either to zero (when W = 0) or to nonzero integers. But the model (9.1) can also be deformed by adding the term [84] Z   1 ¯ ¯ ¯ ¯n ¯ ¯ DZ m DZ n − B¯m¯ ¯ ¯ ¯m dθdθdt Bmn (Z, Z) (9.28) SB = ¯ n (Z, Z) D Z D Z 4 with an arbitrary antisymmetric complex Bmn and its conjugate B¯m¯ ¯ n . This deformation is similar to the deformation (8.52) in the de Rham case, but Bmn carries now holomorphic indices. The component Lagrangian acquires now the contribution ∆LB = − 3i ∂[k Bmn] z˙ k ψ m ψ n − ∂p¯∂k Bmn ψ¯p¯ψ k ψ m ψ n + c.c. (9.29) Note that adding this extra term to the Lagrangian (9.3) gives the Lagrangian of the same form where the torsion tensor now includes besides (9.4) pure holomorphic components Ckmn = 12 ∂[k Bmn] and Ck¯m¯ ¯n = 12 ∂[k¯ B¯m¯ (see also Theorem 9.5 below). Thus, the deformation (9.29) ¯ n] brings about extra torsions and not “quasitorsions” as was the case for the deformation (8.52). The corresponding contributions to the supercharges are ¯ ˆ ˆ¯ = i ∂¯ B¯ ψˆ¯k¯ ψˆ¯m ˆ = i ∂ B ψk ψm ψn , ψ¯n¯ . (9.30) ∆Q ∆Q B

k

B

mn

m

k

n

m¯ ¯n

Consider the (2, 0)-form B = Bmn dz ∧ dz and the complex conjugate ¯ It is easy to see that the following is true [84, 90]: (0, 2)-form B. 7 The

simplest such invariant is the magnetic flux (5.25).

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ˆ W,B on the wave functions Theorem 9.3. The action of a generic Q m m a Ψ(z , z¯ ; ψ ) maps to the action of the nilpotent operator ∂W,B = ∂W − ∂B∧

(9.31)

ˆ¯ in the twisted deformed Dolbeault complex.8 The action of a generic Q W,B m m ¯a on the wave functions Ψ(z , z¯ ; ψ ) maps to the action of the operator ¯ ∂¯W,B = ∂¯W − ∂¯B∧

(9.32)

in the anti-Dolbeault complex. The deformations (9.31) and (9.32) are possible when the complex dimension d is 3 or more. When d ≥ 5, one can introduce a more complicated deformation Z ¯ SE ∝ dθdθdt Emnpq DZ m DZ n DZ p DZ q + c.c. (9.33) Starting from d = 7, one can write a contribution with six covariant derivatives, etc. N = 1 Superfield Description

9.2

Take D = 2d N = 1 real superfields X M = xM + iθΨM [see Eq. (7.8)] and write the action in the following form [77]: Z i dθdt gM N (X )X˙ M DX N S= 2 Z 1 dθdt CKLM DX K DX L DX M . (9.34) − 12 with D given in Eq. (7.14). We added to the action corresponding to (8.3) the extra term involving a totally antisymmetric tensor CKLM playing the R role of the torsion. Doing the integral dθ, we reproduce the Lagrangian (9.3). R The N = 1 supersymmetry of the action (9.34) is manifest. The integral dθdt is invariant under the change of variables (7.1) that generate the shift of the superfields δ0 X M = X M (θ + 0 , t + i0 θ) − X M (θ, t) . The components of X δ0 x 8 Here

M

M

(9.35)

transform as in (7.10):

= i0 ΨM ,

δ0 ΨM = −0 x˙ M .

(9.36)

“twisted” means the presence of a gauge field and “deformed” means the presence of an extra (2,0)-form B.

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We now require the action to be invariant under the following extra supersymmetry transformation: δX M = IN M DX N ,

(9.37)

where IN M (X P ) is a tensor to be specified below. The components in X M transform as δxM = iIN M ΨN ,
δΨM =  IN M x˙ N − i ∂S IN M ΨS ΨN .

(9.38)

The invariances (9.36) and (9.38) give us two real conserved supercharges Q0 and Q. It is rather evident that the transformations (9.35) and (9.37) commute. Indeed, Y M = δX M is a superfield, and hence δ0 (δX M ) and δ(δ0 X M ) coincide, having both the form (9.35) with X M replaced by Y M . A corollary ˆ Q ˆ 0 } of the quantum of this is the vanishing of the anticommutator {Q, supercharges. A nontrivial requirement that we impose is that the commutator of two transformations (9.37) boils down to the time translation. Then the square of the supercharge associated with (9.38) coincides with the Hamiltonian, and we obtain the minimal N = 2 supersymmetry algebra ˆ2 = Q ˆ 2 = H, ˆ Q 0

ˆ 0 , Q} ˆ = 0. {Q

(9.39)

This requirement and the requirement of invariance of the action under (9.37) imposes nontrivial constraints on the complex structure IN M and the torsion tensor9 CKLM . These constraints are unravelled by the following theorem: Theorem 9.4. The action (9.34) is invariant under (9.37) and the algebra (9.39) holds iff the following set of conditions is satisfied: (1) I 2 = −1. (2) The Nijenhuis tensor (2.60) vanishes. (3) The matrix IM N = gN K IM K is skew-symmetric. (B) (B) (4) ∇ I M + ∇ I M = 0, L

(B)

where ∇L (9.5).

N

N

L

(9.40)

is the Bismut covariant derivative with the connection

9 We will see soon that I M has all the properties of an integrable complex structure N and should be interpreted as such. And the tensor CKLM is a true torsion entering the relevant Bismut affine connection (9.5).

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∂L C[M SN IR] L + I[N L ∂R CM S]L − 2CL[M S (∂R IN ] L ) = 0.

(9.41)

This theorem was proved first in [26] by explicit component calculations. We give here, following Ref. [91], an alternative proof based on the language of N = 1 superfields. Proof. The conditions 1 and 2 follow from the algebra (5.1). Note first that the law (9.37) implies δ(DX N ) = D(δX N ) = −D(ILN DX L ) = −(∂K IL N )DX K DX L + iIK N X˙ K .

(9.42)

The commutator of two supersymmetry transformations (9.37) is then derived to be (δ2 δ1 − δ1 δ2 ) X M = 2i1 2 (I 2 )K M X˙ K (9.43) h

Mi L M L K + 21 2 IK ∂L IN + ∂N IK IL DX DX N . ˆ 2 = H], ˆ If we want it to coincide with −2i1 2 ∂t X M [as is dictated by Q 2 the condition I = −1, as well as the identity

(9.44) ∂L I[N M IK] L + ∂[N IK] L IL M = 0 follow. Multiplying this by IM S , using I 2 = −1 and flipping the derivative in the first term, the condition (9.44) can be brought to the form N[KN ] S = 0, with N[KN ] S defined in (2.60). The conditions 3 and 4 follow from the vanishing of the variation of the action under (9.37). The calculation gives Z Z i M ˙N ˙ dθdt PM,SN X˙ M DX S DX N δS =  dθdt IM N X X − 4 Z  + dθdt TRM SN DX R DX M DX S DX N , (9.45) 12 where PM SN = 2∇M ISN + 2∇N (IM S + ISM ) − C L SN IM L ,
TRM SN = (∂L CM SN ) IR L − 3CLSN ∂R IM L .

(9.46)

Note that ∇M entering (9.46) are the ordinary Levi-Civita covariant derivatives. Let us first see what happens if the torsion vanishes. In this case, the second line in (9.45) vanishes identically and the first line vanishes iff I(M N ) = 0 (which means together with the conditions 1 and 2, which

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we already derived, that IM N is an integrable complex structure) and ∇M ISN = 0. We arrive at the K¨ ahler geometry. If CM SN does not vanish, the situation is somewhat more complicated. One of the complications is that the second and the third integrals in (9.45) are not completely independent: a certain part of the second integral gives the same structure as the third one after flipping the derivative D. Indeed, look at the second term in the R.H.S. of Eq. (9.45). Only the antisymmetric in S ↔ N part of PM SN contributes. We represent
1 2PM [SN ] + PS[M N ] − PN [M S] , (9.47) PM [SN ] = P[M SN ] + 3 so that the structure in the parentheses gives zero after complete antisymmetrisation over [M SN ]. After trading X˙ M for iD2 X M and integrating by parts, the integral involving P[M SN ] gives Z dθdt P[M SN ] X˙ M DX S DX N = Z i dθdt ∂R P[M SN ] DX R DX M DX S DX N . (9.48) − 3 We thus present the variation as a sum of three independent structures Z δS =  dθdt IM N X˙ M X˙ N Z i − dθdt (2PM [SN ] + PS[M N ] − PN [M S] ) X˙ M DX S DX N 12 Z
 + dθdt TRM SN − ∂[R PM SN ] DX R DX M DX S DX N . (9.49) 12 It vanishes iff I(M N ) = 0 ,

(9.50)

2PM [SN ] + PS[M N ] − PN [M S] = 0 ,

(9.51)

T[RM SN ] = ∂[R PM SN ] .

(9.52)

and

Equation (9.50) gives the requirement 3 in the list above. The condition (9.51), after taking account of (9.50) and after substituting PM [SN ] from Eq. (9.46), yields (B)

(B)

(B)

2∇M ISN + ∇S IM N − ∇N IM S = 0 .

(9.53)

Symmetrizing this over M ↔ S and raising the index N (we are allowed to (B) do so due to ∇M g QN = 0), we arrive at (9.40).

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Finally, substituting into (9.52) the expression for TRM SN from (9.46) and   1 L 2 (9.54) ∂M ISN − C SN IM L + cycle (M, S, N ) P[M SN ] = 3 3 (the terms ∝ ΓP M N cancel), we derive the identity (9.41). The conditions 1–3 above concern only the tensor I. According to Theorem 2.6 (the Newlander-Nirenberg theorem), they are necessary and sufficient for IM N to be an integrable complex structure. Actually, we have just given an alternative proof of the necessity of these conditions using supersymmetric methods.10 Indeed, suppose the complex coordinates can be introduced. Then the complex supercharges (9.11) can be written. Their real and imaginary parts satisfy the algebra (9.39). As we have just proven, the conditions 1–3 including the integrability condition 2 follow. But the supersymmetric description also allows one to understand why these conditions are sufficient to introduce the complex coordinates z n . The heuristic argument is the following. From Theorem 9.4 we have learned that, if I 2 = −1 and the Nijenhuis tensor vanishes, the generators of the transformations (9.36) and (9.38) satisfy the N = 2 supersymmetry algebra (9.39). We have altogether 2d real dynamical bosonic and 2d real dynamical fermionic variables. These variables provide an infinite-dimensional representation for the N = 2 algebra, but this representation must be reducible, as the only known irreducible N = 2 representation including the equal number of bosonic and fermionic dynamical variables is (2, 2, 0). The representation (2d, 2d, 0) must be decomposable into a direct sum of d representations (2, 2, 0) which transform as in (7.27). But this is tantamount to saying that the solution to the conditions (2.51) for z n exists. In Chap. 2 we did not prove the NN theorem rigourously as we did not prove the convergence of the infinite series (2.73). But a rigourous mathematical proof of this theorem exists. That is why it is probably better to look at the problem from the opposite direction: The NN theorem can be interpreted as a statement that a superfield representation of the N = 2 superalgebra including 2d real dynamical bosonic and 2d real dynamical 10 In fact, our reasoning was simply a translation into the supersymmetric language of the traditional proof of this fact given on p. 42. The equations (2.51) may be interpreted as a requirement that the supersymmetric variations of z n depend only on the complex Grassmann parameter ε = 0 − i, but not on ε¯. And the commutator [δ1 , δ2 ] in the L.H.S. of Eq. (9.43) has the same meaning as the commutator [DM , DN ] in the L.H.S. of Eq. (2.62).

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fermionic variables is necessarily reducible and can be decomposed into a direct sum of d (2, 2, 0) representations [9]. Look now at the conditions (9.40) and (9.41). Equation (9.40) reminds (B) us of the condition ∇N IL M = 0, which uniquely defines the Bismut torsion (2.18) for an integrable complex structure IM N . However, in our case where the covariant derivative of I vanishes only after symmetrization over N ↔ L, the torsion is not rigidly fixed. To understand the geometric meaning of the constraints which the torsion tensor still obeys, consider also the condition (9.41). Its meaning can be clarified if one expresses (9.41) in the language of forms. Introduce the operator ι which acts on a generic P -form α according to the rule: if

α = αM1 ...MP dxM1 ∧ · · · ∧ dxMP ,

then ια = P αN [M2 ...MP I N M1 ] dxM1 ∧ · · · ∧ dxMP .

(9.55)

For a form αp,q with p holomorphic and q antiholomorphic indices with respect to the complex structure I, the action of ι amounts to the multiplication by i(p − q). Then (9.41) is equivalent to the condition ιdC =

2 d(ιC) , 3

(9.56)

for the 3-form C. We can now prove the following theorem [26, 91]. Theorem 9.5. Let (g, I, C) satisfy conditions 1–5 of the preceding theorem. Consider the holomorphic decomposition of the torsion form with respect to the complex structure I, C =

3 X

C3−q,q = C3,0 + C2,1 + C1,2 + C0,3 .

(9.57)

q=0

Then (i) the mixed part C2,1 + C1,2 is the Bismut torsion (2.18); (ii) the holomorphic and antiholomorphic parts are closed, ¯ 0,3 = 0 . ∂C3,0 = ∂C

(9.58)

Proof. We express (9.40) in complex coordinates. The complex structure I acquires a simple form (2.52). Choose L = l, N = n ¯ , M = m. ¯ Look at ¯ the first term in (9.40). We have ∂l δnm ¯ = 0 and the two other contributions cancel out. For the second term, they add up, and we obtain the condition ¯ Gm nl = 0 , n ¯ l = 0 =⇒ Gm,¯

(9.59)

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as in (2.21). The components Gm,n nl ¯ ¯ l are the complex conjugates of Gm,¯ and also vanish; this can also be derived directly from (9.40) if one chooses there L = l, N = n ¯ , M = m. From this, from the definition (9.5), and from the definition of the Christoffel symbols, we may derive that the part in the torsion tensor corresponding to C2,1 + C1,2 has the Bismut form (9.4). When all the three indices in (9.40) are holomorphic or all of them are antiholomorphic, the equality is satisfied automatically. The choice L = l, N = n, M = m ¯ does not bring about new information. One can also notice that the forms C2,1 and C1,2 are exact: C2,1 = 3Cml¯n dz m ∧ dz l ∧ d¯ z n¯ = −6∂ω, ¯ where ω = hm¯n dz m ∧ d¯ z n¯ . Similarly, C1,2 = 6∂ω. Consider now the condition (9.41) and choose all the free indices holomorphic: M = m, N = n, S = s, R = r. Bearing in mind (2.52), we immediately derive ∂[r Cmns] = 0, which means that C3,0 is closed.11 Choosing all the indices antiholomorphic, we derive the closedness of C0,3 . Note finally that the reality of C in (9.57) implies that Cm¯ ¯ ns¯ = Cmns .

We can now return to the N = 2 description of the preceding section. If C3,0 = 0, the torsion is Bismut and the system (9.34) is equivalent to the system (9.1) with W = 0. As we learned in the preceding section, the latter maps to the twisted Dolbeault complex with the gauge potential [cf. (9.15) and (9.26)]   i i . (9.60) AM = − ∂m ln det h, ∂m ¯ ln det h 4 4 And the torsion with nonzero C3,0 gives us the deformed twisted Dolbeault complex involving the extra contribution (9.28) in the action; C3,0 = 12∂B. Bearing in mind the established mapping between the two approaches, it is easy to write the expressions for the real supercharges Q0 and Q. They can be found as real and imaginary parts of the complex classical supercharge, taken with the factor 2.12 When there is no deformation and 11 This is also clearly seen in the language of forms. Substituting C 3,0 in (9.56), we obtain ¯ 3,0 = 2i(∂ + ∂)C ¯ 3,0 . i(4∂ + 2∂)C

And this can be fulfilled only if ∂C3,0 = 0. 12 With the chosen conventions, the complex supercharge is expressed as (Q + iQ)/2 0 [cf. Eq. (5.2)].

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W = 0, we derive, starting from (9.7), the following expressions for the classical supercharges [5, 92, 93]:   i i M A B K P Q0 = Ψ PM − ωAB,M Ψ Ψ + CM KP Ψ Ψ , 2 12   i i N A B K P M Q = Ψ IN , (9.61) PM − ωAB,M Ψ Ψ − CM KP Ψ Ψ 2 4 where ωAB,M are the Levi-Civita spin connections and PM is obtained from the variation of the Lagrangian with fixed ΨA . The terms ∝ CM KP cancel the holomorphic and antiholomorphic connections ωab,m ¯ and ωa ¯¯ b,m , which are present in ωAB,M [see (2.25), (2.26)], but absent in (9.7). Bearing in mind the normalization {ΨA , ΨB }P = iδ AB , the supercharges (9.61) obey the classical supersymmetry algebra. The corresponding covariant quantum supercharges that are obtained by the ordering procedure outlined on p. 165 have the form   ˆ AΨ ˆ B + i CM KP Ψ ˆ KΨ ˆP , ˆ M PˆM − i ∂M ln det g − i ωAB,M Ψ Q0 = Ψ 8 2 12  ˆ N IN M PˆM − i ∂M ln det g − i ωAB,M Ψ ˆ AΨ ˆB Q= Ψ 8 2  i ˆ KΨ ˆP , − CM KP Ψ (9.62) 4 ˆ A are the operators obeying the Clifford algebra where PˆM = −i∂M and Ψ ˆ A, Ψ ˆ B } = δAB {Ψ

⇒

ˆM, Ψ ˆ N } = gM N . {Ψ

(9.63)

Naturally, the supercharges (9.62) coincide up to the factor 2 with the Hermitian and anti-Hermitian parts of the complex quantum supercharge ˆ in Eq. (9.11). Q Adding to the N = 1 action (9.34) the term with five factors DX M in the integrand, one can reproduce the term (9.33) in the N = 2 action, etc. To describe in the N = 1 language the system (9.1) with nonzero W [deformed or not deformed by the extra holomorphic torsion term (9.28)], one has to add to (9.34) an extra term with only one DX M factor: Z SA = −i dθdt AM (X ) DX M . (9.64) In components, this gives   Z i SA = dt −AM (x) x˙ M + FM N (x) ΨM ΨN . 2

(9.65)

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Theorem 9.6. The action (9.64) is invariant under the transformation (9.37) provided FK M IM L − IK M FM L = 0 .

(9.66)

Proof. Bearing in mind (9.42), the variation of SA is Z δSA = −i dθdt [(∂K AM )IL K DX M DX L + AM IL M DDX L + AM (∂K IL M )DX K DX L ] ,

(9.67)

where we replaced −i∂/∂t → DD. Next, we integrate by parts the middle term. The part ∝ (∂I)(DX )(DX ) cancels the third term and we are left with Z δSA = i dθdt (∂K AM ) [IL M DX K − IL K DX M ]DX L . (9.68) After some manipulations, this can be brought to the form Z i dθdt (IM K FKL − FM K IKL ) DX M DX L , δSA = 2

(9.69)

which vanishes provided (9.66) vanishes. In fact, we obtained nothing new, but simply reproduced, using the N = 1 language, the result obtained earlier: to keep N = 2 supersymmetry, the commutator [F, I] must vanish. (See the Remark on p. 180.) 9.3

Dolbeault and Dirac Complexes on S 4 \{·}

We saw in the previous sections that both the classical Dolbeault complex and the Dolbeault complex twisted by the gauge field and/or deformed by the holomorphic torsions can be described in the language of supersymmetry. However, as was repeatedly announced before, the latter is more powerful than traditional mathematical methods. It allows one to describe systems that have not yet attracted much attention from mathematicians. Consider S 4 . It is not a complex manifold. Still, one can try to proceed along the same lines as for S 2 : to trace at its south pole the 4-dimensional tangent hyperplane and to make the stereographic projection. The points of S 4 are then parametrized by the Cartesian coordinates (xM =1,2,3,4 ) of this hyperplane. We introduce the complex coordinates: z1 =

x1 + ix2 √ , 2

z2 =

x3 + ix4 √ . 2

(9.70)

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The following expression for √ the metric can be written [cf. (2.6); the radius of such sphere is equal to 1/ 2]:13 ds2 =

2dzj d¯ zj (1 + zj z¯j )

2

.

(9.71)

This metric is perfectly well defined everywhere except for the north pole of S 4 where one of zj hits infinity. In contrast to S 2 where one can make a similar map for the northern hemisphere in such a way that the transition functions between the two sets of complex coordinates are analytic, one cannot do so for S 4 (see p. 32). But never mind. Just consider S 4 without the north pole and use the expression (9.71) for the metric. Topologically, our manifold is now R4 . In contrast to S 4 , R4 is a complex manifold and one can well introduce there complex coordinates and define the Dolbeault complex. However, R4 is not compact, and one can formulate there essentially different spectral problems for different Hamiltonians associated with different Hilbert spaces distinguished by different measures. Thus, for the simplest spectral problem on R4 , the Hamiltonian is the flat Laplacian. Its spectrum is continuous. What we now want to do, however, is to assume the spheric metric (9.71) and to include the factor det h =

1 (1 + z z¯)4

(9.72)

in the measure R(with the shorthand z z¯ ≡ zj z¯j ). Then the volume of our manifold V = d2 zd2 z¯ det h is finite and the spectral problem is rather similar to spectral problems on compact complex manifolds. Proceeding in a similar way to what we did for compact manifolds, we consider [94] the SQM action (9.1) with the metric (9.71). We choose W = 41 ln det h so that the covariant quantum supercharges of our model, which are given by (9.19), map to the operators ∂ and ∂ † of the standard (untwisted) Dolbeault complex. Their explicit expressions are ˆ = −iψj [(1 + z z¯)∂j + z¯k ψk ψˆ¯j ] , Q ˆ = −iψˆ ¯j [(1 + z z¯)∂¯j − 2zj + zk ψˆ¯k ψj ] . ¯ Q

(9.73)

13 The metric (9.71) of S 4 is conformally flat. In this situation, it is not convenient anymore to distinguish between the covariant and contravariant indices. Neither will we distinguish in this section the indices j and ¯ j. Further, we will choose the vielbein in the conformally flat form, eja = λ δaj and will not distinguish between the world and tangent space indices, displaying explicitly the conformal factor λ(z, z¯) when it is necessary. The summation over the repeated indices is assumed, as usual.

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ˆ¯ act on the Hilbert ˆ = {Q, ˆ Q} The operators (9.73) and the Hamiltonian H space involving square-integrable wave functions with the measure d2 zd2 z¯ 2 2 ¯ −ψj ψ¯j d ψd ψ e (1 + z z¯)4

dµ =

(9.74)

[cf. Eq. (6.26)]. ˆ and There are three zero modes in the spectrum annihilated by both Q ˆ ¯ Q: Ψ(0) = 1, z¯1 , z¯2 .

(9.75)

Thereby, the Witten index is equal to 3. The whole spectrum of the Hamiltonian was found in [94]. It is noteworthy that all the eigenstates in the ˆ¯ are ˆ and Q spectrum are square-integrable and that the supercharges Q bound in this Hilbert space. In other words, for any eigenstate Ψ of the ˆ¯ either vanish or are also squareˆ and QΨ Hamiltonian, the functions QΨ integrable eigenstates.14 As a result, the spectrum of all excited states is (at least) double degenerate, as it should be in a supersymmetric system. One can also solve the same problem for the twisted Dolbeault complex involving q Wq = ln det h (9.76) 4 with an arbitrary integer q. The Witten index is then equal to IW (q) = 2q 2 + |q| .

(9.77)

The value q = 0 is special. The supercharges acquire the form ˆ = −iψ [(1 + z z¯)∂ − z¯ + z¯ ψ ψˆ¯ ] , Q j

j

j

k

k

j

¯j [(1 + z z¯)∂¯j − zj + zk ψˆ¯k ψj ] ¯ = −iψˆ Q

(9.78)

In this case, one can write, however, another pair of supercharges: ˆ = −iε ψ [(1 + z z¯)∂¯ − z − z ψ ψ¯ˆ ] . R jk

k

j

j

l

l

j

ˆ ¯ = −iεjk ψˆ¯k [(1 + z z¯)∂j − z¯j − z¯l ψˆ¯l ψj ] , R

(9.79)

A direct calculation shows that the four supercharges (9.78), (9.79) obey the extended N = 4 supersymmetry algebra (5.36). And this is not a miracle, but a corollary of the fact that S 4 \{·} is in fact an HKT manifold! Indeed, its metric is conformaly flat and belongs to ˆ 14 That is a nontrivial fact. A glance at (9.73) tells one that the operators Q ˆ and Q ¯ bring about an extra power of z and could in principle make a non-integrable at infinity function out of an integrable one. But this does not happen.

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the class (3.67). And we will see in the next chapter and then in Chap. 12 that the HKT manifolds admit the N = 4 supersymmetric description.15 The index (9.77) vanishes in this case, so that there are no zero-energy states in the spectrum. The states carrying nonzero energy split into degenerate quartets. One of such quartets with the supercharge actions is represented schematically in Fig. 9.1. Note that all the states in this quartet (and in all other such quartets) are normalizable with the measure (9.74)—similar to what we had for the Dolbeault complex discussed at the beginning of this section.

O R Q ψj z j Q

R Q

O

R

R ε jkψjψk

1

Q

R R

Q ε jkψj zk

Q R

O

Q

Q R O Fig. 9.1 on S 4 .

A supersymmetric quartet of degenerate eigenstates of the Dirac Hamiltonian

In Chap. 13, we will go back to the supersymmetric sigma model on S 4 with q = 0 16 and show that it can be extended to include a nonAbelian self-dual gauge field (an instanton) in such a way that the N = 4 supersymmetry is left intact. 15 Actually, the expressions (9.78), (9.79) are nothing but the quantum counterparts of the classical supercharges (10.81) written for S 4 . 16 We will learn there that it describes a “purely gravitational” Dirac complex involving extra torsions.

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One can also consider S 6 and still higher even-dimensional spheres. It is an open question whether S 6 is a complex manifold or not. But even if it is, the true complex structure IM N and the correspondning complex coordinates are not known. One can make, however, for S 2d the same 2 4 stereographic the coordinates z 1 = √ projection as for S and S , introduce 2d 1 2 (x + ix )/ 2, etc. and treat the problem for S \{·} in the same way as d−1 we did it for S 4 \{·}. For the untwisted Dolbeault complex, one finds C2d−1 bosonic zero modes. If d = 3, IW = 10.

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Chapter 10

Sigma Models with Extended Supersymmetries

In Chaps. 2,3, we defined some special classes of complex manifolds: the K¨ ahler, hyper-K¨ ahler and HKT manifolds. And now we are going to describe these and some other related geometries in the supersymmetric language. The corresponding sigma models enjoy extended N = 4 and N = 8 supersymmetries. We consider the N = 4 models first. 10.1

K¨ ahler Manifolds and Around

The K¨ ahler geometries admit several different supersymmetric descriptions. To begin with, consider the model (8.6) involving an even number of real N = 2 superfields ¯ M. X M = xM + θψ M + ψ¯M θ¯ + θθF

(10.1)

By construction, the action of this model is invariant under the N = 2 supersymmetry transformations (7.27): δ0 xM = 0 ψ M − ¯0 ψ¯M , δ0 ψ M = ¯0 (F M − ix˙ M ) , δ0 ψ¯M = 0 (F M + ix˙ M ) , δ0 F M = −i(0 ψ˙ M + ¯ψ¯˙ M )

(10.2)

We now impose a constraint and require the action to be invariant under the extra transformations ¯ N) . δX M = IN M (X)(DX N − ¯DX

(10.3)

For the lowest component, this means δxM = IN M (ψ N − ¯ψ¯N ) . 197

(10.4)

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The expressions for δψ M and δF M are somewhat more complicated. By the same reasoning as in the previous chapter [see a comment after Eq. (9.38)], one can deduce that the transformations (10.2) and (10.3) comˆ¯ Q ˆ¯ }, {Q, ˆ¯ Q ˆ¯ } ˆ Q ˆ 0 }, {Q, ˆ 0 }, {Q, ˆ Q mute and hence the anticommutators {Q, 0 0 of the corresponding supercharges vanish. One can now formulate a theorem analogous to Theorem 9.4 of the previous chapter:1 Theorem 10.1. The action corresponding to (8.6) is invariant under (10.3) and the generators of the transformations (10.2), (10.3) satisfy the standard N = 4 supersymmetry algebra iff IN M is an integrable complex structure that is covariantly constant with respect to the Levi-Civita connection, ∇P IM N = 0, i.e. iff the manifold is K¨ ahler. Proof. The proof follows the same lines as the proof of Theorem 9.4. Note first that the conditions I 2 = −1 and NM N K = 0 are necessary and sufficient to close the supersymmetry algebra—to ensure that the Lie bracket of two supertransformations (10.3) boils down to the time derivative: (δ2 δ1 − δ1 δ2 )X M = 2i(2 ¯1 − 1 ¯2 )X˙ M .

(10.5)

This can be checked by a direct calculation. Consider now the supersymmetric variation of (8.6). Let us concentrate on the terms involving . Bearing in mind D2 = 0, we obtain δX M ∼  IN M (X)DX N , δ(DX M ) ∼ −(∂P IN M )DX P DX N ,   N ¯ M ) ∼ − (∂P IN M )DX ¯ P DX N + IN M DDX ¯ δ(DX .

(10.6)

The variation of the action (8.6) reads Z  P Q ¯ M N ¯ dθdθdt[(∂ δS = P gM N )IQ DX DX DX 2 Q ¯ P DX Q DX N − IQN (DDX ¯ − gM N (∂P IQ M )DX )DX N ¯ M DX P DX Q ] . + gM N (∂P IQ N )DX

(10.7)

The first, the second and the fourth terms in this expression have the ¯ same structure ∼ DXDXDX. For the third term, we decompose IQN = I(QN ) + I[QN ] . The part involving I[QN ] can also be brought to the form 1 A similar theorem, formulated in the language of components for a field theory sigma model system, was proven in [95].

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¯ ¯ ∼ DXDXDX by flipping the Grassmann covariant derivative D: Z Q ¯ ¯ − dθdθdt I[QN ] (DDX )DX N = Z 1 ¯ ¯ P DX Q DX N dθdθdt (∂P I[QN ] )DX 2

199

(10.8)

The part involving I(QN ) is a linearly independent structure. For the variation (10.7) to vanish, the condition I(QN ) = 0 should thus hold, which we assume in the following. ¯ The structure ∼ DXDXDX in the variation vanishes provided 1 IQ P ∂P gM N + gP N ∂M IQ P − ∂M IQN + gP M ∂N IQ P 2 −[Q ↔ N ] = 0 .

(10.9)

After some massaging, this can be brought in the form AM [QN ] = ∇M IQN + ∇N IQM − ∇Q IN M = 0 .

(10.10)

If ∇Q IN M = 0, it obviosly vanishes. To show that (10.10) implies the covariant constancy of the complex structure, we symmetrize this relation over M ↔ Q to derive ∇M IQN +∇Q IM N = 0 and hence ∇M IQN +∇N IQM = 0. From this and from (10.10) the condition ∇Q IN M = 0 follows. Now we proceed in the same way as in Chaps. 8,9. At the first step, we go down to components and, using Noether’s theorem, derive the expressions for the conserved classical complex supercharges corresponding to the invariances (10.2) and (10.3). These expressions are especially simple if written in terms of the fermion fields ψ M with the world indices and the canonical momenta ΠM obtained by the variation of L over x˙ M with fixed ψ M [96]:   1 i Q0 = √ ψ M ΠM − ∂M gP K ψ P ψ¯K , 2 2   ¯ 0 = √1 ψ¯M ΠM + i ∂M gP K ψ P ψ¯K , Q 2 2   1 i Q1 = √ ψ N IN M ΠM − ∂M gP K ψ P ψ¯K , 2 2   ¯ 1 = √1 ψ¯N IN M ΠM + i ∂M gP K ψ P ψ¯K . Q (10.11) 2 2 At the second step we quantize it using the recipe of [57]. For the quantum supercharges, the representation via the flat fermion variables ψ A is more

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convenient. We derive the following expressions for the covariant quantum supercharges [97]:

ˆ† Q 0

ˆ 1 = − √i ψ N IN M DM , ˆ 0 = − √i ψ M DM , Q Q 2 2 i M ∂ i † N ˆ = − √ eC IN M ∂ D M , = − √ eC DM , Q 1 ∂ψC ∂ψC 2 2

(10.12)

where the operator of “spinor” covariant derivative, DM =

∂ ∂ + ωAB,M ψA , ∂xM ∂ψB

(10.13)

was defined in (8.27) and we use here the dagger notation rather than bar notation for the conjugated supercharges. We will now prove the following theorem representing an analog of Theorems 8.1 and 9.1 and clarifying the geometric meaning of the supercharges (10.12). Theorem 10.2. For K¨ ahler manifolds, the action of the supercharges Q0 and Q1 on the wave functions Ψ(xM , ψ M ) is isomorphic to the action of the certain linear combinations of the operators ∂ and ∂¯ in the K¨ ahler–de Rham complex of the (p, q) forms. ¯ Proof. Choose the complex coordinates xM → (z m , z¯m ) such that the M ˆ 0 and Q ˆ1 complex structure tensor IN acquires the form (2.52). Then Q 2 can be represented as ¯ ˆ 0 = − √i (ψ m Dm + ψ m Dm Q ¯) 2 ¯ ˆ 1 = − √1 (ψ m Dm − ψ m Q Dm ¯). 2

(10.14)

Consider the action of the operator ψ m Dm on a wave function Ψ(xM , ψ M ) = Am1 ...mp n¯ 1 ...¯nq (z, z¯) ψ m1 · · · ψ mp ψ n¯ 1 · · · ψ n¯ q .

(10.15)

The same calculation as in the proof of Theorem 8.1 yields ψ s Ds Ψ = (∂s Am1 ...mp n¯ 1 ...¯nq )ψ s ψ m1 · · · ψ mp ψ n¯ 1 · · · ψ n¯ q

(10.16)

s N plus some extra terms involving the structures ΓP sN ψ ψ Generically, these structures do not vanish (and that is why this theorem is not valid for generic complex manifolds), but in the K¨ahler case, the 2 We

ˆ ≡ ψˆ† = ∂/∂ψ. ¯ not to confuse it with the operators ψ ¯ do not write a bar over ψ m

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only nonvanishing Christoffel symbols have all holomorphic or all antiholo¯ morphic indices [see Eq. (2.30)]. Hence ΓpsN = 0 and ΓpsN ψ s ψ N → Γpsn ψ s ψ n = 0 due to the symmetry Γpsn = Γpns . Thus, the equality (10.16) is exact and the isomorphism ψ m Dm ↔ ∂ is established. ¯ The same reasoning allows one to establish the isomorphism ψ m Dm ¯ ↔ ¯ And the supercharges (10.12) map to the operators ∂. √ √ ¯ ˆ 1 → (∂¯ − ∂)/ 2 ˆ 0 → −i(∂ + ∂)/ 2, Q (10.17) Q and their Hermitian conjugates. We are now able to pay our old debt and prove Theorem 2.3. It affirms that, for K¨ ahler manifolds, the algebra ¯ = (∂ † )2 = (∂¯† )2 = {∂ † , ∂¯† } = 0 , ∂ 2 = ∂¯2 = {∂, ∂} ¯ ∂¯† }, {∂, ∂¯† } = {∂, ¯ ∂†} = 0 {∂, ∂ † } = {∂,

(10.18)

holds. Proof. The isomorphism established above makes this statement trivial. ˆ 0,1 and their conjugates satisfy the extended N = 4 The supercharges Q supersymmetry algebra. The same concerns the operators in the R.H.S. of Eq. (10.17). This algebra is equivalent to (10.18). To make contact with geometry, the description of this N = 4 model in terms of N = 2 superfields (10.1) is quite convenient. But if one sticks to his/her supersymmetric glasses and does not want or does not need to remove them, the most simple and natural description of the K¨ahler manifolds is based on N = 4 chiral superfields (7.48). Take d such superfields, √ (10.19) Zm=1,...,d = z m (tL ) + 2θj ψ jm (tL ) + 2θ1 θ2 F m (tL ) . Their lowest components will be interpreted as the complex coordinates. Now, ψ jm are their fermion superpartners (in this case, there are two complex fermion fields for each complex coordinate z m ) and F m are complex auxiliary variables. Each superfield (10.19) represents a (2, 4, 2) multiplet, which can be decomposed into a pair of (1, 2, 1) multiplets (10.1). Let us write the action in the form [98] Z 1 kin ¯ ¯j ¯m ¯ 2 θdt K[Zm (θj , tL ), Z S = d2 θd (θ , tR )] . (10.20) 4

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with real K. This is an obvious generalization of the first (kinetic) term in (7.51). The corresponding component Lagrangian is very simple:   i ˙ m ¯j n¯ ¯ m ˙n m¯ jn ¯ m ¯n ¯ ˙ L = κm¯n z˙ z¯ − (ψj ψ − ψj ψ ) + F F , (10.21) 2 where κm¯n = ∂m ∂n¯ K .

(10.22)

In other words, K is nothing but the K¨ahler potential and κm¯n is the Hermitian K¨ ahler metric (see p. 35). F m are nondynamical auxiliary variables. The equation of motion for them following from (10.21) is simply F m = 0. The complex structure has the canonical form (2.52). In real terms, this corresponds to IM N = diag(−ε, . . . , −ε) as in (2.50). One can add to (10.20) a superpotential term as in (7.51): Z Z 1 1 ¯ ¯m ¯ RW ¯ (Z d2 θdtL W (Zm ) + d2 θdt ). S pot. = 2 2

(10.23)

(10.24)

Then ¯ ¯ κm¯n F m F¯ n¯ → κm¯n F m F¯ n¯ − F m ∂m W − F¯ m ∂m ¯W .

¯ . When the auxilIn this case, the equations of motion are F m = κn¯ m ∂n¯ W iary variables are excluded, the component Lagrangian acquires the potential term ¯ . Lpot = −κn¯ m ∂m W ∂n¯ W

(10.25)

Note that Eq. (10.22) provides the full classification of the K¨ahler metrics in a particular chart defining the manifold: invent any real function ¯ K(z m , z¯m ) and the metric will follow. But it says nothing about the topological structure of the manifold—how different charts are glued together and what are the holomorphic transition functions. For completeness, we would also like to mention here that still another description of N = 4 K¨ ahler models is possible [99]. It is a description in terms of N = 2 superfields of different nature: instead of taking D real (1, 2, 1) superfields, we may take half-as-much chiral superfields (2, 2, 0) and the same number of Grassmann superfields (0, 2, 2) including each lowest fermion and bosonic auxiliary component: √ (10.26) Φn = χn + 2 θF n − iθθ¯χ˙ n .

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This description is similar to the description of the de Rham N = 2 model in terms of the bosonic and Grassmann N = 1 superfields given on p. 163. The Lagrangian of the model reads   Z 1 ¯ ¯ n¯ 1 m ¯ n¯ m ¯ ¯ DZ DZ + Φ Φ L = dθdθ ∂m ∂n¯ K(Z, Z) . (10.27) 4 2 This Lagrangian is invariant under the extra N = 2 supersymmetry transformations δZ n = −ηΦn ,

¯ n¯ , δ Z¯ n¯ = η¯Φ

δΦn = i¯ η Z˙ n ,

¯ n¯ = −iη Z¯˙ n¯ . (10.28) δΦ

Its component expansion coincides with (10.21). 10.1.1

Quasicomplex K¨ ahler models

By construction, the deformation (10.24) preserves N = 4 supersymmetry. But it is not the only possible such deformation. Besides the superpotential term (10.24), one can add in the action the following N = 4 supersymmetric term involving time derivatives of the superfields Zm : Z 1 d2 θdtL Am (Z) Z˙ m + c.c. (10.29) SA = 2 with an arbitrary vector holomorphic function Am . This structure, introduced in Ref. [96], is specific for quantum mechanics and does not have a field theory analog. That is probably the reason why it has not attracted attention before. Consider the component Lagrangian derived from the action S = S kin + S A .

(10.30)

The bosonic terms (see Refs. [73, 96] for the full expression) read ¯ ¯ n¯ m Lbos = κm¯n (z˙ m z¯˙ n¯ + F m F¯ n¯ ) + Fnm z˙ n F m + F ¯˙ n¯ F¯ m , ¯z

(10.31)

where Fnm = ∂n Am − ∂m An . Excluding the auxiliary variables, we derive Lbos = hm¯n z˙ m z¯˙ n

(10.32)

¯ ¯ hm¯n = κm¯n + Fml κpl Fp¯ ¯n .

(10.33)

with

The Lagrangian (10.32) describes the motion along a complex manifold with the metric (10.33). In contrast to κm¯n , this metric is not K¨ahler!3 3 Nontrivialities

start when the complex dimension of the manifold is 2 or more.

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Note that, after excluding the auxiliary variables, the Lagrangian does not include the terms linear in z˙ m . The object Am looks as an Abelian gauge field and Fmn as a field strength tensor, but their nature is in fact different.4 One can ask at this point: how is it possible? Our model enjoys the extended N = 4 supersymmetry, and have we not just proven Theorem 10.1 affirming that the extra N = 2 invariance (10.3) of the action (8.6) entails K¨ ahler geometry? The answer is the following. Being expressed in terms of the N = 2 superfields X M , the action (10.30) acquires the form (8.6), indeed, but the tensor gM N (X) is not real and symmetric any more. It has an imaginary antisymmetric part, gM N = g(M N ) + ib[M N ] . (10.34) The second term does not spoil the reality of the action—the only essential and necessary requirement. The model (10.30) belongs to the class of quasicomplex supersymmetric sigma models [85]. The bosonic part of the action of a generic such model describes the motion over a manifold with the metric GM N = g(M N ) + b[M P ] g (P S) b[SN ] (10.35) [cf. Eq. (10.33)]. We will come back to the discussion of the quasicomplex models in the next chapter. A generic quasicomplex model is N = 2 supersymmetric, but the model (10.30) enjoys the extended N = 4 supersymmetry. Using a sloppy language, we will call the model (10.30) the quasicomplex K¨ ahler model. The manifold that it describes is not K¨ahler, however (you might recall at this point that neither the “hyper-K¨ahler with torsion” manifolds are K¨ ahler), and a more exact (but also more long) name for (10.30) would be “a quasicomplex deformation of a K¨ahler–de Rham supersymmetric sigma model”. 10.1.2

Bi-K¨ ahler models

The superfields (10.19) are ordinary chiral N = 4 superfields satisfying ¯ j Zm = 0. But as was mentioned at the end of Chap. 7.3.1, one can D 4 We remind that, being complex, K¨ ahler manifolds also admit the presence of gauge fields. In a supersymmetric language, they are conveniently described in the language of chiral (2, 2, 0) superfields—see Eq. (9.1). For a generic K¨ ahler manifold, this action is N = 2 supersymmetric. We will see in Chap. 12 that for hyper-K¨ ahler and HKT manifolds, the first kinetic term in (9.1) enjoys N = 4 supersymmetry, and also one can ¯ such that N = 4 supersymmetry is preserved. choose some special W (Z, Z)

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introduce also mirror (2, 4, 2) superfields satisfying the constraints5 ¯ 1 U = D2 U = 0 . D

(10.36)

If all the fields entering the action are mirror, this amounts to redefining θ2 ↔ θ¯2 , which makes no difference. However, the situation changes when the action involves both ordinary and mirror multiplets. Consider a system involving d1 ordinary N = 4 chiral multiplets Zm and d2 mirror multiplets Uµ , d1 + d2 = d. Consider the action Z 1 ¯ ¯m ¯ µ¯ ) . ¯ 2 θdt L(Zm , Z d2 θd ; Uµ , U (10.37) S = 4 When going down to components, the bosonic part of the Lagrangian boils down to Lbos = hm¯n z˙ m z¯˙ n¯ + hµ¯ν u˙ µ u ¯˙ ν¯ ,

(10.38)

where hm¯n = ∂m ∂n¯ L,

hµ¯ν = −∂µ ∂ν¯ L .

(10.39)

We see two distinct sectors. In both sectors the metric is obtained as a double derivative of the prepotential L, but this double derivative enters with a negative sign for the mirror sector. The mixed components of the metric hm¯ν and hµ¯n vanish. This does not mean, however, that the corresponding manifold is a direct product of two K¨ahler manifolds: hm¯n may depend not only on z, z¯, but also on u, u ¯, and hµ¯ν may depend not only on u, u ¯, but also on z, z¯. Both the ordinary chiral superfields Zm=1,...,d1 and the mirror superfields Uµ=1,...,d2 can be expressed via the real (1, 2, 1) superfields X M =1,...,2d1 and Y M =2d1 +1,...,2d , respectively. The action (10.37) expressed in these terms acquires the form (8.6) with a real tensor gM N (X, Y ). One can ask again: how come the geometry of our model is not K¨ahler (which Theorem 10.1 seems to dictate), but has a more complicated bi-K¨ahler nature? The answer is that the law of extra N = 2 supersymmetry transformations is now more complicated than (10.3) and is different in the ordinary and mirror sectors. Generic such transformations can be brought to the following canonical form [72, 73]: ¯ N, M, N = 1, . . . , 2d1 ⇒ δX M = (Id1 )N M (D − ¯D)X ¯ N , (10.40) M, N = 2d1 + 1 + 1, . . . , 2d ⇒ δY M = (Id2 )N M (¯ D − D)Y 5 D2

≡ Dj=2 , not a square of anything.

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where Id1 and Id2 are the canonical matrices (10.23) of dimensions 2d1 and 2d2 . Indeed, once θ2 and θ¯2 have been interchanged, the same concerns  and ¯, which realize the shifts of θ2 and θ¯2 . Introducing now  = + + i− , D = D+ + iD− (10.41) with real ± and Hermitian D± , one can rewrite (10.40) in the form  M  N Id1 0 X δ = 2i+ D− 0 Id2 N Y   M  N Id1 0 X +2i− D+ , (10.42) 0 −Id2 N Y Both diag(Id1 , Id2 ) and diag(Id1 , −Id2 ) can be chosen for a complex structure of a bi-K¨ ahler manifold. The basic action (10.37) can be deformed by adding a potential or a quasicomplex term. Let us concentrate on the latter. Obviously, we can add the terms like in Eq. (10.29) independently in each sector and write A Sbi−K = Z Z 1 1 m ˙ dθ1 dθ2 dtL Am (Z) Z + dθ1 dθ¯2 dt˜L Bµ (U) U˙ µ + c.c., (10.43) 2 2 where tL = t − i(θ1 θ¯1 + θ2 θ¯2 ) , t˜L = t − i(θ1 θ¯1 − θ2 θ¯2 ) (10.44) ¯ 1,2 and t˜L vanishes under the action of (tL vanishes under the action of D ¯ 1 and D2 ). D But one can also write N = 4 invariants of more complicated structure. Consider the superfield ¯ 2 [Am (Z, U)D2 Zm ] . R = D ¯ 1 Zm = D ¯ 1 Uµ = 0 and (D ¯ 2 )2 = 0, we deduce that Bearing in mind thatR D ¯ Dj R = 0 and hence dtL dθ1 dθ2 R represents a supersymmetric invariant. Note that in the case when Am does not depend on Uµ , one can pull ¯ 2 through to obtain R = 2iAm Z˙ m so that the structure in (10.29) is D reproduced. The most general quasicomplex deformation reads A,B Sbi−K = Z n o i 2 m (2) ¯ 1 m ¯ 2 [A(1) ¯ dθ1 dθ2 dtL D − m (Z, U)D Z ] + D1 [Am (Z, U)D Z ] 4 Z n o i ¯ 1 Uµ ] ¯ 2 Uµ ] + D ¯ 1 [B(2) (U, Z)D − dθ1 dθ¯2 dt˜L D2 [B(1) (U, Z) D µ µ 4 + c.c. (10.45) 

X Y

M



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Symplectic Sigma Models

The K¨ ahler and related models of the preceding section were formulated for complex manifolds, whose real dimension is even. But for certain special manifolds of dimension D = 3n or D = 5n with integer n, one also can formulate supersymmetric sigma models that enjoy the extended N = 4 and even N = 8 (in the case D = 5n) supersymmetry. The N = 8 models will be discussed at the end of the chapter. Here we concentrate on N = 4 models describing 3n-dimensional manifolds. Consider the (3, 4, 1) superfield (7.62) with the component expansion (7.66). We can generalize the action (7.67), the bosonic part of which described free 3-dimensional motion, to write [75] 6 Z ¯ 2 θdt F (ΦM ) S = − d2 θd (10.46) with arbitrary F (ΦM ), M = 1, 2, 3. The bosonic part of this action reads 1 ˙ 2, h(A)A 2

(10.47)

h(A) = 24F (A) .

(10.48)

Sbos = where

It describes the motion along a conformally flat 3-dimensional manifold.7 The component expressions for the quantum supercharges and the Hamiltonian may be found following the standard procedure described above. The supercharges read [74] ˆ j = √1 f (A)PˆM (σM )j k ψk + Q 2 1 ˆ j = √ Pˆ f (A)ψˆ¯k (σ ) j − ¯ Q M M k 2

i ˆ¯ ψ ψ , √ (∂M f ) ψσ M j 2 i ˆ¯ ψ , √ (∂M f ) ψˆ¯j ψσ M 2

(10.49)

where f (A) is an arbitrary function giving the metric, h(A) = f −2 (A), σM ˆ¯ j = (Q ˆ j )† . are the Pauli matrices, and Q We see that the supercharges carry spinorial indices belonging to the fundamental representation of the group Spin(3) ≡ Sp(1). The same is true 6 The opposite sign compared to [75] is due to the different sign convention in the fermion measure. 7 Note that here 4 is not the covariant Laplace-Beltrami operator on this manifold, but simply 4 = ∂M ∂M .

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for the (5, 8, 3) sigma model to be considered in Sect. 11.4. Four complex supercharges of that model belong to the fundamental representation of Spin(5) ≡ Sp(2). That is why we called the models of this type symplectic [100, 101]. These models describe the motion over a certain class of manifolds that we cannot call symplectic (this term is already taken) and, having not found a better word, we will call these manifolds spinorial. The anticommutator of the supercharges (10.49) gives the Hamiltonian: ˆ¯ ψ f (∂ f )Pˆ ˆ = 1 f (PˆM )2 f + εM N P ψσ H M N P 2 1 ˆ¯ ψ ψσ ˆ¯ ψ. + f (4f ) ψσ (10.50) M M 6 An obvious generalization of the action (10.46) is the action Z ¯ 2 θdt F (Φk ) S = − d2 θd

(10.51)

including several (3, 4, 1) superfields Φk=1,...,n . Its bosonic part describes the motion along a 3n-dimensional manifold with the metric ∂2F . (10.52) gM k,N l = 2δM N ∂AP k ∂AP l It is difficult for us to say whether this new geometry is interesting or not from the mathematical point of view, but the system (10.51) has interesting applications in physics: • For some special set of Φk and for some special function F (Φk ), it gives the effective action for a 4-dimensional supersymmetric gauge theory placed in a small spatial box [74, 102]. • Surprisingly, the same model describes the dynamics of extremal Reissner-Nordstr¨ om black holes [103]. Having started to talk about physics, I cannot resist mentioning another interesting class of (3, 4, 1) models that describes the effective Lagrangian of 4-dimensional chiral (i.e. having a left-right-asymmetric fermion content) supersymmetric gauge theories squeezed in a small box. The simplest such SQM system was found in [104]. It has a pair of complex supercharges: h i ˆ j = √1 (σM )j k ψk (PˆM + AM ) − iψj D , Q 2 i h 1 j ˆ ¯ = √ ψˆ¯k (σM )k j (PˆM + AM ) + iψˆ¯j D , (10.53) Q 2 where A and D are the functions of A such that ∇ × A = B = −∇D .

(10.54)

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The standard N = 4 supersymmetry algebra (5.36) holds, giving the Hamiltonian ˆ¯ ˆ + A)2 + 1 D2 + B ψσψ ˆ = 1 (P H . (10.55) 2 2 A physically interesting case is D = 1/(2|A|) ≡ 1/(2A), where the Hamiltonian (10.55) describes the 3-dimensional motion of a spin- 12 particle in the field of a Dirac magnetic monopole BM = AM /(2A3 ) supplemented by the scalar potential V (A) = 1/8A2 . Remarkably, this Hamiltonian describes also the dynamics of 4-dimensional chiral supersymmetric electrodynamics (the theory involving the photon, a left-handed fermion of charge 2, eight right-handed fermions8 of charge 1, and their superpartners) placed in a small spatial box.9 Well, the statement above was not quite correct. In fact, the effective Hamiltonian for such a theory describes the motion in an infinite cubic lattice of monopoles carrying the magnetic charges +1 and -7 in such a way that the average magnetic charge is zero. But we do not want to harass our reader-mathematician more. A reader-physicist can consult the original paper [105]. It is possible to construct generalizations of this model. First of all, we can endow the 3-dimensional manifold, along which the particle moves, with a nontrivial conformally flat metric. The quantum supercharges and the Hamiltonian acquire the form [74] h i ˆ¯ ψ ψ , ˆ j = √f (σM )j k ψk (PˆM + AM ) − iψj D + i∂√M f ψσ Q M j 2 2 h i f ˆ¯ ψ , (10.56) ˆ ¯k (σM )k j (PˆM + AM ) + iψˆ¯j D √ − i∂√M f ψˆ¯j ψσ ¯ j = ψˆ Q M 2 2

+εM N P

  1 2 1 ˆ 2 ˆ ¯ ˆ (P + A) + D + BM ψσM ψ f H = f 2 2 ˆ ˆ¯ ψ ¯ ˆ ψσM ψ f (∂N f )(PP + AP ) − f (∂M f )D ψσ M

1 ˆ¯ ψ ψσ ˆ¯ ψ , f (4f ) ψσ (10.57) M M 6 One can also take several (3, 4, 1) superfields and describe in this way the effective Hamiltonians of non-Abelian chiral gauge theories in small +

8A

remark addressed to experts: one has to take eight of them to cancel the anomaly. avoid confusion, note that the dynamical variables A have nothing to do in this case with the physical spatial coordinates. They have the meaning of the zero Fourier components of the physical vector potential [having again nothing to do with A(A) entering Eqs. (10.53) and (10.55)]. 9 To

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boxes [106]. We present here the simplest nontrivial such Hamiltonian for the SU (3) chiral theory. To derive it, it is sufficient to take two symplectic superfields. The Hamiltonian reads ˆ a + Aa )2 + (P ˆ b + Ab )2 + (P ˆ a + Aa )·(P ˆ b + Ab ) ˆ = 2 [(P H 3 + (Da )2 + (Db )2 + Da Db ] ¯a σM ψ a + B b ψˆ¯b σM ψ b + B ab (ψˆ¯a σM ψ b + ψˆ¯b σM ψ a ) , (10.58) + B a ψˆ M

M

M

where Da,b , Aa,b , and Ba,b are certain functions of the dynamical variables Aa,b . The former satisfy the constraints a a a BM = εM N P ∂N AaP = −∂M Da , b b b BM = εM N P ∂N AbP = −∂M Db , ab a b a b BM = εM N P ∂N AbP = εM N P ∂N AaP = −∂M Db = −∂M Da . a

(10.59)

b

The physical meaning of A and A are the zero Fourier modes of the components of gauge potential belonging to the Cartan subalgebra of su(3)—their projections on the simple roots a = diag(1, −1, 0) and b = diag(0, 1, −1). The superfield description of the symplectic models including gauge fields is only possible in the framework of the harmonic superspace approach [79]. One has to introduce a set of Grassmann-analytic double-charged superfields Vk++ subject to the constraints D++ Vk++ = 0 [see Eq. (7.105) for their component expansion] and write the action as the integral over the analytic superspace, Z S kin = d2 θ+ d2 θ− dt du L(Vk++ , Vk+− , Vk−− , u) , (10.60) where 1 1 −− ++ D Vk , Vk−− = (D−− )2 Vk++ (10.61) 2 2 and L carries zero harmonic charge. After integration over harmonics, the action (10.60) coincides with (10.51) and gives the metric (10.52). The contribution including the gauge fields A is reproduced, if one adds the structure Z Vk+− =

S gauge =

d2 θ+ dtA du L++ (Vk++ ) ,

(10.62)

with an appropriate function10 L++ of harmonic charge +2. We refer the reader to the paper [79] for further details. 10 To obtain the effective small-box effective Hamiltonian for a particular chiral theory, one should take some particular L and L++ . But the system is evidently N = 4 supersymmetric with arbitrary L and L++ .

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Bi-symplectic models

We have seen in the previous section that the supermultiplet (2, 4, 2) exists in two modifications—the ordinary and the mirror one. Both multiplets ¯ 1Z = D ¯ 1 U = 0, but the second constraint is differsatisfy the constraint D ¯ ent: D2 Z = 0 for the ordinary multiplet and D2 U = 0 for the mirror one. This amounts to the interchange θ2 ↔ θ¯2 in the θ-expansion. By the same token, on top of the ordinary symplectic multiplet Φ whose ˜ expansion is given in Eq. (7.66) one can consider the mirror multiplet Φ 2 ¯ with the same expansion up to the interchange θ2 ↔ θ . If the model includes only mirror multiplets, the component action has the same form as with the ordinary multiplets. But if we take some number of ordinary and some number of mirror multiplets, we arrive at the sigma-model of new type, which is natural to call a bi-symplectic sigma model. Consider the simplest such model involving a single ordinary and a single mirror multiplet. Write the action as Z ¯ 2 θdt F (Φ, Φ) ˜ . S = − d2 θd (10.63) The bosonic part of the component Lagrangian reads in this case [cf. (10.38), (10.39)] Lb =

2 ˜ ˙ ˜ ∂ 2 F (A, A) ˜2 ˙ 2 − ∂ F (A, A) A A 2 2 ˜ (∂A) (∂ A)

(10.64)

The corresponding 6-dimensional metric represents a composition of two conformally flat blocks,  2  ∂ F ∂2F gM N = 2 diag 1,− 1 . (10.65) ˜ 2 (∂A)2 (∂ A) A generalization to the case when one has several ordinary and mirror multiplets is obvious. The metric of a generic bi-spinorial manifold is block diagonal with the structure (10.52), taken with the positive or negative sign, depending on whether the corresponding multiplet is ordinary or mirror, in each block. 10.3

HKT and bi-HKT Models

Interesting geometries arise from sigma-models involving (4, 4, 0) superfields. As was the case for the (2, 4, 2) and (3, 4, 1) multiplets, the multiplets (4, 4, 0) exist in two modifications: the ordinary and the mirror

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ones [11, 26, 108]. A mirror multiplet is distinguished from an ordinary one by the interchange θ2 ↔ θ¯2 in the θ-expansion. The models involving such multiplets of the same type describe HKT geometry. To obtain a full description of HKT metrics, one should use the harmonic superspace approach and take several nonlinear (4, 4, 0) multiplets satisfying the constraint (7.100). We will treat such models in Chap. 12. In this chapter, we restrict ourselves by the case L+3a = 0. This defines linear multiplets, which can be alternatively described in the ordinary N = 4 superspace. As was shown in Chap. 7, an ordinary linear multiplet can be described by a pair of (2, 4, 2) superfields ZA related by the constraint (7.69): ¯B = 0 , ¯jZ Dj ZA − εAB D

(10.66)

An obvious generalization of the free action (7.71) reads Z 1 ¯ A) , ¯ 2 θdt L(ZA , Z d2 θd S = k k 8

(10.67)

where k labels the multiplets and L is an arbitrary real function. For the linear multiplets discussed in this chapter, the solution to the constraints (7.100) is easy to find. It is given by Eq. (7.70), where the multiplet ZA is expressed in terms of a pair of unconstrained chiral (2, 2, 0) superfields Z A . Now we can substitute (7.70) into (10.67) to obtain Z
1 A 1 ¯ ¯B S = dθ1 dθ¯1 dt 4kl (10.68) AB L D Zk D1 Zl , 4 where11 4kl AB L

1 = 2



∂2L ∂2L AC BD + ε ε ∂ZkA ∂ Z¯lB ∂ Z¯kC ∂ZlD

 .

(10.69)

Integrating further over dθ1 dθ¯1 , we obtain the component Lagrangian. Its bosonic part reads12
A B Lbos = 4kl ¯˙l . (10.70) AB L z˙k z In other words, the expression (10.69) is nothing but the Hermitian metric hkA,lB on our complex manifold. If we have only one multiplet, the Lagrangian (10.70) reduces to Lbos =

1 ∂2L z˙ B z¯˙ B . 2 ∂z A ∂ z¯A

(10.71)

11 As we do not need to care much in this case about the position of the indices, we adopt the convention ε12 = ε12 = 1. 12 See Ref. [73] for the full expression.

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This Lagrangian describes the motion along a conformally flat manifold of real dimension 4. In this way of description, a half of the supersymmetries of the action (10.68) associated with the coordinate transformations in the superspace (t, θ1 , θ¯1 ) is manifest and another half is “hidden”, being realized as ¯ 1 Z¯kB , δZkA = ηεAB D δ Z¯kA = −¯ η εAB D1 ZkB , (10.72) where η is a complex Grassmann parameter associated with the shift of θ2 . Theorem 10.3. The metric (10.69) describes an Obata-flat HKT manifold. Lemma 1. The manifold with the metric (10.69) is hypercomplex. Proof. We go one more step down and express each chiral multiplet ZkA as a pair of N = 1 multiplets, as what was done in Chap. 9.2. The actions (10.67) and (10.68) can be then rewritten in the form (9.34). By construction, this action enjoys four real supersymmetries. One of them is manifest, it is given by Eq. (9.36). And three others are hidden. One of the latter has the following component form δ1 xM = i1 IN M ΨN , δ1 ΨM = 1 IN M x˙ N , (10.73) M where the complex structure IN has the canonical form (2.50) with an even number of blocs. Note that (2.50) is a constant matrix and we could suppress by that reason the second term in (9.38). Two remaining real supersymmetries follow √ from the complex supersymmetry (10.72). By posing η = i(2 + i3 )/ 2, using the expansion (7.28) and expressing the complex bosonic and fermionic coordinates via the real ones as x1 + ix2 x3 + ix4 √ √ z11 = , z12 = , 2 2 Ψ3 + iΨ4 Ψ1 + iΨ2 √ √ , ψ12 = (10.74) ψ11 = 2 2 (then z2A are expressed via x5,6,7,8 , etc.), we can bring (10.72) into the same form as (10.73): δ2 xM = i2 JN M ΨN , δ3 xM = i3 KN M ΨN , δ2 ΨM = 2 JN M x˙ N , δ3 ΨM = 3 KN M x˙ N (10.75) with the constant matrices J, K having the canonical form (3.2). The triple (I, J, K) is quaternionic and hence the manifold is hypercomplex.

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Lemma 2. The form ω+ = ωJ + iωK is closed with respect to the exterior holomorphic derivative associated with the complex structure I: ∂I ω+ = 0. Proof. According to Theorem 3.4, the only nonzero components of I + = (J + iK)/2 are (I + )m n¯ . In our case, they are very simple:13 (I + )m n¯ = diag(−ε, . . . , −ε) .

(10.76)

Then the form ω+ reads ¯ n p ω+ = 2(I + )n m hpm = 2εAC ¯ dz ∧ dz

∂2L dz B ∧ dzlA ∂ z¯lC ∂zkB k

(10.77)

(the two terms in (10.69) give the same contribution). When acting upon this with the exterior holomorphic derivative ∂pD dzpD ∧, we obtain zero. Two lemmas above and Theorem 3.5 allow us to conclude that the manifold is HKT. As the complex structures IN M , JN M and KN M are constant matrices in this case, the Obata connections (3.61) vanish. Remark. We have proven our theorem using certain specific features [like the constancy of the complex structures (I, J, K)N M ] of the action (10.67) including the linear (4, 4, 0) multiplets. In Chap. 12 we will prove a more general theorem: any hypercomplex manifold described by an N = 4 supersymmetric model is HKT. The invariances (9.37), (10.73), (10.75) bring about four conserved real supercharges. They have the form [93]   i i A B K P M , Q0 = Ψ PM − ωAB,M Ψ Ψ + CM KP Ψ Ψ 2 12   i i N M A B K P Q1 = Ψ IN PM − ωAB,M Ψ Ψ − CM KP Ψ Ψ , 2 4   i i Q2 = ΨN JN M PM − ωAB,M ΨA ΨB − CM KP ΨK ΨP , 2 4   i i N M A B K P Q3 = Ψ KN PM − ωAB,M Ψ Ψ − CM KP Ψ Ψ . (10.78) 2 4 In other words, two extra supercharges to be added to the supercharges (9.61) have the same form as Q1 where the complex structures J, K are 13 This [and the related matrix I − = (J − iK)/2, which in our case has the same form (10.76) modulo the interchange n ↔ n ¯ ] are the hypercomplex structures defined on p. 63. Note also that the matrices I ± enter the law (10.72) of the extra supersymmetry transformation by the same token as the matrix of the ordinary complex structure enters the laws of the extra supersymmetry transformations in (9.37) and (10.3).

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substituted for I. All the supercharges involve the Levi-Civita spin connection ωAB,M and the universal Bismut torsion CM KP . It is also instructive to write the explicit expressions for the complex supercharges [96], Q0 ± iQ1 , S, S¯ = 2

¯ = Q2 ± iQ3 . R, R 2

(10.79)

We derive   ¯ S = ψ m pm − iωa¯b,m ψ a ψ¯b ,   ¯ a ¯¯ b S¯ = ψ¯m pm , ¯ − iωa¯ b,m ¯ψ ψ  ¯ a ¯¯ b R = ψ k (I + )k m pm ¯ − iωa¯ b,m ¯ψ ψ −  ¯ ¯ k − m ¯ ¯ pm − iωa¯b,m ψ a ψ¯b − R = ψ (I )k¯

 i l k ¯¯ Cmk ψ ψ , ¯ 2 ¯ l  i ¯ C ¯ ψ k ψ¯l , 2 mkl

(10.80)

where I ± = (J ± iK)/2 are the matrices of hypercomplex structure. These expressions become somewhat more elegant if written in terms of the momenta πm and πm ¯ as in (9.6):   i S = ψ m πm − ∂m hnp¯ ψ n ψ¯p¯ , 2   i m ¯ n ¯p¯ ¯ ¯ S=ψ πm , ¯ + ∂m ¯ hnp¯ ψ ψ 2   i n ¯p¯ k + m ¯ , R = ψ (I )k πm ¯ hnp¯ ψ ψ ¯ − ∂m 2   ¯ = ψ¯k¯ (I − )k¯ m πm + i ∂m hnp¯ ψ n ψ¯p¯ , R (10.81) 2 The first pair of the supercharges in (10.81) coincides with (9.6), which is not surprising: HKT manifolds are complex and HKT models belong to the broad class of N = 2 Dolbeault models (9.1). In the second pair of the supercharges, the hypercomplex structures appear in exactly the same fashion as the complex structure appears in the expression (10.11) for the classical K¨ ahler–de Rham supercharges. This is also not surprising: as was mentioned in the Geometry part, HKT manifolds relate to generic hypercomplex manifolds roughly in the same way as K¨ahler manifolds relate to generic complex manifolds. The hypercomplex structure plays for the HKT manifolds the same role as the complex structure for the K¨ahler manifolds. An important remark is in order. The derivation of the expressions for ¯ in (10.80), (10.81) involved a nontrivial step that we now want R and R

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¯ in R, ¯ the to elucidate. Besides the structures ∼ ψψ ψ¯ in R and ∼ ψ¯ψψ holomorphic structure in R proportional to 1 ¯ p l + m ¯ k p l X = ψ k (I + )k m Cmpl (10.82) ¯ ψ ψ = (I )k (∂p hlm ¯ )ψ ψ ψ 2 ¯ seem to pop out. and the similar structure ∼ ψ¯ψ¯ψ¯ for R Theorem 10.4. For an HKT manifold, the expression X vanishes identically. Proof. At the first step, we flip the derivative, capitalizing on the fact that the form (10.77) is closed with respect to ∂I : k p l ¯ X = −[∂p (I + )k m ] hlm ¯ ψ ψ ψ .

(10.83)

The complex structures I, J and K are covariantly constant with respect to one and the same Bismut connection. It follows that (B)

(B)

¯ ¯ ¯ ∂p (I + )k m = hq¯s Gq¯,pk (I + )s m − hms Gs,p¯q (I + )k q¯ ¯ ¯ = hq¯s (∂k hp¯q ) (I + )s m + hms (∂s hp¯q − ∂p hs¯q ) (I + )k q¯,

(10.84)

where we used (2.20) and (2.18). Then   X = ψ k ψ p ψ l (∂p hl¯q − ∂l hp¯q )(I + )k q¯ + ∂k hp¯q (I + )l q¯ = 3ψ k ψ p ψ l ∂p hl¯q (I + )k q¯ .

(10.85)

Comparing with (10.82), we deduce X = 3X and hence X = 0. Note that this proof and hence the expressions (10.81) for the supercharges do not depend on the assumption of Obata-flatness. These expressions hold for a generic HKT manifold. The quantum supercharges that correspond to (10.78) have a simple interpretation in the language of differential forms:

Q1

† , Q0 = ∂˜I + ∂˜I† = ∂˜J + ∂˜J† = ∂˜K + ∂˜K † † † ˜ ˜ ˜ ˜ ˜ ˜ = i(∂I − ∂I ), Q2 = i(∂J − ∂J ), Q3 = i(∂K − ∂K ) ,

(10.86)

where ∂˜I,J,K are the twisted operators of holomorphic exterior derivatives with respect to the complex structures I, J, K. These operators were de† fined in Eq. (9.26), where one should set W = 0. The operators ∂˜I,J,K are their Hermitian conjugates. Note finally that HKT models can be deformed by adding a gauge field. If we want to keep N = 4 supersymmetry, this gauge field should have a special form. Generic description of such fields requires using the harmonic superspace formalism, and we postpone it till Chap. 12.

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10.3.1

217

Bi-HKT models

Consider a model [73] involving some number of the ordinary linear (4, 4, ¯ A 0) multiplets ZA=1,2 k=1,...,d1 /2 satisfying Dj Zk = 0 and the constraints (10.66), and some number of mirror multiplets UA α=1,...,d2 /2 (d1,2 are even). The latter satisfy the constraints obtained from the constraints for ZA k by the ¯ 2: interchange D2 ↔ −D AB ¯ ¯ B ¯ 2 UA D D1 Uα = 0 , α +ε 1 A AB 2 ¯ B D Uα − ε D Uα = 0 .

(10.87)

The constraints (10.66), (10.87) can be resolved to express each field ZA k via a pair of chiral multiplets ZkA as in (7.70), and each field UA via a pair α of chiral multiplets UαA as A ¯2 AB D ¯ 1U ¯ B + iθ2 θ¯2 U˙ A , UA α = Uα + θ ε α α ˙A, ¯A = U ¯ A − θ2 εAB D1 U B − iθ2 θ¯2 U ¯ U α α α α

(10.88)

A generic action reads 1 S = 8

Z

¯ A ; UA , U ¯ A) . ¯ 2 θdt L(ZA , Z d2 θd k k α α

After integrating over dθ2 dθ¯2 , it acquires the form Z 1 S = dθ1 dθ¯1 dtL0 4

(10.89)

(10.90)

with αβ 1 A ¯ ¯B 1 A ¯ ¯B L0 = (4kl AB L) D Zk D1 Zl − (4AB L) D Uα D1 Uβ ,

1 ¯ 1 Z¯ A D ¯ 1U ¯ C − (∂ ∂ L) D1 Z A D1 U C ], (10.91) + εAB εCD [(∂Bk ∂Dα L) D k α k α Bk Dα 2 αβ where 4kl AB L was defined in (10.69), 4AB L is defined analogously, ∂Bk = ¯ Z, ¯ ∼ ∂/∂ZkB , etc. The Lagrangian (10.91) involves the terms ∼ DZ D ¯ ¯ ¯ ¯ ¯ ¯ DU DU and also the terms ∼ DZDU and ∼ DZ DU . It belongs to the class of the Dolbeault SQM models twisted by holomorphic torsions that we discussed in Chap. 9.1.1. A half of the supersymmetries of the action (10.89) is realized in this formulation manifestly, while another half is “hidden”. The latter is implemented as the invariance with respect to the following transformations of the N = 2 superfields:

¯ 1 Z¯kB , δZkA = η εAB D ¯ 1U ¯αB , δUαA = η¯ εAB D

δ Z¯kA = −¯ η εAB D1 ZkB , ¯αA = −η εAB D1 U ¯αB . δU

(10.92)

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With the Lagrangian (10.91) at hand, we can go down to components. The second derivatives of the prepotential L in the first line in (10.91) give the metric so that the bosonic part of the component Lagrangian reads ¯ ¯ ¯˙ B Lb = hAk,Bl z˙kA z¯˙lB + hAα,Bβ u˙ A (10.93) αu β with hAα,Bβ = −4αβ (10.94) hAk,Bl = 4kl AB L , AB L . A A The derivatives are taken now with respect to the bosonic fields zk , uα . The second line in (10.91) involving holomorphic torsions does not contribute in the bosonic action, but contributes in the terms involving fermions, like in (9.29). The full expression for the component Lagrangian is written in [73]. Remark. The reader has noticed, of course, that the whole construction here is quite parallel to the construction of bi-K¨ahler and bi-symplectic models discussed above. In particular, formula (10.92) is an analog of (10.40) and the expression (10.94) for the metric is an analog of expressions (10.39) and (10.65). This is by no means accidental. In fact, the models (4, 4, 0), (3, 4, 1) and (2, 4, 2) are interrelated: two last models can be derived from the first one by the procedure of Hamiltonian reduction.14 We will discuss this in detail in the next chapter. Our next task is to determine the complex structures associated with the supersymmetry transformations in our model. We proceed in the same way as we did for the HKT models. One of the complex structures is associated with the manifest N = 2 invariance of the action (10.90). It has the standard canonical form I = diag(I, . . . , I) (10.95) with the matrix I given √ in (3.3). To find two other complex structures, we pose η = i(2 +i3 )/ 2, substitute it in (10.92) and go down to components: to real coordinates xM and their superpartners ΨM . Then the law of the supersymmetry transformations has the same form as in Eq. (10.75) with the following complex structures: J = diag(J, . . . , J, −J, . . . , −J) | {z } | {z } d1 /2

d2 /2

K = diag(K, . . . , K) . (10.96) In contrast to the HKT case, the triple of complex structures (10.95), (10.96) is not quaternionic. On the other hand the matrices I, J, K still satisfy the Clifford algebra, I p I q + I q I p = −2δ pq 1 . (10.97) 14 That

is why the multiplet (4, 4, 0) is sometimes called a root multiplet.

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That is why the models of this type were called “Clifford K¨ahler with torsion” in Ref. [11]. We find, however, the name bi-HKT more adequate. Indeed, the algebra of the matrices (10.95), (10.96) is not closed under multiplication, and it is a known mathematical fact (see e.g. Ref. [107]) that its closure represents a direct sum of two quaternion algebras H+ ⊕H− . The subalgebras H+ and H− correspond in our case to the ordinary and the mirror HKT sectors. Indeed, the whole set of matrices obtained by multiplying the matrices (10.95), (10.96) is 14n = diag(14 , . . . , 14 ), ∆ = diag(14 , . . . , 14 , −14 , . . . , −14 ), {z } | {z } | d1 /2

d2 /2

I1 = diag(I, . . . , I), I2 = diag(I, . . . , I, −I, . . . , −I), | {z } | {z } d1 /2

d2 /2

J1 = diag(J, . . . , J), J2 = diag(J, . . . , J, −J, . . . , −J), | {z } | {z } d1 /2

d2 /2

K1 = diag(K, . . . , K), K2 = diag(K, . . . , K, −K, . . . , −K). (10.98) | {z } | {z } d1 /2

d2 /2

Consider the linear combinations 1± 4n =

14n ± ∆ ± I1 ± I2 J1 ± J2 K1 ± K2 , I = , J± = , K± = . (10.99) 2 2 2 2

It is clear that the members of the “plus” quartet and the “minus” quartet satisfy separately the quaternion algebras and that the product of any “plus” matrix to any “minus” matrix vanishes. The diagonal subalgebra of H+ ⊕ H− involves the standard quaternionic matrices (3.2), which can be chosen for the complex structures not only in the HKT, but also in the bi-HKT case.15 That means that all bi-HKT manifolds are hypercomplex. However, • This quaternion triple cannot be associated with the supersymmetry transformations as in (10.75). • The exterior derivative ∂I ωJ+iK does not vanish in this case. 15 If we want to be precise, we must remind here that, for the HK and generic HKT manifolds, only the tangent space projections of the complex structures (I, J, K)AB have the form (3.2), while the tensors (I, J, K)M N look more complicated. However, for the Obata-flat HKT manifolds, which we only study in this chapter, also (I, J, K)M N are the simple constant matrices (3.2). Generic HKT models with nonvanishing Obata curvature will be studied in Chap. 12. We do not know now whether generic not Obataflat bi-HKT models exist.

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In view of Theorem 3.5, the latter assertion means that, though bi-HKT manifolds are hypercomplex, they are not HKT. Examples of hypercomplex, but not HKT manifolds were found earlier in Ref. [29], but they were essentially more complicated. In Ref. [73] the explicit expressions for the conserved supercharges in the bi-HKT models were given. They are not so simple, however, and we have chosen not to present them here. 10.4 10.4.1

N = 8 Models Hyper-K¨ ahler models

The most simple way to describe these models is to use the N = 2 superfield language. Consider the model (8.6) involving 4n such superfields X M and require the action to be invariant under three different supersymmetry transformations ¯ N) , δ1 X M = IN M (1 DX N − ¯1 DX ¯ N) , δ2 X M = JN M (2 DX N − ¯2 DX ¯ N) δ3 X M = KN M (3 DX N − ¯3 DX (10.100) with three different complex Grassmann parameters 1 , 2 , 3 . We know from Theorem 10.1 that the action is invariant under each of these transformations iff I, J, K represent integrable complex structures that are covariantly constant with the ordinary Levi-Civita connection, i.e. iff the manifold is K¨ ahler with respect to any one of these structures. The model involves eight supercharges: ˆ † = − √i eM ∂ DM , ˆ 0 = − √i ψ M DM , Q Q C 0 ∂ψC 2 2 i i † M ∂ ˆ I = − √ ψ N IN M DM , ˆ = − √ eN DM , Q Q C IN I ∂ψC 2 2 M ∂ ˆ J = − √i ψ N JN M DM , ˆ † = − √i eN Q Q DM , C JN J ∂ψC 2 2 M ∂ ˆ † = − √i eN ˆ K = − √i ψ N KN M DM , Q Q DM (10.101) C KN K ∂ψC 2 2 with DM defined in (8.27) and (10.13). Being translated into the language ˆ0, Q ˆ I,J,K acquire the form of differential forms, the supercharges Q ˆ 0 → ∂I + ∂¯I = ∂J + ∂¯J = ∂K + ∂¯K , Q ˆ I → ∂¯I − ∂I , Q

ˆ J → ∂¯J − ∂J , Q

ˆ K → ∂¯K − ∂K . Q

(10.102)

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where ∂I,J,K and ∂¯I,J,K are the exterior derivatives that are holomorphic and antiholomorhic with respect to the corresponding complex structures. It follows from the analysis of Sect. 11.1 that ˆ 0 } = {Q ˆ† , Q ˆ I } = {Q ˆ† , Q ˆ J } = {Q ˆ† , Q ˆK } = H ˆ ˆ† , Q {Q 0 I J K

(10.103)

and the anticommutators ˆ0, Q ˆ I,J,K }, {Q

ˆ† , Q ˆ† {Q 0 I,J,K },

ˆ† , Q ˆ I,J,K }, {Q 0

ˆ0, Q ˆ† {Q I,J,K } (10.104)

vanish. Now we will prove the theorem: Theorem 10.5. For the anticommutators involving different complex ˆI , Q ˆ J }, {Q ˆI , Q ˆ † }, etc., to vanish and the standard N = 8 structures, {Q J supersymmetry algebra with the only nontrivial anticommutators given in (10.103) to hold, it is necessary and sufficient that the triple (I, J, K) of the covariantly constant complex structures is quaternionic, i.e. the manifold is hyper-K¨ ahler. Proof. Sufficiency. Introduce the operators ∂ ∂ , FˆJ = JM N ψ M , FˆI = IM N ψ M ∂ψ N ∂ψ N ∂ FˆK = KM N ψ M . (10.105) ∂ψ N When acting on a wave function including p holomorphic and q antiholomorphic (with respect to I ) Grassmann factors, the operator FˆI multiplies it by i(q − p). Using the physical slang, FˆI may be called the operator of fermion charge. Lemma 1. The following identities hold: ˆ0] = Q ˆ I,J,K , [FˆI , Q ˆ I ] = [FˆJ , Q ˆ J ] = [FˆK , Q ˆ K ] = −Q ˆ0 , [FˆI,J,K , Q ˆ J ] = −[FˆJ , Q ˆI ] = Q ˆ K , [FˆJ , Q ˆ K ] = −[FˆK , Q ˆJ ] = Q ˆI , [FˆI , Q ˆ I ] = −[FˆI , Q ˆK ] = Q ˆJ [FˆK , Q (10.106) ˆ† and similarly for Q I,J,K . Proof. Consider the commutator   ˆ 0 ] = IM N ψ M ∂ , − √i ψ P DP . [FˆI , Q ∂ψ N 2

(10.107)

P ˆ I . It also includes It involves the term ∝ ∂ψ P /∂ψ N = δN , and this gives Q ˆ the term where the operator DP acts on FI . However, capitalizing on

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ˆ P by the full covariant derivative the scalar nature of FˆI , one can trade D A operator ∇P , which does not act on ψ , but acts on IM N and also on the vielbeins eM A as in Eq. (1.50): M M Q M ∇P e M A = ∂P eA + ΓP Q eA + ωAB, P eB .

(10.108)

B ˆ ˆ Bearing in mind that ∇P eM A = ∇P eN = 0, DP FI involves only the strucN ture ∇P IM . But our manifold is K¨ahler with respect to the complex ˆ P FˆI vanstructure I (as well as J and K), hence ∇P I = 0 and hence D ishes. The commutator   i P Q N M ∂ ˆ ˆ , − √ ψ JP DQ (10.109) [FI , QJ ] = IM ψ ∂ψ N 2

is treated in a similar way. The contribution involving ∂ψ P /∂ψ N gives16 ˆ K and the contribution ∝ DQ FˆI vanishes. All other commutators in Q (10.106) are derived analogously. ˆI , Q ˆ J } follows then from the vanishing of the The vanishing of, say, {Q anticommutators in (10.104) and from the Jacobi identity: ˆI , Q ˆ J } = {Q ˆ I , [FˆJ , Q ˆ 0 ]} = {Q ˆI , Q ˆ 0 }] − {Q ˆ 0 , [FˆJ , Q ˆ I ]} = 0 + {Q ˆ0, Q ˆ K } = 0 . (10.110) [FˆJ , {Q Necessity. Consider the Lie bracket (δ2 δ1 − δ1 δ2 )X M . The calculation gives [cf. Eq. (9.43)] (δ2 δ1 − δ1 δ2 )X M = 1 ¯2 [i(IJ + JI)P M X˙ P + independent structures] + 2 ¯1 [· · · ] + 1 2 [· · · ] + ¯1 ¯2 [· · · ]

(10.111)

Look at the term ∝ 1 ¯2 . If it vanishes, the structure ∝ X˙ must also vanish. This implies IJ + JI = 0. And a manifold admitting two anticommuting covariantly constant complex structures is hyper-K¨ ahler: the third structure chosen as K = IJ is also covariantly constant and the whole triple is quaternionic (see the Remark on p. 54). P

The assertion that the hyper-K¨ahler geometry implies the N = 8 supersymmetry of the action (8.6) was first made in [95]. The simple proof of the emergence of the N = 8 algebra for hyper-K¨ahler manifolds that we outlined above represents an adaptation to the N = 8 case of the similar proof given in [93] for the fact that the Obata-flat HKT geometry implies the N = 4 supersymmetry algebra for the supercharges (10.78). A similar 16 The

quaternionic nature of the triple (I, J, K) is used at this step.

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but not identical proof of the emergence of N = 8 supersymmetry, which used, instead of the fermion charge operators (10.105) and the commutators (10.106), the Lefschetz operators Λ, L and the Kodaira identities, was given in [28]. 10.4.2

N = 8 supersymmetric bi-K¨ ahler models

We have proven that the model (8.6) with a hyper-K¨ahler metric gM N enjoys N = 8 supersymmetry. But the condition for the manifold to be hyper-K¨ ahler is not necessary for the N = 8 supersymmetry of the action (8.6) to emerge. Certain special bi-K¨ahler models also have this property [72]. Consider a system (10.37) with an equal number of ordinary and mirror multiplets: Z 1 ¯ α ; Uα , U ¯ α) . ¯ 2 θ L(Zα , Z d2 θd (10.112) S = 4 Consider the following extra supersymmetry transformations mixing ¯ α ) and (Uα , U ¯ α ): (Zα , Z ¯ α + ηD ¯ α = −¯ ¯α , ¯ 1U ¯ 2 Uα ⇒ δ Z δZα = D D1 Uα − η¯D2 U 2 1 ¯ α + η¯D Zα ⇒ δ U ¯ α = ¯D Zα − η D ¯ α . (10.113) ¯ 1Z ¯ 2Z δUα = −D ¯ j δZα = 0 and the condition These transformations respect the condition D (10.36) for δUα . Now,  and η are two different complex Grassmann numbers giving four extra supersymmetries. The transformations (10.113) play exactly the same role as the transformation (9.37) for the N = 2 Dolbeaux model when it was expressed in terms of N = 1 superfields or the transformation (10.3) for the N = 4 model when it was expressed in terms of real N = 2 superfields. Theorem 10.6. The action (10.112) is invariant under the transformations (10.113) iff the conditions   ∂2 ∂2 (10.114) ¯ β + ∂Uα ∂ U ¯β L = 0 ∂Zα ∂ Z are fulfilled for all α, β. Proof. Necessity. Consider the terms ∝  in the variation δL,   ∂L ¯ ¯ ∂L ¯ ¯ D1 Uα − D1 Zα . (10.115) δ L =  ∂Zα ∂Uα

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For the action to stay intact, this variation should represent a total supersymmetric covariant derivative of some complex function R:   ¯ α + ∂R D ¯α . ¯ 1Z ¯ 1U ¯ 1 R =  ∂R D (10.116) δ  L = D ¯α ¯α ∂Z ∂U Comparing this with (10.115), we obtain the system of differential equations ¯ α ; Uα , U ¯ α ) representing a kind of generalized Cauchy-Riemann for R(Zα , Z conditions: ∂R ∂R ∂L ∂L , . (10.117) = = − ¯ ¯ ∂Zα ∂Uα ∂ Uα ∂ Zα ¯ β , expressing ∂R/∂ Z ¯ β from the Differentiating the first equation over Z second equation and using the reality of L, we arrive at (10.114) as the integrability conditions for the system (10.117). The same integrability conditions follow if one considers the terms ∝ ¯ or the terms ∝ η or ∝ η¯ in δL. Sufficiency. The condition (10.114) means that the form ∂L ¯ ∂L ¯ dUα − dZα (10.118) ∂Zα ∂Uα ¯ α, U ¯ α ) is trivial and hence, is closed. But the topology of the space (Z according to Theorem 1.2, this form is also exact. And this gives us (10.117) and (10.116). ω =

The pair (Zα , Uα ) represents a large (4, 8, 4) N = 8 supermultiplet. If the system has only one such multiplet, the condition (10.114) means that L is a 4D harmonic function. The simplest nontrivial such function is 1 L = ¯ ¯ . ZZ + UU

10.4.3

(10.119)

N = 8 bi-HKT models

Very similar theorems can be formulated and proven for the bi-HKT and bi-symplectic models. We dwell on the bi-HKT case [73,108,109]. Consider a bi-HKT model including an equal number of ordinary multiplets ZA α=1,...,n subject to the constraints (10.66) and the mirror multiplets UA α=1,...,n subject to the constraints (10.87). The action (10.89) acquires the form Z 1 ¯ A ; UA , U ¯ A) . ¯ 2 θdt L(ZA , Z d2 θd (10.120) S = α α α α 8

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Generically, it describes N = 4 dynamics on a certain 8n-dimensional biHKT manifold. But for some special Lagrangians L, the symmetry extends A to N = 8. In such a case, the set (ZA α , Uα ) constitutes for a given α one large N = 8 multiplet (8, 8, 0). The following theorem holds: Theorem 10.7. The action (10.120) is invariant under four extra supersymmetry transformations AB ¯ B AB B ¯A ¯ ¯ 2 (η3 UA δZA Uα ) + iD Uα ) , α = iD1 (η1 Uα + η2 ε α + η4 ε ¯ A = iD1 (η1 UA + η2 εAB UB ) + iD2 (η3 U ¯ A + η4 εAB U ¯B) , δZ α α α α α ¯ A + η2 εAB Z ¯ B ) − iD2 (η3 ZA + η4 εAB ZB ) , ¯ 1 (η1 Z δUA = −iD α

α

α

α

α

¯ A = −iD1 (η1 ZA + η2 εAB ZB ) − iD ¯ A + η4 εAB Z ¯ B ) (10.121) ¯ 2 (η3 Z δU α α α α α with real Grassmann η1,2,3,4 iff the generalized 8-dimensional harmonicity conditions ! ∂2 ∂2 (10.122) ¯ A + ∂UA ∂ U ¯A L = 0 ∂ZA α ∂ Zβ α β are fulfilled. Proof. At the first step, one should be convinced that the superfield variations (10.121) are consistent with all the constraints that we imposed on A ZA α and Uα . This can be checked directly. Consider the terms ∝ η1 in the variation of the Lagrangian: δ1 L =   ∂L 1 A ∂L ¯ ¯ A ∂L 1 A ∂L ¯ ¯ A −iη1 D U + D U − D Z − D Z (10.123) 1 α 1 α α α ¯A ¯A ∂ZA ∂UA ∂Z ∂U α α α α For the variation of the action to vanish, this variation should represent a total derivative ¯ 1 R + D1 Q) = δ1 L = −iη1 (D   ∂R ¯ ¯ A ∂Q 1 A ∂R ¯ ¯ A ∂Q 1 A −iη1 D Zα + D Uα . (10.124) ¯ A D1 Zα + ∂ U ¯ A D1 Uα + ∂ZA ∂UA ∂Z α α α α Comparing (10.123) and (10.124), we derive in particular that the function R should satisfy the equations ∂L ∂R ∂L ∂R = , = − A. (10.125) A A A ¯ ¯ ∂Z ∂U ∂ Uα ∂ Zα α α By the same reasoning as in the bi-K¨ahler case, we arrive at (10.122) as the necessary and sufficient conditions for the integrability of the system (10.125). A similar equation system for Q and the equation systems that one derives by considering the terms in δL proportional to η2 , η3 and η4 lead to exactly the same conditions.

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A If we have only one pair of multiplets (ZA α , Uα ), the conditions (10.122) D=8 are reduced to the harmonicity condition 4 L = 0. The simplest nontrivial solution to this condition is

L =

10.4.4

1 A A ¯ ¯ A UA )3 . (Z Z + U

(10.126)

Symplectic model of the second kind

The whole zoo of N = 8 models was systematically studied in [110]. And not all the animals there belong to the “bi” genus. As the last example, consider the model involving two different N = 4 superfields: the (3, 4, 1) superfield (7.66) and the chiral (2, 4, 2) superfield (7.48). Consider the action Z ¯ Z) ¯ 2 θdt L(ΦM , Z, S = d2 θd (10.127) with a real prepotential L. We require this action to be invariant under the following extra N = 4 supersymmetry transformations: 2i j ¯ k ΦM , ¯ (σM )j k D 3 ¯ = 2i (σM )j k k Dj ΦM , δZ 3 ¯ − i(σM )j k k Dj Z . ¯ kZ δΦM = −i¯ j (σM )j k D δZ =

(10.128)

One can show that, up to coefficients, these are the only transformations ¯ and respect the constraints D ¯ j Z = 0 and the that mix ΦM ↔ (Z, Z) constraints ΦM = ΦM and (7.63) for ΦM . If the invariance under (10.128) holds, we are dealing not just with two separate N = 4 supermultiplets, but with a single N = 8 multiplet of type (5, 8, 3). Theorem 10.8. [100,111] The action (10.127) is invariant under (10.128) iff the condition ∂2L ∂2L + 2 ¯ = 0 ∂ΦM ∂ΦM ∂ Z∂Z is fulfilled.

(10.129)

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Proof. The variation of the Lagrangian under (10.128) reads   ∂L j 2 ∂L j D Φ − δL = ik (σM )j k D Z + c.c. M ¯ 3 ∂Z ∂ΦM

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(10.130)

We now proceed in the full analogy with what we did before. The variation of the action vanishes iff the variation (10.130) represents a total derivative,   ∂L j k j k j k 2 ∂L (σM )j ¯ D ΦM − ∂ΦM D Z = D R + (σM )j D LM , (10.131) 3 ∂Z ¯ Bearing in where R and LM are some complex functions of ΦM , Z and Z. j j j k ¯ ¯ j Z = D Z = 0 and that D ΦM = (σM )k Ξ ¯ with some Ξ ¯k mind that D [see Eq. (7.64)], we can write ∂R k ∂R ¯p , D Z+ (σM )p k Ξ ∂Z ∂ΦM ∂LM ∂LM j ¯p . D Z+ (σN )p j Ξ Dj LM = (10.132) ∂Z ∂ΦN Then, comparing the coefficients of the independent structures in both sides of (10.131), we end up with the following set of linear partial differential equations: Dk R =

∂L ∂LM = 0, 2 ¯ − ∂ Z ∂ΦM ∂LM ∂L = 0, + ∂ΦM ∂Z ∂LM ∂R iεM N P − = 0, ∂ΦN ∂ΦP ∂R = 0. (10.133) ∂Z Differentiating the first equation over Z, the second over ΦM and adding the results, we arrive at (10.129) as an integrability condition for the system (10.133). This proves the necessity of (10.129). Two other equations impose some constraints on the function R and LM , which present for us no particular interest. Now, let L satisfy the constraint (10.129). This is nothing but a 5dimensional harmonicity condition

with

∂2L = 0, ˜ M ∂Φ ˜M ∂Φ

(10.134)

  √ √ ˜ M = ΦM , 2 Re(Z), 2 Im(Z) . Φ

(10.135)

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˜ M plus a sum of A generic real solution to (10.134) is a linear function of Φ singular “Coulomb” functions, like 1 , ˜ ˜ (ΦM ΦM )3/2

(10.136)

and their derivatives. It is then not difficult to write for any such L an expression for the harmonic function Q satisfying the additional requirement −∂Q/∂Z = L. Then one of the possible solutions for the system (10.133) that guarantees the invariance of the action under (10.128) reads R = 0,

LM =

∂Q . ∂ΦM

(10.137)

This proves the sufficiency of (10.129). The system (10.127) with harmonic L has 4 complex supercharges. In a complete analogy with the ordinary symplectic (3, 4, 1) system where the supercharge lies in the fundamental representation of Spin(3) ≡ Sp(1), in this case, the supercharge lies in the fundamental representation of Spin(5) ≡ Sp(2). One may thus call the model (10.127) a symplectic model of the second kind. We have presented here its description in terms of N = 4 superfields. But one can also describe the whole (5, 8, 3) multiplet as a single N = 8 superfield living in harmonic superspace [100]. As was the case with all other models considered in this section, multidimensional versions of the symplectic models of the second kind exist. Take some number n of (5, 8, 3) multiplets and write the action Z ¯ 2 θdt L(Φ ˜α S = d2 θd (10.138) M) , α = 1, . . . , n. Proceeding in the same way as before, one can prove the theorem [100, 111]: Theorem 10.9. The action (10.138) is invariant under the transformations (10.128) for each supermultiplet (10.135) iff the conditions ∂2L ˜ α ∂Φ ˜β ∂Φ M

= 0

M

are fulfilled for all α, β, M, N .

and

∂2L = 0 ˜ [α ∂ Φ ˜ β] ∂Φ M

N

(10.139)

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Chapter 11

Taming the Zoo of Models

A reader who has just finished reading the previous chapter could be somehow confused. A large hip of different models were described, and it might not be immediately clear whether there is a relationship between them, whether this hip has a structure. In this chapter we answer positively to this rhetoric question. 11.1

Similarity Transformations

Suppose we have a N = 2 SQM model involving the algebra (5.1). Consider the operator ˆ 0 = eRˆ Qe ˆ −Rˆ Q (11.1) ˆ As the supercharge Q ˆ is nilpotent, the same obviously with an arbitrary R. ˆ 0 . The Hermitially conjugated supercharge concerns the new supercharge Q ˆ ¯0 ≡ Q ˆ 0† is Q ˆ¯ 0 = e−Rˆ¯ Qe ˆ¯ Rˆ¯ . Q

(11.2)

The new Hamiltonian is ˆ = {Q ˆ0, Q ¯ˆ 0 } . H (11.3) ˆ ˆ ˆ ¯ If R is anti-Hermitian, Q and Q are rotated with the same unitary matrix and the same concerns the Hamiltonian. This is just a unitary transformaˆ¯ 6= −R ˆ , the tion, which boils down to a redefinition of the basis. But if R transformations (11.1) and (11.2) produce a different Hamiltonian (11.3) with a different spectrum. ˆ in (8.26) of a generic de Rham Theorem 11.1. [97] The supercharge Q supersymmetric sigma model is obtained from the flat supercharge ˆ flat = √1 ψA PˆA , Q (11.4) 2 229

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PˆA = −i∂/∂xA , by a similarity transformation (11.1). Proof. Choose ˆ = αAB ψA ψˆ¯B R

(11.5)

¯B = ∂/∂ψB and a real symmetric function αAB (x) of all the cowith ψˆ ordinates (an antisymmetric αAB would lead to an uninteresting unitary ˆ = eRˆ Q ˆ flat e−Rˆ . We use the Hadamard formula transformation). Evaluate Q ˆ ˆ −R ˆ ˆ + [R, ˆ X] ˆ + 1 [R, ˆ [R, ˆ X]] ˆ + ... eR Xe = X 2 In our case, this implies ˆ

(11.6)

ˆ

eR ψA e−R = ψB (eα )BA , ˆ ˆ eR ∂A e−R = ∂A + (eα ∂A e−α )CD ψC ψˆ¯D

(11.7)

and h i ˆ = √1 ψB (eα )BA PˆA − i(eα ∂A e−α )CD ψC ψˆ¯D . Q 2

(11.8)

Introducing the vielbeins as α eM A = (e )AM ,

eM A = (e−α )M A ,

(11.9)

which give the metric gM N = (e−2α )M N ,

g M N = (e2α )M N ,

(11.10)

one can show by a direct calculation that the result (11.8) can be represented as ˆ = √1 ψ M (PˆM − iωCD,M ψC ψˆ¯D ) , (11.11) Q 2 where ωCD,M is the standard Levi-Civita connection N P ωCD,M = eCN (∂M eN D + Γ M P eD ) .

(11.12)

The expression (11.11) for the supercharge coincides with that in (8.26). As we know from Theorem 8.1, the supercharge (11.11) maps to the exterior derivative operator d of the de Rham complex. In other words, the SQM sigma model describing the de Rham complex is “genetically” related to the free model living on RD and involving D real ˆ a simple function −W (x), we may superfields (1, 2, 1). By adding to R

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ˆ W in Eq. (8.42). An additional also obtain the deformed supercharge Q similarity transformation, ˆ → eBM N ψM ψN Q ˆ e−BM N ψM ψN Q (11.13) generates the model with quasitorsions discussed at the end of Chap. 8. Another family of SQM sigma models involves the chiral multiplets (2, 2, 0). These models discussed in Chap. 9 can also be obtained by a similarity transformation from flat models. Consider a flat model living on Cd with the supercharges1 ˆ¯ flat = ψˆ¯a¯ pˆ ˆ flat = ψ a pˆa , Q Q (11.14) a ¯ (ˆ pa = −i∂/∂z a , pˆa¯ = −i∂/∂ z¯a , ψˆ¯a¯ = ∂/∂ψ a ). Consider a similarity transformation with ˆ = βa¯b ψ a ψˆ¯¯b R (11.15) such that the matrix βa¯b (z, z¯) is Hermitian (an anti-Hermitian βa¯b amounts to a unitary transformation). Theorem 11.2. The similarity transformation (11.1) of the supercharge ˆ flat in (11.14) with R ˆ written in Eq. (11.15) gives the supercharge that Q maps to the exterior holomorphic derivative ∂ of the Dolbeault complex with the complex vielbeins β m em a = (e )a ,

eam = (e−β )m a

(11.16)

Proof. The line of reasoning is the same as for the preceding theorem. Using the Hadamard formula, we derive for the transformed supercharge ¯ ˆ = ψ b (eβ )b a [ˆ Q pa − i(eβ ∂a e−β )c d ψ c ψˆ¯d ] . (11.17) One can verify by a direct calculation that the supercharge (11.17) can be represented in the form ¯ ˆ = ψ m (ˆ Q pm − iωa¯b,m ψ a ψˆ¯b ) , (11.18) where ωa¯b,m = −ω¯ba,m = −ebp (∂m epa + Γpmk eka )

(11.19)

are the relevant components of the Levi-Civita spin connection. The result (11.18) coincides with the sum of the contributions in (9.11) and in (9.13) with W = (1/4) ln det h. Theorem 9.1 tells us that this supercharge maps to the operator ∂ of the untwisted Dolbeault complex. 1 We

are writing the indices at the two levels now—see the footnote on p. 33.

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One can also make an additional transformation (additional deformation) (9.14). This brings about an extra gauge field [or it may cancel the field, derived from W = (1/4) ln det h, that is incorporated in (11.18)]. As was mentioned in Chap. 9 and will be discussed in more details in Chap. 13, the function W (z, z¯) is not arbitrary here, for the topological charge associated with this gauge field should better be integer. And the deformation ˆ → eBmn ψm ψn Q ˆ e−Bmn ψm ψn Q (11.20) amounts to the shift (9.30) and generates the Dolbeault complex with holomorphic torsions. A similarity transformation makes a N = 2 supersymmetric model out of a N = 2 supersymmetric model. But flat models often enjoy extended supersymmetries. For example, any model with the supercharge (11.4), involving an even number of variables (xA , ψA ), also admits another complex supercharge 1 (11.21) Sˆflat = √ ψA IAB PˆB , 2 where IAB is a complex structure: I = −I T , I 2 = −1. In fact, many such supercharges with different complex structures can be written, but one cannot take all of them, they would not mutually anticommute. But a single ˆ¯ flat , ˆ flat and Q extra supercharge in the form (11.21), anticommuting with Q 2d can always be chosen on R . This gives N = 4 supersymmetry. By rotations, one can always bring I in the canonical form I = diag(−ε, . . . , −ε). ˆ flat and We now apply a similarity transformation for the supercharges Q flat ˆ ˆ S with the same Hermitian operator R in (11.5). By construction, the transformed complex supercharges ˆ ˆ ˆ = eRˆ Q ˆ flat e−Rˆ , Q Sˆ = eR Sˆflat e−R (11.22) given by the expressions in (10.12), are nilpotent and anticommute. But ˆ¯ S} ˆ ˆ do not necessarily coincide and the ¯ Q} ˆ and {S, the anticommutators {Q, ˆ ˆ¯ Q} ¯ S} ˆ and {S, ˆ do not necessarily vanish. anticommutators {Q, However, there are some special deformations when the latter property holds. If the complex structures IM N = eM A eN B IAB thus obtained turn out to be covariantly constant with respect to the Levi-Civita connections, we are led to the K¨ ahler geometry, and in this case the supercharges (10.12) satisfy the N = 4 superalgebra.2 2 This is clear from the considerations in Chap. 10.1, especially from Theorem 10.2 demonstrating the isomorphism between the supercharges (10.12) and the operators of the K¨ ahler–de Rham complex, but a dedicated reader can also consult the Appendix in Ref. [97], where it was proven by a direct calculation of the anticommutators.

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If the number of variables D is an integer multiple of 4, one can write not one, but three additional complex supercharges 1 SˆIflat = √ ψA IAB PˆB , 2

1 SˆJflat = √ ψA JAB PˆB , 2 1 flat SˆK = √ ψA KAB PˆB , 2

(11.23)

where three quaternionic complex structures I, J, K can be brought to a canonical form (3.2). The supercharges (11.4), (11.23) and their conjugates obey the N = 8 algebra. An arbitrary similarity transformation with the operator (11.5) keeps N = 2, but not N = 8 supersymmetry. However, if the metric (11.10) thus obtained turns out to be hyper-K¨ahler and all three complex structures are covariantly constant, the anticommutators ˆ ¯ SˆI,J,K }, {Q, ˆ Sˆ ¯I,J,K } and {Sˆ¯I , SˆJ6=I } all vanish, the anticommutators {Q, ˆ ¯ Q}, ˆ {Sˆ ¯I , SˆI } all coincide, and the deformed model is N = 8 supersym{Q, metric. As the last example, consider the SQM model on Cd=2k involving k (4, 4, 0) supermultiplets. This model involves two pairs of complex supercharges satisfying the N = 4 algebra: ˆ flat = ψ a pˆa , Q ¯ b

Sˆflat = ψ a (I + )a pˆ¯b , ¯

ˆ¯ flat = ψˆ¯a¯ pˆ , Q a ¯ flat a ¯ ˆ ˆ ¯ ¯ S = ψ (I − )a¯ b pˆb ,

(11.24)

¯

where (I + )a b = (J + iK)a b /2 and (I − )a¯ b = (J − iK)a¯ b /2 are the flat hypercomplex structures, which can be both chosen as diag(−ε, . . . , −ε). ˆ flat and Perform now a simultaneous similarity transformation (11.1) of Q ¯¯b . For a generic Hermitian β, N = 4 supersymmetry ˆ = βa¯b ψ a ψˆ Sˆflat with R of the flat model will be lost. But for some special β giving a hyper-K¨ahler or an HKT metric hm¯n = (e−2β )m¯n , it is left intact. The deformed supercharges will represent in this case the covariant quantum counterparts of (10.80), with the extra one-fermion contributions involving the derivatives of ln det(e) being removed. This truncation corresponds to endowing the system with a certain self-dual gauge field (see the end of Chap. 12 and Sec. 13.4.2). One can get rid of this extra gauge field and arrive at the pure HKT model by performing an additional similarity transformation, ˆ → eW Qe ˆ −W , Q with W = −(1/4) ln det h.

ˆ −W , Sˆ → eW Se

(11.25)

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Hamiltonian Reduction

The procedure of Hamiltonian reduction is well-known to physicists. It is indispensible for accurate formulation of gauge theories and was worked out by Dirac 90 years ago [112]. For the benefit of our reader-mathematician, we have chosen to describe it here in some details. 11.2.1

A toy model

Consider the quantum Hamiltonian 2 ˆ 0 = 1 (Pˆx2 + Pˆy2 ) + ω (x2 + y 2 ) , (11.26) H 2 2 which describes the 2-dimensional oscillator. This Hamiltonian admits an ˆ = 0. It has the meaning of integral of motion Pˆφ = xPˆy − y Pˆx : [Pˆφ , H] the angular momentum [cf. Eq. (4.24)]. The spectrum of (11.26) may be expressed as

Enr ,m = ω(1 + 2nr + |m|) ,

(11.27)

where an integer m is the eigenvalue of Pˆφ and an integer non-negative nr is the radial quantum number. Now suppose that we are not interested in the full spectrum of (11.26), but only in its part including the states Ψ having zero angular momentum: Pˆφ Ψ = 0. The spectrum E = ω(1 + 2nr )

(11.28)

coincides up to an irrelevant shift with the spectrum of the one-dimensional oscillator describing radial motion on the half-line 0 ≤ r < ∞. The corresponding eigenfunctions do not depend on the angular variable φ. Thus, the number of dynamical degrees of freedom is effectively reduced from two to one. The word “dynamical” is important here. In fact, the angular variable φ has not disappeared completely. It has become a gauge degree of freedom. To understand the latter statement and the meaning of the word “gauge”, consider the classical Hamiltonian describing our reduced system. It reads H = H0 + λPφ .

(11.29)

∂H = Pφ = 0 ∂λ

(11.30)

The equation

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is our constraint. The other Hamilton equations of motion (4.3) are P˙i + ω 2 xi + λεij Pj = 0 , x˙ i − Pi + λεij xj = 0 .

(11.31)

Performing the standard Legendre transformation (4.8), we derive the canonical Lagrangian L =

ω2 2 1 (x˙ i + λεij xj )2 − x . 2 2 i

(11.32)

The equations of motion following from (11.32) are equivalent to (11.31). Theorem 11.3. The Lagrangian (11.32) is invariant under the gauge transformation xi → Oij xj λ → λ + χ˙

(11.33)

with  O =

cos χ − sin χ sin χ cos χ

 .

(11.34)

Proof. The simplest way to see it is to consider an infinitesimal transformation δxi = −χεij xj ,

δλ = χ˙

(11.35)

and show that the variation δL vanishes. The variable λ(t) may be interpreted as a time-dependent angular velocity with which the reference system rotates. Note the difference with rigid symmetries like in (4.19) or in (5.49), where the parameters of the transformations were ordinary or Grassmann constants. The same concerns the rigid rotational symmetry of the original system (11.26). But in (11.33) the parameter of the transformation χ(t) is time-dependent.3 Noether’s theorem does not apply in this case. In fact, gauge symmetry is not quite a symmetry, but rather a manifestation of the fact that the chosen description of the system involves redundant unphysical variables (in our case, λ and φ). In our trivial case, it is simple to get rid of λ and φ altogether (in the Hamiltonian language, the constraint Pˆφ = 0 3 And that is what the word “gauge” means—it refers to the situation when the action is invariant under the transformations whose parameters depend on time. In field theories, also on spatial coordinates.

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allows one to fix the gauge: to set φ = 0) and describe the radial motion of the system by the Hamiltonian 2 2 ˆ2 ˆ reduced = Pr + ω r (11.36) H 2 2 with the boundary condition Ψ(r = 0) = 0 to be imposed on the wave function. But in many cases (especially, in field theories), getting rid of the redundant gauge variables is not practically feasible, and one has to work with the Lagrangians of the type (11.32) where the redundant variables are present.

We will not need it in the following, but having started this discussion, I cannot resist giving an example of such a nontrivial system. Consider the a=1,2,3 following quantum Hamiltonian involving nine dynamical variables Aj=1,2,3 ˆ a = −i∂/∂Aa : and the corresponding momenta E j j h i 1 ˆ a )2 + (B a )2 , ˆ = 1 (E (11.37) H Bja = εabc εjkl Abk Acl . j j 2 2 It is actually the Hamiltonian of Yang-Mills theory with the SU (2) gauge group in the quantum mechanical limit (the vector potentials Aaj depend only on time, but not on spatial coordinates). Impose the constraints ˆ a Ψ = εabc Abj E ˆjc Ψ = 0 . G

(11.38)

One can do so because: ˆ a commute with the Hamiltonian; • The operators G • The algebra of the constraints is closed: ˆa, G ˆ b ] = iεabc G ˆc , [G

(11.39)

ˆ a Ψ = 0 is self-consistent. so that the requirement G The constraints (11.38) kill 3 out of 9 dynamical variables. For the quantum mechanical system (11.37), one can in principle resolve the constraints, i.e. get rid of three gauge degrees of freedom and find a Hamiltonian that depends only on gauge-invariant variables [113] (for the “parent” field theory, it is not technically possible), but this Hamiltonian is complicated and not so convenient. It is better to introduce three Lagrange multipliers λa ≡ Aa0 and write an analog of (11.32)—the Lagrangian that depends on Aaµ = (Aa0 , Aaj ). It has a nice form 1 a µν a L = − Fµν F , 4

(11.40)

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a where the nonzero components of Fµν are

a a Fjk = −Fkj = εabc Abj Ack ,

a a F0j = −Fj0 = A˙ aj + εabc Ab0 Acj

(11.41)

and the indices are lifted and lowered with the Minkowski metric ηµν = (1, −1, −1, −1) (a birthmark from the field-theory origin of this model). This Lagrangian is invariant under certain gauge transformations. Infinitesimally, under the variations δAa0 = χ˙ a + εabc Ab0 χc ,

11.2.2

δAaj = εabc Abj χc .

(11.42)

(2, 2, 0) −→ (1, 2, 1)

We are going to show now that the two types of SQM models discussed above—the models involving the chiral supermultiplets (2, 2, 0) and the models involving the real supermultiplets (1, 2, 1) are genetically related: the latter are obtained from the former by Hamiltonian reduction. The main idea is very simple. Take the chiral supersymmetric sigma model (9.1) with W = 0 and assume that the metric hm¯n does not depend on the imaginary parts of the complex coordinates z m . If nothing depends on Im(z m ), we can get rid of it and obtain a model defined on the real manifold with the coordinates Re(z m ). What is this model? One way to derive it is to go down to components, to the Hamiltonian (9.8) (or rather to its quantum counterpart), to observe that, under the assumption that hm¯n does not depend on Im(z m ), the Hamiltonian commutes with Im(ˆ pm ), and to impose the constraint Im(ˆ pm )Ψ = 0. Fixing the gauge Im(z m ) = 0, one can thus derive the Hamiltonian involving only Re(z m ) and the corresponding canonical momenta. Observing further that Im(ˆ pm ) commutes not only with the Hamiltonian, but also with the supercharges, we derive that the reduced model enjoys the same N = 2 supersymmetry as the original one. However, this procedure does not allow us to trace back the superspace and superfield structure of the model thus derived. To do so, one should perform the Hamiltonian reduction in the Lagrangian superfield formalism. The appropriate technique was developed in [114]. Consider the supersymmetry transformations (7.31) of the components of a chiral superfield. We distinguish the real and imaginary parts of

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√ z = (x + iy)/ 2 to obtain δx = ψ − ¯ψ¯ , ¯ , δy = −i(ψ + ¯ψ) δψ = ¯(y˙ − ix) ˙ , ¯ δ ψ = (y˙ + ix) ˙ .

(11.43)

One can now observe that the transformations (11.43) coincide with the transformations (7.27) for the real superfield if one identifies y˙ ≡ F . Consider now the Lagrangian Z 1 ¯ hm¯n (Z, Z) ¯ D ¯ Z¯ n¯ DZ m L = dθdθ (11.44) 4 and suppose that the metric does not depend on y m . Also the component Lagrangian (9.2) does not depend in this case on y m , but only on y˙ m . Suppose first that the metric hm¯n is real and symmetric. One can be convinced that, by setting hm¯n → gM N and trading y˙ m → F M , we reproduce the component Lagrangian (8.7) of the de Rham supersymmetric sigma model. Heuristically, the superfield expression of this Lagrangian, Z 1 ¯ gM N (X) DX ¯ M DX N , dθdθ (11.45) L = 2 √ ¯ is obtained from (11.44) by setting Z m , Z¯ m → 2 X M , h → g. However, the Hermitian metric hm¯n need not be necessarily real. It might contain an imaginary antisymmetric part. The Hamiltonian reduction of a generic model (11.44) gives the quasicomplex N = 2 model mentioned in the previous chapter. The superfield form of its Lagrangian is4 Z 1 ¯ [g(M N ) (X) + ib[M N ] (X)] DX ¯ M DX N , dθdθ (11.46) L = 2 with real symmetric g(M N ) and real skew-symmetric b[M N ] . The bosonic part of the component Lagrangian (the full expression is written in [85]) is  1  L = g(M N ) + b[M P ] g (P Q) b[QN ] x˙ M x˙ N . (11.47) 2 In other words, the true metric of the manifold is not g(M N ) , but depends also on b[M N ] , being given by the combination (10.35). The conserved supercharges of this model read [cf. (8.13)]   1 M i N ¯P Q= √ ψ ΠM − ∂M (g(N P ) + ib[N P ] ) ψ ψ , 2 2   i 1 ¯M N ¯P ¯ ΠM + ∂M (g(N P ) − ib[N P ] ) ψ ψ Q= √ ψ . (11.48) 2 2 R ¯ (BM N DX M DX N one adds to this Lagrangian also the quasitorsion term ∼ dθdθ + c.c.), one arrives at the model mentioned in passing in Ref. [26]—see Eq. (3.10) there. 4 If

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In the discussion above we assumed that the metric did not depend on Im(z m ). But reduction can be performed not only in this restricted case, but for any metric with isometries. To illustrate this, consider the model involving a single chiral superfield Z with the free action, Z 1 ¯ ¯ ZDZ ¯ dθdθdt D . (11.49) S = 4 One of the isometries of the flat metric are rotations z → zeiφ . In the same way as for the toy model of the preceding section, we can impose the constraint pˆφ Ψ = 0 on the wave functions.5 The action of the reduced model is Z 1 ¯ ¯ DR , S = dθdθdt DR (11.50) 2 where √ R is the real (1, 2, 1) superfield whose lowest component is r = |z| 2. The motion goes in the case along the ray [0, ∞) with the boundary condition Ψ(0) = 0. 11.2.3

(4, 4, 0) −→ (2, 4, 2)

Consider an Obata-flat HKT model given by the Lagrangian (10.68). Suppose that its metric (10.69) depends only on zkA=1 , but not on zkA=2 . Then = the quantum Hamiltonian and the supercharges commute with pˆA=2 k A=2 and the Hamiltonian reduction preserving the N = 4 super−i∂/∂zk symmetry of the original model is possible. From the N = 2 viewpoint, a (4, 4, 0) multiplet represents a pair of (2, 2, 0) multiplets. As we have learned above, the supersymmetry transformations in the multiplet (2, 2, 0) are √ identical to those in the ˙ ≡ F ; the reduced multiplet (1, 2, 1) under the identification 2 Im (z) model involves only the real parts of z, while their imaginary parts show up as auxiliary variables. The same holds for the large multiplets (4, 4, 0). A direct calculation shows that one can choose there any two real coordinates, identify their time derivatives with two real auxiliary variables of a (2, 4, 2) multiplet and observe that, under this identification, the supersymmetry transformations in the two multiplets coincide. The reduced model must therefore be expressed in terms of (2, 4, 2) multiplets and represent a K¨ahler model. Generically—a quasicomplex K¨ ahler model whose action (10.30) is given by the sum of standard kinetic 5 For sure, one can also introduce the variable w = ln z. In this case, the angular variable φ is presented as Im(w) and we come back to the situation analyzed above. But it is not always easy to choose the variables in such a way, and it is actually not necessary.

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term (10.20) and the extra term (10.29). The only question is how the K¨ ahler potential K and the functions Am determining the action of the reduced model depend on the HKT prepotential L of the original model. ¯ 2 , with wk , ξ k being their bosonic Let us pose Wk = Z1k and Ξk = Z k components. In these terms, the bosonic part (10.70) of the component HKT Lagrangian can be represented as ¯ ¯ ¯ ¯¯ w ˙ k¯ ¯˙¯l LHKT ¯˙ l + ξ˙k ξ¯˙l ) + Fkl w˙ k ξ˙l + F = κk¯l (w˙ k w kl ¯ ξ , bos

(11.51)

where κk¯l =

1 2



∂2 ∂2 + k ¯l ¯ k l ∂ξ ∂ ξ ∂w ∂ w ¯

 L

(11.52)

and   1 ∂2 ∂2 − L, 2 ∂wk ∂ξ l ∂ξ k ∂wl   ∂2 1 ∂2 − = L. 2 ∂w ¯ k¯ ∂ ξ¯¯l ∂ ξ¯k¯ ∂ w ¯ ¯l

Fkl = ¯ ¯¯ = Fkl F kl

(11.53)

For the model to be reducible, κk¯l and Fkl should not depend on ξ p and ξ¯p¯. What constraints does it impose on the prepotential? Theorem 11.4. In one of the charts with trivial topology, into which our complex manifolds is subdivided, the prepotential L, satisfying the requirements ∂ ∂ ∂ ∂ κk¯l = ¯p κk¯l = Fkl = ¯p Fkl = 0 , p p ∂ξ ∂ξ ∂ξ ∂ξ

(11.54)

can be chosen in the form L ¯ Ξ ¯ + Ak (W) Ξk + A¯¯ (W) ¯ k¯ , = K(W, W) k 2

(11.55)

where K and Ak are such that κk¯l = ∂k ∂¯l K,

Fkl = ∂k Al − ∂l Ak

(11.56)

(∂k ≡ ∂/∂wk ). Proof. We prove first the following Lemma: ¯ ¯ ¯ k ¯ ¯ ¯ dw Lemma. The forms κk¯l dwk ∧ dw ¯ l , Fkl dwk ∧ dwl and F ¯ l are kl ¯ ∧ dw closed under the condition (11.54).

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¯ ¯ k ¯ ¯ ¯ dw Proof. The closeness of the forms Fkl dwk ∧ dwl and F ¯ l is an kl ¯ ∧ dw immediate corollary of the definitions (11.53). Consider now κk¯l . The first term in (11.52) gives a closed form and the only problem is with the ¯ second term. Expand L in ξ p and ξ¯p¯. A“dangerous” (ξ, ξ)-independent contribution to κk¯l may come from the structure X = Rm¯n (w, w) ¯ ξ m ξ¯n¯ (11.57)

in the prepotential. But the same structure gives the contribution ∝ (∂k Rl¯n − ∂l Rk¯n )ξ¯n¯ in Fkl . Bearing in mind (11.54), this contribution must vanish, and we derive ∂k Rl¯n − ∂l Rk¯n = 0. Considering the contribution ¯ ¯¯, we derive ∂¯ R ¯ − ∂¯R ¯ = 0. It follows that the of (11.57) into F kl k ml l mk ¯ ¯ contribution (1/2)Rk¯l dwk ∧ dw ¯ l in the form κk¯l dwk ∧ dw ¯ l is closed. If the forms are closed and the topology of a chart is trivial, they are also exact (see Theorem 1.2). And that means that κk¯l and Fkl may be presented in the form (11.56). Remark. One can choose L in the form (11.55), but L does not necessarily have this form. One can always add to L the structures like ¯ − (∂k ∂¯ Q) Ξk Ξ ¯ l , (11.58) ¯ k or ∆L = Q(W, W) ∆L = Bk (W) Ξ l which do not affect κk¯l and Fkl . And it is the latter objects that determine geometry and that we are finally interested in. The shifts (11.58) are analogous to gauge transformations of electromagnetic vector potential which do not affect the observable field density. We are ready now to prove: Theorem 11.5. A reducible HKT SQM model with the prepotential (11.55) gives after reduction the quasicomplex K¨ ahler model with the action Z  Z 1 ¯ +1 ˙ k + c.c. . (11.59) ¯ 2 θdt K(W, W) S = d2 θd d2 θdtL Ak (W) W 4 2 Proof. One of the ways to prove it is to compare the bosonic terms in the original HKT model and in the model (11.59) [viz. (11.51) and (10.31)] and observe that they coincide under the identification ξ˙ ≡ F . All other terms in both models are reproduced by N = 4 supersymmetry. They should also coincide, and one can check that they do. One can also formulate the following theorem: Theorem 11.6. In a topologically trivial chart, the prepotential ¯A A ¯ A L(ZA k , Zk ; Uα , Uα ) of a generic reducible bi-HKT model (10.89) can be chosen, modulo gauge transformations that do not affect geometrical properties,

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in the form h i 1 ¯ V, V) ¯ + A(1) (W, V) + A(2) (W, V) ¯ Ξk + L = K(W, W; k k 2 h i h i ¯ (1) (2) ¯ V) ¯ + A¯¯ (W, ¯ V) Ξ ¯ Σα ¯ k + Bα(1) (V, W) + Bα(2) (V, W) A¯k¯ (W, k h i (1) ¯ ¯ (2) ¯ ¯ α¯ , (11.60) + B¯α¯ (V, W) + B¯α¯ (V, W) Σ ¯ A=2 ; Vα = UA=1 , Σα = U ¯ A=2 . The reduced where Wk = ZA=1 , Ξk = Z α α k k model is the generic quasicomplex bi-K¨ ahler model with the action Z 1 ¯ V, V) ¯ − ¯ 2 θdt K(W, W; d2 θd S= 4 Z o n i ¯ 1 Wk ] + ¯ 2 [A(1) (W, V)D2 Wk ] + D ¯ 1 [A(2) (W, V)D dθ1 dθ2 dtL D k k 4  Z n o ¯ 1 Vα ] − c.c. ¯ 2 Vα ] + D ¯ 1 [B(2) (V, W)D dθ1 dθ¯2 dt˜L D2 [B(1) (V, W) D α α (11.61) with tL and t˜L defined in (10.44). Proof. The second part of this theorem [that the Hamiltonian reduction of (11.60) gives (11.61)] was proved in Ref. [73] in exactly the same way as Theorem 11.5: by comparison of the bosonic Lagrangians in the two models, bearing in mind the identification of the time derivatives of the reduced coordinates in the bi-HKT Lagrangian with the auxiliary variables in the bi-K¨ ahler Lagrangian. We address the reader to that paper for further details. The first statement [that the choice (11.60) is always possible] can be justified by noting that (11.61) is the most general form of the quasicomplex bi-K¨ ahler action. If the original bi-HKT action involves the terms like ¯ k and is reducible (its metric does not depend on ξ and σ) it Ak (W, V, V)Ξ should still give (11.61) after reduction. And that means that the bi-HKT action can be brought to the form (11.60) by a gauge transformation. A dedicated reader is welcome to construct a more rigourous proof of this fact in the spirit of Theorem 11.4. 11.2.4

Symplectic and other models

Up to now we discussed models where the metric did not depend on a half of real coordinates. But one can also consider the situation when the

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metric involves an arbitrary number of isometries. Performing a reduction, we may obtain a vast number of different supersymmetric sigma models. We can start e.g. with an HKT model involving some number of root (4, 4, 0) multiplets and consider the case when the metric does not depend on a quarter of coordinates—of one real coordinate in each superfield. Quite naturally, we obtain in this way a symplectic model involving the multiplets (3, 4, 1). Or one can consider a model where the reduction is performed with respect to three quarters of the coordinates and obtain a model involving the multiplets (1, 4, 3), where each dynamical coordinate has two different complex superpartners. A certain class of such models was studied in recent [115]. One can also start with the model involving N = 8 (8, 8, 0) root multiplets and perform different reductions. The symplectic model of the second kind can be obtained in this way. And many other models. Not always a Hamiltonian reduction relates the manifolds of different kind. In Chap. 3, we briefly mentioned the multidimensional generalizations of the Taub-NUT metric that describe the dynamics of N BPS magnetic monopoles. These manifolds enjoy the isometries belonging to SU (N ). The Hamiltonian reduction with respect to a part of these isometries may lead to a hyper-K¨ahler manifold of a lower dimension, whose isometries belong to a subgroup of SU (N ) [20, 22, 23]. Finally, I would like just to mention an interesting relationship between the bi-K¨ ahler manifolds of a special type and hyper-K¨ahler manifolds. Their metrics are interrelated by nontrivial Legendre duality transformations [20].

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Chapter 12

HK and HKT through Harmonic Glasses

In Chap. 10 we discussed, among other extended supersymmetric models, the hyper-K¨ ahler and HKT models, but the efficiency and generality of our discussion was limited. • A hyper-K¨ ahler model was defined as an N = 2 model with three extra complex supersymmetries (10.100). We proved that the metric is hyper-K¨ ahler iff the complex structures are quaternionic, but the extra supersymmetries had an external nature; they did not appear as superspace symmetries. As a result, we did not have tools that could allow us to construct the models that do enjoy these extra symmetries. • We showed that the models with the superspace action (10.67) are HKT, and their metrics are given by the double derivatives of the prepotential, as in (10.70). But all these metrics belonged to a rather special class of HKT metrics, being Obata-flat. The complex structures (I, J, K)M N are constant matrices in this case. How to construct a generic HK or HKT metric? We have already briefly touched upon this question in Chap. 7.4. We wrote the harmonic superfield action (7.110) involving 2n N = 8 harmonic G-analytic pseudoreal superfields Q+a and claimed that the corresponding geometry is hyper-K¨ahler. We also claimed that a model, involving 2n N = 4 harmonic G-analytic pseudoreal superfields q +a subject to the nonlinear constraints (7.100), describes HKT geometry. In this chapter, we will justify these claims and show how, given a superspace action, to derive the corresponding HK or HKT metric. Generically, it is much more difficult than for e.g. K¨ahler manifolds, where the metric is always given by the double derivative of the K¨ahler potential, hm¯n = ∂m ∂n¯ K. For hyper-K¨ahler and not Obata-flat HKT manifolds, the 245

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metric can only be found as a result of solution of rather complicated harmonic differential equations. Only in rare cases it is possible to present the solutions in a closed analytical form. 12.1

The Hyper-K¨ ahler Model

The harmonic N = 8 superspace construction which describes hyper-K¨ahler geometry was worked out in Ref. [116] (see also Chap. 5 of the book [78]). For reader’s convenience we remind here some definitions that we gave at the end of Chap. 7. Our starting point is the ordinary N = 8 superspace that involves time, four complex odd coordinates θjα (α, j = 1, 2) and their conjugates θ¯jα = θjα . We harmonize it by introducing θ± = θj u± , θ¯±α = θ¯jα u± , tA = t + i(θ+ θ¯−α + θ− θ¯+α ), (12.1) α

α j

α

j

α

where tA is Grassmann analytic time. The analytic subspace (ζ, u) = (tA , θα+ , θ¯+α ; u± j ) of the large superspace stays invariant under the supertransformations and under the so-called pseudoconjugation defined in Eq. (7.79).1 Now we take 2n G-analytic superfields Q+a (ζ, u) = F +a (tA , u) + θ+ χαa (tA , u) + θ¯+α κa (tA , u) α

θα+ θ¯+α A−a (tA , u) + . . . + (θα+ θ¯+α )2 P −3a (tA , u) ,

α

a = 1, . . . , 2n

(12.3)

(only some terms relevant for us in what follows are displayed in the expansion). The superscripts like “+”, “-” or “-3” refer to the harmonic charges: each field entering Eq. (12.3) is an eigenstate of the operator D0 , defined in (7.84), with the corresponding eigenvalue. To make the representation (12.3) irreducible, we impose the pseudoreality constraint g + g +a = −Q+ , Q = Ωab Q+ ≡ Q+a ⇒ Q (12.4) a

b

a

ab

where Ω = −Ωab is the symplectic matrix, and e is the pseudoconjugation. We write the harmonic superspace action as an integral over the analytic superspace   Z 1 +4 +a + ++ +a 2 ¯+ 2 + 1 Q D Q + L (Q , u) , (12.5) S = dtA du d θ d θ 8 a 4 1 The presence of the index α does not change much. The odd coordinates transform now as + +α = −θ + . ¯+α , θf θ¯g (12.2) α = θ α

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with2 ∂ ∂ (12.6) D++ = ∂ ++ + 2iθα+ θ¯+α , ∂ ++ = u+j − . ∂tA ∂uj The equations of motion derived from it are ∂L+4 D++ Q+a = Ωab . (12.7) ∂Q+b Substituting here the expansion (12.3), we obtain a set of equations for the components. In particular, we derive ∂ ++ F +a = Ωab

∂L+4 ∂F +b

(12.8)

and ∂ 2 L+4 A−c = −2iF˙ +a + fermionic terms. (12.9) ∂F +b ∂F +c Consider Eq. (12.8). If L+4 were absent, the linear equation ∂ ++ F +a = 0 would have a simple solution, F +a ≡ F0+a = xja u+ (12.10) j , ja with x obeying the pseudoreality condition (3.15) that follows from (12.4). To be quite precise, xja depends on tA which includes the odd coordija = x . nates, and one should rather say that xja satisfies the condition xf ja However, bearing in mind that tA is invariant under pseudoconjugation (which plays the role of the ordinary complex conjugation in this approach), one can still write the condition (3.15) and treat tA as an ordinary real time t. In the following, the index “A” will be suppressed. When L+4 6= 0, it is a rather difficult task to find the solution to (12.8). To date, it was found in a closed form only for a few particular choices of L+4 [78, 116, 117], including the choice corresponding to the Taub-NUT manifold to be discussed below. In the general case, the solution to (12.8) can be found by iterations: redefine L+4 → λL+4 and represent the solution as a formal series in λ: F +a = F0+a + λF1+a + λ2 F2+a + . . . . (12.11) We obtain the chain of equations ∂L+4 , ∂ ++ F1+a = Ωab ∂F0+b ∂ 2 L+4 ∂ ++ F2+a = Ωab F1+c , ∂F0+b ∂F0+c ... = ... . (12.12) ∂ ++ A−a − Ωab

2 This is Eq. (7.98) where we added the summation aver α and suppressed the terms − ∝ ∂/∂θα , ∂/∂ θ¯−α —they do not work due to analyticity of Q+a .

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These are in fact algebraic equations, as becomes clear if one expands their left-hand and right-hand sides in a proper harmonic basis. For example, we represent + − (ijklp)a + + + − − F1+a = A(ijk)a u+ ui uj uk ul up + . . . i uj uk + B

(the linear term ∝ u+ j does not contribute in the left-hand sides of it is attributed to F0+ ) and ∂L+4 + + (ijklp)a + + + + − = C (ijk)a u+ ui uj uk ul up + . . . . Ωab i uj uk + D ∂F0+b

(12.13) (12.12);

(12.14)

Then the first equation in (12.12) implies 1 D, etc. A = C, B = 2 We first solve the equation for F1+a , then we substitute its solution to the equation for F2+a , solve it, substitute into the equation for F3+a , etc. As a result, F +a (t, u) is expressed via the harmonic-independent coefficients xja (t), which have the meaning of the coordinates on the hyper-K¨ahler manifolds that we are set to describe. Note that the pseudoreality conditions (3.15) for xja imply that the vectors i (12.15) xM = √ (ΣM )ja xja 2 [with the constant matrices ΣM = ΣM ≡ ΣA defined in Eq. (3.8)] are real. But any other choice of 4n real coordinates xM is possible. If we suppress in Eq. (12.9) the fermion terms, this truncated equation can be solved in a similar way. Its solution A˜−a (t, u) is also expressed via xja (t). It follows from (12.9) that A˜−a = −2ix˙ ja u− + nonlinear in harmonics terms . (12.16) j

To find the metric of the manifold that our action describes, we substitute the solutions thus obtained for F +a (t, u) and A˜−a (t, u) into the bosonic part of the action (12.5). Indeed, if one expresses the latter via the components, one obtains, using the equation of motion (12.9), a very simple expression Z i ˙ +a , S = dt du A˜− (12.17) aF 2 and the other bosonic components of the superfield (12.3) do not contribute! +a After expressing A˜− through xja (t), the action (12.17) takes the a and F generic form Z 1 S = dt gja, kb x˙ ja x˙ kb , gja, kb = εjk Ωab + O(λ) . (12.18) 2

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Note the following. The action (12.18) was derived using the equations of motion (12.8) and (12.9). As was mentioned, these equations have algebraic nature and allow one to express the multitude of auxiliary variables dwelling in F +a (t, u) and A−a (t, u) via the dynamical variables xja (t). But the superfield equation of motion (12.7) contains also the dynamical equation for xja (t). The latter is hidden in the component equation ∂ 2 L+4 P −3c = −2iA˙ −a + ferm. terms (12.19) ∂F +b ∂F +c that we did not discuss before. Substituting there the solution (12.16), we could derive, in the flat case L+4 = 0, the equation x ¨ja = 0, giving the straight geodesic. A similar program can be carried out for the fermionic components. In particular, comparing the terms ∝ θα+ on both sides of Eq. (12.7), we derive ∂ ++ P −3a − Ωab

∂ 2 L+4 χαc = 0 . (12.20) ∂F +b ∂F +c The variables κaα satisfy a similar equation, but they are not independent, being expressed via χαa in virtue of (12.4): def

D++ χαa = ∂ ++ χαa − Ωab

αa . g καa = χ

(12.21) 3

A general solution to (12.20) can be presented as

χαa (t, u) = (M −1 )a b (u) ψ αb (t) , where (M

(12.22)

−1 a

) b , which satisfies the equation

∂ 2 L+4 (M −1 )c b = 0, (12.23) ∂F +b ∂F +c is a very important object called the bridge. It relates the fields carrying the world symplectic index a to those carrying the tangent space underlined symplectic index a.4 The inverse matrix M b a satisfies the equation D++ (M −1 )a b ≡ ∂ ++ (M −1 )a b − Ωab

∂ 2 L+4 M bc = 0 (12.24) ∂F +a ∂F +b Note that the field ψ αb (t) (through which the field χαa is expressed) does not carry harmonic dependence. It is natural to associate it with the fermion field in tangent space. One can also define the harmonicindependent fermion field ψ αa (t) carrying the world symplectic index (we will do so a little bit later). ∂ ++ M b a + Ωcb

3 The notation M −1 → M would probably be more natural, but we follow the conventions of Refs. [80, 91]. 4 See Ref. [80] for more details.

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There are infinitely many solutions to Eqs. (12.23), (12.24) interrelated by the transformation (a kind of gauge transformation) M b a → Rb c M c a

(12.25)

with an arbitrary harmonic-independent Rb c . We will make a particular convenient choice for R and M (pick up a particularly convenient gauge) in Sects. 12.3 and 12.4. One can show that all other components of Q+a in the expansion (12.3) are auxiliary variables, they are algebraically expressed via xja and ψ αa . We have altogether 4n complex dynamical fermionic variables—one complex fermionic variable for each real bosonic coordinate. Their bilinear ˙ The variables can contribution to the Lagrangian has the structure5 ∝ ψ¯ψ. be chosen such that the coefficients in the fermion kinetic term and in the bosonic kinetic term are given by the same metric tensor. There is also a four-fermionic term coming from the expansion of L+4 . One thus obtains the action of a supersymmetric sigma model that can be expressed in terms of 4n (1, 2, 1) multiplets, a particular case of the generic action (8.9). However, our model possesses by construction extended N = 8 supersymmetry. As we know from the analysis in Chap. 10.4, this is only possible if the manifold is hyper-K¨ahler or bi-K¨ahler of a special type. Theorem 12.1. The action (12.5) describes a hyper-K¨ ahler supersymmetric sigma model. Proof. It is sufficient to study in this model the supersymmetry transformations of the coordinates xM defined in (12.15) and show that they coincide with (12.26), δxM = IN M (1 ψ N − ¯1 ψ¯N ) , δxM = JN M (2 ψ N − ¯2 ψ¯N ) , δxM = KN M (3 ψ N − ¯3 ψ¯N ) ,

(12.26)

with quaternionic complex structures and properly defined ψ M . This will exclude the bi-K¨ ahler possibility—we have seen that in the latter case the law of supersymmetry transformations is more complicated [see Eq. (10.40)]. + αa comes from the first term in (12.5), where one has to substitute θα χ + θ¯+α κa α +a for Q . 5 It

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At the level of the harmonic superfield (12.3), the supersymmetry trans¯+α = ¯+α . This gives, formations are generated by the shifts δθα+ = + α , δθ bearing in mind (12.21), αa α . g δF +a = δ F +a + δ¯F +a = + (12.27) − ¯+α χ a αχ +a Let us concentrate on δ F . Using (12.22) and multiplying both sides of (12.27) by M b a , we may write αb M b a (∂kb F +a ) δ xkb = + (12.28) α ψ (t) . + The right-hand side of this relation is proportional to uk (we define + α = α k u + ), while higher-order terms in the harmonic expansion are absent. k The same should be true for the left-hand side.6 We can thus define kb M b a (∂kb F +a ) = ekb (t) u+ (12.29) k . Then the transformation law (12.28) acquires the form k αb , (12.30) δ xkb = ekb kb α ψ kb

where ekb kb is the inverse of ekb . Later we will see that these matrices are nothing but the ordinary vielbeins under a particular choice of the matrix R in Eq. (12.25). Next, we rename the dummy index α → j so that ψ αb → ψ jb and define jb ψ jc = ejc jb ψ ,

k

j k = 0 δj + ip (σp )j k ,

where σp are the Pauli matrices. We derive h i jb k jc δ xkb = ejc 0 δj + ip (σp )j k ekb kb ψ .

(12.31) (12.32)

At this stage, we may go to the vector notation (12.15) and also i ψ M = √ (ΣM )ja ψ ja , (12.33) 2 restore the variation δ¯xM from the requirement for xM and δxM to be real, and derive δxM = 0 ψ M + p (I p )N M ψ N − ¯0 ψ¯M − ¯p (I p )N M ψ¯N (12.34) with i jb M (12.35) (I p )N M = − (ΣN )jc ejc (σp )j k ekb kb (Σ )kb . 2 M (we remind that Σ = ΣM are flat). One can observe, using (3.14), that the complex structures (12.35) are quaternionic [in fact, they are nothing but the flat structures (3.16) dressed with the vielbeins]. The terms involving 0 and ¯0 give the standard N = 2 supertransformation of the lowest component of the superfield X M and the terms involving p and ¯p coincide with the lowest component of (10.100). 6 And

it is. See Ref. [91] for an independent derivation.

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We have shown that the model (12.5) describes hyper-K¨ahler geometry. Two remarks are in order. (1) A legitimate question is whether any hyper-K¨ahler metric can be derived this way. The answer to this question is positive. In the paper [118] (see also Chap. 11 in [78]) the problem was solved in a different way—not invoking supersymmetry, but solving instead the constraint RA B ∈ sp(n) for the curvature form (see Theorem 3.2). It was shown that a general solution to this constraint depends on an arbitrary harmonic function L+4 (Q+a , u) and that this solution coincides with the solution following from (12.5). (2) Our discussion was local—we described all hyper-K¨ahler metrics in a given chart with trivial topology. We said nothing about how different charts are glued together and thus we have not obtained the classification of hyper-K¨ahler manifolds. Similarly, the expression hm¯n = ∂m ∂n¯ K gives a complete classification of K¨ahler metrics in a given chart, but does not give a classification of K¨ahler manifolds.

12.2

The HKT Model

This model enjoys N = 4 supersymmetry and, to give its superspace description, one should take N = 4 superspace including time, a doublet θj and a doublet θ¯j , and harmonize it. The basic definitions were given in Chap. 7. They coincide with the definitions of the preceding section, where one only has to suppress the index α. We take 2n G-analytic superfields (7.95), q +a (ζ, u) = f +a (t, u) + θ+ χa (t, u) + θ¯+ κa (t, u) + θ+ θ¯+ A−a (t, u) (12.36) (t ≡ tA ) and impose the nonlinear constraint (7.100), D++ q +a = L+3a (q +b , u)

(12.37)

+a = with arbitrary L+3a , supplemented by the pseudoreality constraint qg + −qa . Substituting (12.36) into (12.37), we derive for the components:

∂ ++ f +a = L+3a (f +b , u) , D

++ a

χ =D

(12.38)

++ a

κ = 0,

D++ A−a = −2if˙+a +

(12.39) 2

+3a

∂ L κb χc , ∂f +b ∂f +c

(12.40)

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where the action of the covariant harmonic derivative D++ on any symplectic vector X a is now defined as7 ∂L+3a b X . (12.42) ∂f +b Note that the constraint (12.37) plays completely different role than (12.8). The latter was the superfield equations of motion following from the action (12.5). And the constraint (12.37) is an external constraint imposed to make the representation (12.36) irreducible. At this stage, we have not written yet the action describing the dynamics of q +a . We will do so soon. The pseudoreality constraint implies that D++ X a = ∂ ++ X a −

f +a (t, u) = xja u+ j + higher harmonic terms

(12.43)

ja

with pseudoreal x . The latter can be traded for 4n real coordinates xM as in (12.15). In the fermion sector, we derive a^± κa (t, u± j ) = χ (t, uj ) .

(12.44) ¯j

By the same (group theory) reasons that we defined θj = θ , it is convenient to define8 χa = χ ¯a

⇒

χ a = −χ ¯a ,

(12.45)

the fundamental representation of Sp(n) becomes antifundamental after conjugation and the other way round. The fermion components in q +a acquire thus the form (q +a )ferm = θ+ χa (t, u) − θ¯+ χ ¯a (t, u ˜) . (12.46) We have 2n complex or 4n real fermion fields—a real fermion field ψ M for each real coordinate. As for A−a , they are expressed via xM and ψ M in virtue of (12.40). The most general N = 4 supersymmetric action involving the fields q +a reads Z S = S0 + Sgauge = dtdu d2 θ+ d2 θ− L(q +a , q −b , u± ) Z + i dtA du d2 θ+ L++ (q +a , u± ) . (12.47) 7 We

note right now that in the particular case L+3a = Ωac

∂L+4 ∂f +c

D++ acts as in Eq. (12.20). 8 Note that in Refs. [80, 91] the opposite sign convention χa = χ ¯a was chosen.

(12.41)

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The term S0 represents an integral over the whole superspace. The integrand L depends on the harmonics, on q +a and also on q −a = D−− q +a . To make the action real, L should be real. The second term represents an integral over the analytic superspace. To make Sgauge real, the integrand L++ should be odd under the pseudoconjugation (7.79). We can now formulate the central statement to be proven in this section. The action S0 in (12.47), where the pseudoreal superfields q +a are subject to the constraints (12.37), describes HKT geometry. The second term Sgauge describes the gauge field of some special form, living on the manifold. The proof of this fact involves several steps. To begin with, we have just seen that the dynamical degrees of freedom in our model include 4n real coordinates xM and 4n their real fermionic superpartners. It is exactly the right number of degrees of freedom that corresponds to the model (9.34) including D N = 1 superfields X M with the possible extra term (9.64). We want to emphasize that the action Z S= Z −i



1 i dΘdt gM N (X )X˙ M DX N − CKLM DX K DX L DX M 2 12 dΘdtAM (X ) DX M



(12.48)

describes a most general N = 1 model including an equal number of bosonic and fermionic dynamic variables and not involving higher derivatives. To construct such a model, we have only the superfields X M = xM + iΘΨM at our disposal. The fermionic superfields like Φ = φ + ΘF do not fit: if we want φ to enter the component Lagrangian with a derivative, the field F is necessarily auxiliary, entering without derivatives (see the discussion on p. 163) leading to the mismatch of bosonic and fermionic dynamic variables. Let us first concentrate on S0 and prove that Theorem 12.2. The action S0 can be expressed in terms of X M as in the first line of (12.48) without an admixture of the second line. Proof. The component Lagrangian corresponding to (9.34) [the first line in (12.48)] was given in Eq. (9.3). In contrast to the Lagrangian in (9.65) that follows from (9.64), it involves only the terms ∝ x˙ 2 , but not the terms ∝ x. ˙ Thus, we only have to show that the action S0 expressed in components involves only the quadratic in velocities term.

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Using the expansion (12.36) and the expansion q −a = D−− q +a = ∂ −− f +a + 2iθ− θ¯− f˙+a + θ+ θ¯+ ∂ −− A−a + (θ− θ¯+ + θ+ θ¯− )A−a + 2iθ+ θ¯+ θ− θ¯− A˙ −a + fermion terms , (12.49) we derive9  ∂2L dudt (f˙+b A˜−a − f˙+a A˜−b ) +a −b ∂f ∂f    i ∂2L , (12.50) + f˙+a ∂ −− A˜−b + A˜−a A˜−b − f˙−a A˜−b 2 ∂f −a ∂f −b where we defined f −a = ∂ −− f +a and A˜−a is the solution to the truncated constraint (12.40) with suppressed fermionic term in the right-hand side, like in (12.9). We see that A˜−a is proportional to f˙+a and finally to x˙ ja . This means that the component Lagrangian derived from S0 involves only the terms ∝ x˙ 2 . S0bos = 2i

Z

After doing the harmonic integral in (12.50), we may derive the metric. We will do so later. We know from Theorem 9.4 and the remark at the end of Sect. 10.2 that if the metric gM N , the torsion CKLM , and the gauge potential AM satisfy certain conditions, the model enjoys N = 2 supersymmetry. In such models, one of the supersymmetries mixes the components of X M according to (7.10) and another supersymmetry transforms the whole N = 1 superfield X M as in (9.37). The model that we are now interested in has, however, not one, but three extra supersymmetries. According to Theorem 9.4, each of them should have a form (9.37) with its own complex structure (I p )N M p p that satisfies the standard conditions: (I p )2 = −1, IM N = −IN M and the vanishing of the Nijenhuis tensor. The action should thus be invariant under δ1 X M = 1 (I 1 )N M DX N , δ2 X M = 2 (I 2 )N M DX N , δ3 X M = 3 (I 3 )N M DX N .

(12.51)

9 Note that the expansion (12.36) was written in the analytic basis, and the expansion (12.49), which was derived by acting on (12.36) with the operator D−− in the form (7.98), is written also in the analytic basis. Thus, we have actually derived (12.50) from the superfield expression for S0 in (12.47) where the integral was done over dtA rather than over dt. But this makes no difference because the super-Jacobian of the transformation t → tA with other variables fixed is equal to 1. After the integral over θ± is done, we can suppress the subscript A.

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Theorem 12.3. For the N = 4 supersymmetry algebra to hold, the complex structures I p should satisfy the Clifford algebra I p I q + I q I p = −2δ pq .

(12.52)

Proof. Consider the Lie bracket (δ2 δ1 − δ1 δ2 )X M . The calculation gives (δ2 δ1 − δ1 δ2 )X M = −i1 2 (I 1 I 2 + I 2 I 1 )N M X˙ N + independent terms. For the supercharges Q1 and Q2 to anticommute, this Lie bracket must vanish. This gives {I 1 , I 2 } = 0. In the same way we derive {I 1 , I 3 } = {I 2 , I 3 } = 0. In Chap. 10 we discussed N = 4 supersymmetric models with this set of variables (a real fermion for each real coordinate) and found two types of such models: the HKT models and the bi-HKT models. The HKT models were distinguished compared to bi-HKT ones by quaternionic nature of their complex structures. In Chap. 10, only Obata-flat models were discussed, but this property holds also in the general case. Theorem 12.4. A model (9.34) with three extra invariances (12.51) where the complex structures are quaternionic describes HKT geometry. Proof. The proof is purely geometric. The invariance under (12.51) and Theorem 9.4 imply in particular the condition (9.40) for each complex structure: (B)

(B)

∇L (I p )N M + ∇N (I p )L M = 0 .

(12.53)

We will show that in the case when the complex structures are quaternionic, they satisfy the stronger condition (B)

∇N (I p )L M = 0 ,

(12.54)

which means the existence of the universal Bismut connection so that the manifold is HKT. The integrability allows us to choose the complex coordinates associated with any of the complex structures. Let us do so for I 1 ≡ I. Then the only nonzero components of the tensor IN M are displayed in Eq. (2.52): n n ¯ n ¯ Im n = −iδm , Im ¯ = iδm ¯,

n Im n¯ = Im ¯ = 0.

(12.55)

Consider the tensors10 I 2 ± iI 3 ≡ J ± iK. As follows from Theorem 3.4 [specifically, from the relations (3.46)], their only nonzero components are n n (J + iK)m n¯ = 2Jm n¯ and (J − iK)m ¯ = 2Jm ¯ . 10 (J

± iK)/2 are the hypercomplex structures I ± defined on p. 63.

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Note that the properties J 2 = K 2 = −1 and JK = −KJ = I [this is the step where the quaternionic nature of the triple (I, J, K) is used] together with (12.55) imply ¯

¯

n l l (J − iK)m ¯ (J + iK)n = −4δm ¯ , ¯

(J + iK)n l (J − iK)¯l m = −4δnm .

(12.56)

Considering the constraint (12.53) for the complex structure I, we derived earlier (see the proof of Theorem 9.5) the properties ¯ = 0 Gnl¯s = G¯nls

(12.57)

for certain components of the Bismut affine connections. Consider now the constraint (12.53) for J + iK: (B)

(B)

∇L (J + iK)N M + ∇N (J + iK)L M = 0 .

(12.58)

We choose L = l and M = m, but do not specify the holomorphicity of N . Using the fact that the only nonzero components of J + iK have an antiholomorphic upper index, we see that in this case most of the terms in (12.58) vanish and we derive s¯ m s¯ Gm = 0. l¯ s (J + iK)N + GN s¯(J + iK)l

(12.59)

However, the first term in the left-hand-side vanishes due to (12.57), and the s¯ l constraint boils down to Gm N s¯(J + iK)l = 0. Multiplying this by (J − iK)r¯ and using the first relation in (12.56), we arrive at the constraint Gm N r¯ = 0

(12.60) ¯ Gm Ns

for all N , holomorphic and antiholomorphic. The constraint = 0 can be derived in a similar way by exploring the condition (12.53) for the structure J − iK. Consider now the covariant derivative of I without symmetrization, (B)

M Q ∇L IN M = GQ − GM LQ IN , LN IQ

(12.61)

and choose again M = m (the case M = m ¯ is treated analogously). We note that if N = n, the right-hand-side of (12.61) vanishes identically due to the cancellation of the two terms and if N = n ¯ , it boils down to −2iGm L¯ n, which vanishes due to (12.60). (B) Thus, ∇L IM N = 0. In fact, we have proven that if the complex structures are quaternionic, the holomorphic and antiholomorphic parts C3,0 and C0,3 in the decomposition (9.57) (Theorem 9.5 only said that the corresponding forms must be closed) are not present, and the torsion C is the Bismut torsion for I. Choosing the complex coordinates associated with (B) J or with K, we can repeat all the arguments and prove that ∇L JM N = (B) ∇L KM N = 0. The Bismut torsion is universal.

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Remark. The term “HKT” in the formulation of this theorem should be understood in a generic sense including also hyper-K¨ahler manifolds with zero torsion. In the latter case, the second term in the action (9.34) is absent. It is also in this sense that the term “HKT” is used in the formulation of the following theorem. Theorem 12.5. The model S0 involving pseudoreal superfields q +a subject to the constraints (12.37) is HKT. Proof. We only need to show that the complex structures are quaternionic. The proof of that is similar to the proof of Theorem 12.1. Supertransformations of the bosonic components f +a of the superfields q +a are generated by the shifts δθ+ = + and δ θ¯+ = ¯+ . They read δ f +a (t, u) = + χa (t, u) − ¯+ χ ¯a (t, u ˜) .

(12.62)

Similarly to (12.22), we express the solution of Eq. (12.39) as χ ¯a (t, u ˜) = (M −1 )a b ψ¯b (t) ,

χa (t, u) = (M −1 )a b ψ b (t) , where the bridge M

−1

(12.63)

satisfies the equation

D++ (M −1 )a b ≡ ∂ ++ (M −1 )a b −

∂L+3a (M −1 )b b = 0 , ∂f +b

(12.64)

Then (12.62) may be rewritten in a way analogous to (12.28): M b a (∂kb f +a ) δ xkb (t) = + ψ b (t) − ¯+ ψ¯b (t) .

(12.65)

The right-hand side, and hence the left-hand side of Eq. (12.65) involve only the linear term ∝ u+ k in the harmonic expansion. Defining the vielbeins as in (12.29), kb

M b a (∂kb f +a ) = ekb u+ k ,

(12.66)

we obtain the transformation law for harmonic-independent fields: k b δ xkb = ekb ¯k ψ¯b ) . kb ( ψ −  b

¯b

The fields ψ and ψ can be joined into the quartet ψ

(12.67) kb

according to

ψ 2b ≡ −ψ¯b .

ψ 1b ≡ ψ b , We also define the matrix l k as 1 k ≡ k ,

2 k ≡ ¯k

(12.68)

l k = i0 δl + p (σ p )l k

(12.69)

and represent it as k

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with real 0 , p . The matrix (12.69) satisfies the identity l k = εki εlj j i . In this notation, the transformation law (12.67) is expressed as k lb kb p k lb δ xkb = ekb = i0 ekb + p ekb kb l ψ kb ψ kb (σ )l ψ .

(12.70)

If we define kb ψ kb ≡ ekb kb ψ ,

(12.71)

the first term in (12.70) reads simply δxkb = i0 ψ kb . This is a N = 1 supersymmetry transformation indicating that xkb and ψ kb are superpartners: they represent the components of the N = 1 superfield [cf. (9.36)]. The second term in (12.70) can be represented as lc

lc δ xkb = i p elc (I p )lc kb ekb ≡ i p (I p )lc kb ψ lc , kb ψ

(12.72)

(I p )lc kb = −i (σ p )l k δcb .

(12.73)

where

Lifting the indices in (12.73), we reproduce the flat quaternionic complex structures (3.18). The complex structures (I p )N M = eN A (I p )AB eM B , or those in the spinor notation, lc

(I p )lc kb = −ielc (σ p )l k ekb kc ,

(12.74)

which enter (12.72), are also quaternionic. Hence, according to Theorem 12.4, our manifold is HKT. Remark. The fact that our action can be expressed in the language of N = 1 superfields and its invariance under the transformations (12.51) implies the existence of the conserved supercharge Q0 associated with the transformations within the N = 1 multiplets and three extra conserved supercharges Qp . Each of the latter depends on the corresponding complex structure as in (9.61). We arrive at the expressions (10.78) for the supercharges, which are thus valid not only for Obata-flat models discussed in Chap. 10, but also in the general case. These expressions are especially nice and simple in the hyper-K¨ ahler case when the torsion CM KP vanishes. 12.3

Metric



Obata Families



HKT → HK.

The theorems proved in the previous sections would be useless if one could not use for practical purposes the harmonic technique presented there: could not derive explicit expressions for the metric of the HKT and HK manifolds. In this section, we describe a general method how this can be

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achieved and we will show in the next section how this method works in particular cases. Consider the expression (12.50) for the bosonic Lagrangian. To find the metric, we should do the harmonic integral and express it through the harmonic-independent coordinates xja . Bearing in mind (12.66), we derive ja f˙+a = (∂kb f +a )x˙ kb = (M −1 )a a ekb x˙ kb u+ j .

(12.75)

The solution to the truncated constraint (12.40) reads ja A˜−a = −2i(M −1 )a a ekb x˙ kb u− j ,

(12.76)

as one can be convinced using D++ M −1 = 0 and ∂ ++ u− = u+ . The calculation is simple when L is quadratic, 1 1 L = − q +a qa− = − Ωab q +a q −b . (12.77) 8 8 Then the term ∝ ∂−a ∂−b L in (12.50) is absent, ∂+a ∂−b L = −Ωab /8, and we obtain Z i bos S0 = Ωab dudt f˙+a A˜−b . (12.78) 2 Substituting there (12.75) and (12.76), we derive Z − bos jc kd ja kb −1 a L0 = x˙ x˙ ejc ekd Ωab du u+ ) a (M −1 )b b . j uk (M

(12.79)

One can observe by symmetry considerations that the expression multiply− ing the structure u+ j uk is antisymmetric under j ↔ k. Then we can replace − u+ j uk → εjk /2 to derive

Lbos = 0

1 gjc,kb x˙ jc x˙ kb , 2

(12.80)

where ja kb

gjc,kb = ejc ekb εjk Ωab

Z

def

ja kb

du (M −1 )a a (M −1 )b b = ejc ekb εjk Gab . (12.81)

When L+3a are absent, (M −1 )a a = δaa , the vielbeins can be chosen as ja

j

a

ejc = δj δc , and we obtain the flat metric (3.14). For an arbitrary L, one can obtain after some work [80] the result for the metric that has the same structure as in (12.81), but the harmonic integral for Gbc involves now an extra factor in the integrand: Z du Fab (M −1 )a a (M −1 )b b , (12.82) Gab =

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where   Fab = −8 ∂+[a ∂−b] − (M −1 )c c (∂ −− M c [a ) ∂−b] ∂−c L .

(12.83)

We see that, generically, the metric is not just a convolution of the matrices ja eja , but involves an extra factor—the antisymmetric matrix Gab carrying the symplectic indices. However, one can bring Gab to the form Ωab using the gauge freedom (12.25) with a harmonic-independent matrix R. Indeed, (0) a let Gab = Ωab and Ra b = δb + λa b with λ  1. We can then write Gab = Rc a Rd b Ωcd = Ωab + 2λ[ab] + o(λ) .

(12.84)

It is clear that a finite version of this transformation can produce an arbitrary antisymmetric matrix Gab . Note that the metric (12.81) is invariant under the gauge rotations with the matrix Ra b defined in (12.25). It is immediately seen from the definition ja

(12.29): The objects eja are rotated by R in the same way as M a a , whereas the factors M −1 in Eq. (12.82) are rotated by R−1 . If the gauge Gab = Ωab ja

is chosen, eja acquire the meaning of the true vielbeins and the metric is given by the standard expression ja kb

gja,kb = eja ekb εjk Ωab .

(12.85)

It is simply the familiar identity gM N = eAM eAN expressed in spinorial notation. In the generic gauge where the expression (12.81) for the metric ja involves an additional structure Gab , one may call eja quasivielbeins. We have shown how, given two harmonic functions L+3a and L, one can derive the metric. Generically, this metric is HKT. We have proven this indirectly, and in Ref. [80] the reader can find the direct calculations of the complex structures, the vielbeins and the connections. With these expressions in hand, one can check explicitly that the Bismut connections for the three complex structures coincide and the metric is HKT. These explicit calculations are, however, not at all simple. Again, one can ask at this point: is it true that any HKT metric can be derived this way? We can conjecture that yes, it can, but, in contrast to the hyper-K¨ ahler case, an exact proof of this assertion has not been given yet. Another remark is that our description has a high level of redundancy. One can perform a variable change q +a → q1+a (q +a , u)

(12.86)

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to obtain another pair of prepotentials (L+3a , L1 ) that describe the same 1 geometry. Let us prove this. Theorem 12.6. The metric (12.85) is invariant under the transformations (12.86). Proof. The transformation (12.86) induces the transformation of the bosonic components of q +a . Infinitesimally, δλ f +a = λ+a (f, u)  f +a . We have ∂

++ +a

f

+3a

=L

(12.87) +3a

(f ). After the transformation, L

is modified:

δλ L+3a = ∂ ++ λ+a = L+3b (∂+b λ+a ) + ∂ˆ++ λ+a , (12.88) +a ++ where ∂+a = ∂/∂f and ∂ˆ is the part of the full harmonic derivative ++ ∂ due to the explicit harmonic dependence: ∂ ++ = ∂ˆ++ + L+3b ∂+b (12.89) (the identity ∂ ++ f +b = L+3b was used). Consider now the equation (12.64) for the bridge (M −1 )a b and determine how its solution transforms under (12.87). To this end, one has first to find out the law of transformation for the structure ∂L+3a E +2a b = . (12.90) ∂f +b Using (12.87), (12.88) and (12.89), we derive11 δλ E +2a b = −E +2a c (∂+b λ+c ) + E +2c b (∂+c λ+a ) + ∂ ++ (∂+b λ+a ) . (12.91) For D++ M −1 to stay zero after the transformation and bearing in mind (12.91), the bridge should transform as δλ (M −1 )a b = (∂+b λ+a )(M −1 )b b .

(12.92)

The inverse matrix transforms as δλ M b a = −(∂+a λ+b )M b b . +2a

(12.93) −1

In other words, in contrast to E and M transform as b , the bridges M contravariant and covariant vectors under the shifts of analytic harmonic coordinates.12 11 The last term in (12.91) comes from the derivative of the second term ∂ ˆ++ λ+a in the variation (12.88) and from the derivative of the factor ∂+b λ+a in the first term there. These contributions add up to produce the total harmonic derivative ∂ ++ . 12 Speaking of E +2a , it plays the role of the connection associated with the analytic b diffeomorphism (12.87). Its law of transformation includes, besides the first two tensorial terms, the extra harmonic gradient term, in the full analogy with the transformation law δAM = ∂M X for the (Abelian) gauge potential or with the second term in the transformation law (1.11) for the ordinary affine connection.

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The same concerns the structure ∂kb f +a —it transforms as δλ (∂kb f +a ) = ∂kb λ+a = (∂kb f +c )(∂+c λ+a )

(12.94)

and represents a contravariant vector. Note that the index “b” in the spatial derivative ∂kb does not “feel” the analytic shift (12.87)! It follows that the L.H.S. of Eq. (12.66) is a scalar under (12.87) and the same concerns the vielbeins in its R.H.S. The metric (12.85) is also invariant. Consider now a set of models characterized by the same potential L+3a and hence the same harmonic constraints (12.37), but different actions S0 . Different L’s result in different Fab ’s in (12.83), which affect Gab . As we have just seen, this modification may be compensated by the appropriate gauge rotations of the bridges and, correspondingly, of the vielbeins. We thus obtain a family of models whose vielbeins are interrelated by the transformations ja

e˜ja = R

a

jb b eja

(12.95)

where all the dependence on L is encoded in the matrix R. To keep eAM real, the matrix Ra b should satisfy pseudoreality conditions, like in (3.15). Then it involves 4n2 real parameters. 2n2 + n of them correspond to the action of Sp(n) group, which does not affect the metric (12.85). So we are left with 2n2 − n essential parameters. For n = 1, only one such parameter is left, which corresponds to multiplying the metric by a conformal factor. An important observation is that the complex structures in all such models coincide. Indeed, looking at the expression (12.74), we observe that it is invariant under R transformations because the upper index c in (12.74) is rotated by the matrix R, while the contracted lower index c is rotated by the inverse matrix R−1 . So we have (I˜p )lc kb = (I p )lc kb . (12.96) In particular, in all the models expressed in terms of linear (4, 4, 0) multiplets with vanishing L+3a , the complex structures keep their flat form (3.16). An important corollary of this observation is Theorem 12.7. The Obata curvature invariants for a family of the HKT metrics, characterized by a particular nonlinear constraint (12.37) but having different actions S0 in (12.47), coincide.13 13 One

may call such a family an Obata family.

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Proof. This statement follows from Theorem 3.7, which says that there exists a frame where the components of the Obata connection are expressed via the complex structures, and the fact that the latter do not depend on L. Then this connection also cannot depend on L and the same is true for the Obata curvature invariants. It follows, in particular, that all the models based on linear (4, 4, 0) multiplets are Obata-flat. Up to now, we discussed here and in the preceding section a generic HKT model characterized by arbitrary L+3a and L. But there is an important particular case. Theorem 12.8. Consider a limited class of models where the function L+3a in (12.37) has the form L+3a = Ωab

∂L+4 , ∂q +b

(12.97)

as in (12.41), and the function L in (12.47) is quadratic, L = Aq +a qa− . In this case, the metric is hyper-K¨ ahler. Proof. The point is that the bosonic metric following from the quadratic S0 in (12.47) with the constraints (12.37) involving (12.97) exactly coincides with the metric derived from the equation of motion (12.7) for the model (12.5) involving the “large” multiplet (12.3). And, as we know from Theorem 12.1, the latter model enjoys N = 8 supersymmetry and gives rise to a hyper-K¨ ahler metric. To prove that the metrics of the two models are, indeed, the same, compare the bosonic action of the two models. We see that in both cases, it is given by the expression (12.17), which is equivalent to (12.78) (one only has to replace F +a by f +a ). If the condition (12.97) is fulfilled, F +a and f +a satisfy the same equations (12.8) ≡ (12.38). Hence they are expressed through the coordinates xja in the same way. The same is true for A˜−a : if (12.97) holds, Eqs. (12.9) and (12.40) coincide. Thus, the bosonic actions and hence the metrics in the two models coincide.

Note that Theorem 12.8 only gives sufficient conditions for the metric to be hyper-K¨ ahler, but not the necessary ones. As was discussed above, our description has a high level of redundancy. We can take the quadratic L and L+3a that satisfies (12.97) and perform a variable change. After that

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L is not quadratic and (12.97) is not fulfilled anymore, but the metric is still hyper-K¨ ahler. On the other hand, for the quadratic L, the condition (12.97) is necessary for the metric to be HK. Heuristically, this is more or less clear: Indeed, let L = q +a qa− and suppose that one cannot represent L+3a as in (12.97), but the metric is still HK. We know, however, from the results of [118] that any HK metric is derived from some prepotential L+4 which enters the N = 8 Lagrangian (7.110). As we have just seen, the same metric can be derived from the N = 4 model (12.47) with the quadratic L and the constraint involving +4

∂L L˜+3a = Ωab +b . ∂q

(12.98)

But it is hardly possible that, given the quadratic L and hence the expression (12.78) for the bosonic action, one and the same metric follows from two different prepotentials L˜+3a 6= L+3a . It is clear from the derivation above that the bridge M and hence the expressions of f +a and A−a [to be substituted into (12.78)] via the coordinates xja depend on the prepotential in an essential way. This heuristic argument is confirmed by the explicit calculations in Ref. [80]. Note first of all that the “analytic connection” E +2a b that enters Eq. (12.64) belongs to the algebra sp(n) iff the representation (12.97) holds.14 Then and only then Eq. (12.64) admits the solutions for the bridge that belong to the group Sp(n). The analysis of the explicit formulas of Ref. [80] exhibits that, given a quadratic L, in this and only in this case the Levi-Civita spin connection ωAB,M dxM also belongs to sp(n), which is a necessary and sufficient condition for the metric to be hyper-K¨ahler, in virtue of Theorem 3.2. We keep now L quadratic and allow for an arbitrary L+3a . A noteworthy fact is Theorem 12.9. The torsion form for an HKT model with L = Aq +a qa− is closed, dC = 0. In other words, we are dealing in this case with the so-called strong HKT geometry [25, 26]. Proof. A generic N = 4 component Lagrangian has the form (9.3) and includes a 4-fermion term. The latter vanishes iff the torsion form is closed. 14 Indeed, a 2n-dimensional matrix E a belongs to sp(n) iff the identity Ωh + hT Ω is b fullfilled, i.e. iff the matrix Ωac E c b is symmetric. And this is true iff (12.97) holds.

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Thus, it is sufficient to prove that the full component action of the HKT model with quadratic L does not involve such term. Let us substitute in L = −q +a qa− /8 the expansions (12.36) and (12.49), where now the fermion terms are also taken into account. The explicit form of the latter is (q +a )ferm = θ+ χa − θ¯+ χ ¯a , (qa− )ferm = θ− χa − θ¯− χ ¯a + −− ¯˙ a ), + θ ∂ χa − θ¯+ ∂ −− χ ¯a + 2iθ− θ¯− (θ+ χ˙ a − θ¯+ χ

(12.99)

but we do not need it so much, the only thing that we need is the fact that both q +a and q −a involve only the linear in fermion field terms. Then the quadratic L may generate only bi-fermion terms [note that they are also present in the term ∼ f˙+ A− including the field A− a (t, u) that satisfies the untruncated constraint (12.40)], while 4-fermion terms are absent.

L = Aq +a qa− L+3a = Ωab

L+3a 6= Ωab

∂L+4 ∂q +b

∂L+4 ∂q +b

L= 6 Aq +a qa−

HK C=0

weak HKT

strong HKT

dC 6= 0 (generically)

dC = 0

When L is not quadratic, the torsion form is not closed in most cases. But in some cases it can happen to be closed. For example, in the linear case L+3a = 0 the metric coincides for n = 1 with the 4-dimensional flat metric multiplied by a certain conformal factor λ(x) [see Eq.(10.71); in fact, we discussed this example in Chap. 3 on p. 68 ]. The torsion form is [109] C ∼ εM N P Q ∂Q λ(x) dxM ∧ dxN ∧ dxP .

(12.100)

If ∂Q ∂Q λ(x) = 0, this form is closed. The results outlined above are illustrated in Table 12.1. The Obata families of the metrics can be divided into two classes: reducible metrics, when the metric of at least one member of the family is

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some L1 , L1+3a =

iab var

267

L+4

ge han c e l

L0 = q 2 , L0+3a ≠ Ω∂L+4

Ob at tra a-pre nsf s orm ervin atio g n L2 = q 2 , L2+3a = L1+3a HK metric

Fig. 12.1

A reducible HKT metric playing hide and seek with analytic frames.

hyper-K¨ ahler, and irreducible metrics which cannot be reduced to hyperK¨ ahler ones by a rotation of the vielbeins as in (12.95). A nontrivial statement is Theorem 12.10. Irreducible metrics exist. This statement is nontrivial due to the redundancy of description mentioned above. The metric is determined by a pair {L+3a , L}, but many (an infinity) of such pairs are equivalent under the analytic variable change (12.86). As a result, one and the same metric can belong to an infinity of apparently different, but in fact equivalent Obata families. Even though the “stem member” of a given Obata family with quadratic L may have a nontrivial L+3a not given by (12.97) and the corresponding manifold is not hyperK¨ ahler, it is not obvious that the constraints L+3a cannot be brought to a form which satisfy (12.97) by a transformation (12.86), after which L would not be quadratic any more. Choosing then quadratic L, while not modifying L+3a , we would arrive at a hyper-K¨ahler metric (see Fig. 12.1). There are many such metrics which seem to be irreducible, but are in fact reducible in disguise. To prove the theorem, one has to give at least one example of a metric that is definitely irreducible. We will do it in the next section. 12.4 12.4.1

Examples Taub-NUT metric

To show how the procedure described in the preceding section really works, we present here a detailed harmonic derivation of this metric.

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Consider the n = 1 model involving two harmonic fields q +(a=1,2) with the prepotential L+4 chosen as 1 L+4 = − (q +1 )2 (q +2 )2 , 2

(12.101)

from which L+3a = εab ∂+b L+4 is derived. In this case, the constraints (12.38)–(12.40) can be solved analytically. We start with Eq. (12.38) for f +a . It reads15 ∂ ++ f +1 = (f +1 )2 f +2 ,

∂ ++ f +2 = −f +1 (f +2 )2 .

(12.102)

The solution to this equation has the form f +1 = exp{J}x+1 ,

f +2 = exp{−J}x+2 ,

(12.103)

ia where x±a = xia u± and i with harmonic-independent x

J =


1 +1 −2 x x + x−1 x+2 . 2

(12.104)

When the nonlinearity is switched off, f +a → x+a . To find the metric, we need to know the bridge. The bridge M satisfies the homogeneous equation (12.24). As was mentioned before, the analytic connection E +2a b = Ωac

∂ 2 L+4 ∂f +b ∂f +c

(12.105)

that enters this equation belongs to the algebra sp(n) and hence there are solutions for the bridge that belong to the group Sp(n). In our case, Sp(1) ≡ SU (2). Even after we impose the condition for the bridge M to belong to SU (2), there is a freedom associated with multiplication by constant SU (2) matrices. We choose a solution that is reduced to the unit matrix in the limit when the nonlinearity associated with L+4 is switched off. It reads ! e−J (1 − x−1 x+2 ) −eJ x−1 x+1 1 b , (12.106) M a = √ 1 + x1 ·x2 e−J x−2 x+2 eJ (1 + x−2 x+1 ) where x1 ·x2 = −x2 ·x1 = xj1 x2j . The index b marks the lines and the index a marks the columns. 15 We

remind that ε12 = −ε12 = −1.

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The vielbeins can be found from the solutions (12.103), (12.106) and from the definition (12.66). They are16   p 1 1 k2 1 1 2 k1 k 1 2 1 + x ·x ei1 = − xi x + δi 1 + x ·x , 2 2 p p 1 1 k1 k2 1 + x1 ·x2 ei2 = − x1i xk1 , 1 + x1 ·x2 ei1 = x2i xk2 , 2 2   p 1 2 k1 1 1 2 k2 k 1 2 1 + x ·x ei2 = xi x + δi 1 + x ·x , (12.108) 2 2 where the world spinor index k in the coordinate components on the righthand side has the same value as the tangent space index k on the left-hand side. A generic expression for the HKT metric given in Eq. (12.81) involves an kb extra factor Gab . Thus, generically, the objects eia found through the definition (12.66) are not quite vielbeins, they are quasivielbeins, and one should rotate them by the matrix R as in (12.95) to obtain the true vielbeins. But in the hyper-K¨ ahler case when M, M −1 ∈ Sp(n) and L = −q +a qa− /8, we kb derive Fab = Ωab and Gab = Ωab . Then eia are the true vielbeins from which the metric is constructed according to (12.85). In our case, the components of the metric are 1 + 21 x1 ·x2 , 1 + x1 ·x2 1 + 21 x1 ·x2 gi1,j2 = gj2,i1 = εij (1 + x1 ·x2 ) + x2i x1j . (12.109) 1 + x1 ·x2 The metric (12.109) is equivalent to the Taub-NUT metric written in a more familiar form (3.38), which we rewrite here: gi1,j1 = x2i x2j

1 + 12 x1 ·x2 , 1 + x1 ·x2

gi2,j2 = x1i x1j

2

ds2 = V −1 (r) (dΨ + AdX) + V (r) dXdX ,

(12.110)

where Ψ ∈ (0, 2π), V (r) =

1 +λ r

(12.111)

(r = |X|) and A1 =

X2 , r (r + X 3 )

A2 = −

X1 , r (r + X 3 )

A3 = 0

(12.112)

16 The higher terms in the harmonic expansion in the L.H.S. of Eq. (12.66) cancel, as they should. The reader may be amused to check it explicitly using the identity + + − + − u+ m un uk = u(m un uk) +

1 + (εmk u+ n + εnk um ) . 3

(12.107)

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The correspondence is established by setting in (12.111) λ = 1 and changing the variables as17 X 1 = x21 x22 − x11 x12 , X 2 = i(x11 x12 + x21 x22 ), X 3 = x11 x22 + x21 x12 , x22 (12.113) Ψ = i ln 11 . x The relations
i r = x1 ·x2 , dΨ + AdX = 1 2 x2 ·dx1 + x1 ·dx2 , x ·x (dX)2 = (x1 ·dx2 )2 + (x2 ·dx1 )2 + 2(x1 ·dx1 )(x2 ·dx2 ) + 2(x1 ·x2 )(dx1 ·dx2 )

(12.114)

18

hold.

Inserting (12.111), (12.112) and (12.113) in (12.110), we obtain 2 + x1 ·x2 2 ds2 = (x ·dx1 + x1 ·dx2 )2 + 4(1 + x1 ·x2 )dx1 ·dx2 , (12.115) 1 + x1 ·x2 which coincides up to the extra irrelevant factor 2 with what the metric (12.109) gives. Another example where the constraints can be explicitly solved and the metric explicitly found is the model described by + 2 (ξ jk u+ j uk ) (12.116) L+4 = 2 (q +a u− a) with an arbitrary symmetric ξ jk . After performing the program outlined above and choosing the coordinates in an appropriate way, one arrives [78, 117] at the hyper-K¨ ahler Eguchi-Hanson metric (3.41). 12.4.2

Delduc-Valent metric

Consider now a 4-dimensional model (n = 1) with L = q +a qa− and the constraints [33] D++ q +1 ≡ L(+3)1 = (q +1 )2 q +2 (λ + iρ) D++ q +2 ≡ L(+3)2 = −q +1 (q +2 )2 (λ − iρ)

(12.117)

with real λ, ρ. When ρ = 0, we go back to the Taub-NUT system, but if ρ 6= 0, ∂a L+3a 6= 0, the constraints (12.117) are not expressed as in (12.97) and we are dealing with a nontrivial HKT manifold. Using a general technique described above, one can derive its metric19 : 4dτ 2 = V −1 (s)(dΨ + ω)2 + V (s)Γ , 17 A

Xp

−(σ p )ij

(12.118) xi1 xj2

more compact form of the first line in (12.113) is = with (σ p )ij written in (3.17). 18 Note that the variable r as defined in (12.114) is real and non-negative in virtue of pseudoreality of xia . 19 We quote Eq. (24) of Ref. [33] where we set γ = 1. 0

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where ρ(1 + λs) 1 ds , + λ, ω = cos θ dφ − s 1 + ρ2 s2 ds2 s2 Γ = + (dθ2 + sin2 θ dφ2 ) . 2 2 2 (1 + ρ s ) 1 + ρ2 s2

V (s) =

(12.119)

In the limit ρ → 0, this metric goes over to the Taub-NUT metric (12.110). For ρ 6= 0, the Bismut connection of this model involves a nontrivial torsion. The explicit expression for the torsion form is C =

ρs(1 + λs) ds ∧ sin θdθ ∧ dφ . (1 + ρ2 s2 )2

(12.120)

In accord with Theorem 12.9, this form is closed. Let us show now that this metric is irreducible and prove thereby Theorem 12.10. Proof. If the HKT metric (12.118) were reducible, a certain conformal transformation d˜ τ 2 = G(s, Ψ, θ, φ)dτ 2

(12.121)

making the metric d˜ τ 2 hyper-K¨ahler would exist [see the remark after Eq. (12.95)]. Equivalently, the vielbeins would √ be conformal to the hyperK¨ ahler ones, with a real conformal factor G. Then the ordinary LeviCivita covariant derivatives of the complex structures [the latter do not change under such conformal transformations—see Eq. (12.96)] with the Christoffel symbols following from (12.121) must vanish: ∇M INP = ∇M JNP = ∇M KNP = 0 .

(12.122)

This condition can be solved to determine the conformal factor G [119], but it turns out20 that in the particular case of the Delduc-Valent metric (12.118), this procedure gives a complex function G(s, Ψ, θ, φ) and a complex “hyper-K¨ ahler metric”. Real solutions to the constraints (12.122) do not exist.

20 We

address the reader to the papers [120, 121] for details.

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Gauge Fields

In the three preceding sections we discussed and analyzed only the first term in the generic N = 4 action (12.47). Let us now look at the second term, Z Sgauge = i dtA du d2 θ+ L++ (q +a , u± ) . (12.123) Substitute there the component expansion (12.36) and integrate over d2 θ+ = dθ¯+ dθ+ . The bosonic term in the action reads Z ∂L++ (f +b , u) ˜−a bos A , (12.124) Sgauge = i dtdu ∂f +a where A˜−a is expressed via x˙ kb as in (12.76). Thus, we can write Z bos Sgauge = − dt Akb x˙ kb (12.125) with the Abelian gauge potential Z ja Akb (t) = −2ekb du (∂+a L++ )(M −1 )a a u− j .

(12.126)

The full component expression for Sgauge is given by the equation (9.65), which we also reproduce here:   Z i M M N SA = dt −AM x˙ + FM N ψ ψ (12.127) 2 √ √ (xM = i(ΣM )ja xja / 2, AM = i(ΣM )kb Akb / 2 and FM N = ∂M AN − ∂N AM ). Theorem 12.11. For the action (12.127) to enjoy N = 4 supersymmetry, N the field density FAB = eM A eB FM N should belong to the algebra sp(n). Proof. In Theorem 9.6, we have established that the action (12.127) enjoys the extra supersymmetry (9.37), provided the field strength tensor commutes with the complex structure. And to enjoy three extra supersymmetries (12.51), it should commute with all three complex structures: p p FAC ICB − IAC FCB = 0 .

(12.128)

This condition has exactly the same form as (3.20), and we can use the result of Theorem 3.2 that any matrix satisfying this condition has the form (3.25) and belongs to sp(n).

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This fact is confirmed by the direct calculations [80]. Let us show how it works for the flat model with L+3a = 0 and L ∝ q + q − . In this case, (12.126) is reduced to Z Akb = −2 du (∂+b L++ ) u− (12.129) k . We have 

Z −2

du

Fja, kb = ∂ja Akb − ∂kb Aja =  ∂ 2 L++ − u+ j uk − (ja ↔ kb) . ∂f +a ∂f +b

(12.130)

Bearing in mind the symmetry of ∂ 2 L++ /(∂f +a ∂f +b ), the field density is − + − proportional to u+ j uk − uk uj = εjk and has, indeed, the form (3.25). For n = 1, FAB ∈ su(2). That means that FAB is anti-self-dual, 1 FAB = − εABCD FCD , 2 if the complex structures (with which FAB commutes) are chosen self-dual, as in (3.3). Naturally, a choice when the complex structures are anti-selfdual while FAB is self-dual, is also possible. The fact that, to keep N = 4 supersymmetry of a 4-dimensional HKT model after inclusion of gauge fields, the latter must be (anti-)self-dual, was noticed in [79, 122].

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Chapter 13

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What physicists call gauge fields, mathematicians call principal Gconnections. We gave the mathematical definition of the latter in Chap. 1.3. We discussed systems including gauge fields in several places in this book— in particular, we did so in Chaps. 5,6,9,12. We mentioned there an essential feature of such systems: gauge fields are not arbitrary; certain integrals that involve field densities, like the magnetic flux (6.65), may acquire only a particular set of discrete (quantized) values. However, we did not discuss this remarkable feature (quantization of topological charge) in detail. The time has come to do so. Gauge fields may live both on non-compact and compact manifolds. For example, the Landau problem discussed in Chap. 5 was defined on R2 (and even on R3 if one also takes into account free motion along the direction of the magnetic field). However, in this and the next chapter we will mostly limit our discussion to compact manifolds, where both physics and mathematics are simpler and more interesting in our opinion.

13.1

Spinors



Dirac Operator

In Chap. 1.4, we introduced the object ωAB,M and called it the spin connection. But the “etymology” of this name was left obscure there. We will elucidate it now. Consider first flat D-dimensional space. The group SO(D) [or, better to say, its double covering Spin(D)—see below] involves spinor representations. For odd D, there is the unique irreducible spinor representation, but, to make contact with supersymmetry, we will only be interested in even D, where there are two irreducible representations SL and SR —left and right spinors. Consider the set of complex functions Φα (x) that belong to the 275

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sum SL ⊕ SR (the bispinor representation1 ). The index α takes ND = 2D/2 different values. We now introduce a convenient object: Definition 13.1. Hermitian matrices γ A=1,...,D of dimension ND × ND satisfying the Clifford algebra γ A γ B + γ B γ A = 2δ AB

(13.1) 2

are called Euclidean D-dimensional gamma matrices.

Remark. We are interested only in even-dimensional manifolds, but gamma matrices can also be defined for odd D. Their dimension in this case is Nodd D = 2(D−1)/2 . Explicit choices for γ A may be different, and we need not specify it. In these terms, the law of infinitesimal global rotations for the bispinor Φα acquires a nice form: 1 (13.2) δΦα = θAB (γ A γ B − γ B γ A )αβ Φβ , 8 where θAB = −θBA is a small rotation angle in the plane [AB]. The reader can be convinced that the generators tAB ∼ i(γ A γ B − γ B γ A ) commute as they should in so(D), [tAB , tCD ] ∼ i(δ AC tBD − δ BC tAD + δ AD tBC − δ BD tAC ),

(13.3)

and that the norm Φ∗α Φα is invariant under the rotations (13.2). One can also note that the matrix nπ o exp (γ A γ B − γ B γ A ) 2 of the finite rotation by 2π in the plane [AB] coincides with −1. That is why mathematicians call the group involving spinor representations Spin(D) rather than SO(D), with Spin(D) representing a two-fold covering of SO(D). But we need not plunge further into these group theory details. Consider the expression (D / Φ)α = (γ M )αβ ∂M Φβ , 1 For

(13.4)

D = 2, physicists habitually call this representation “spinor” rather than “bispinor”, which is not quite exact—the complex 2-dimensional representation of SO(2) is reducible. We also remind that what physicists call “representation” is “representation space” in the mathematical language. 2 They were first introduced ninety years ago by Dirac in Minkowski 4-dimensional space. In this case, Dirac matrices satisfy the relation {γ µ , γ ν } = 2η µν = 2 diag(1, −1, −1, −1).

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Assuming that γ M transform under rotations as a vector, in the same way as the gradient operator ∂M , it is easy to see that (D / Φ)α transforms in the same way as Φα , i.e. it represents a bispinor. • The operator (13.4) is called the Dirac operator in flat space in the absence of the gauge fields. The gauge field is described by the vector potential (alias, a principal G-connection) AM (x). It can be Abelian (mathematicians call the corresponding bundle a line bundle) or non-Abelian. In the latter case, it represents a Hermitian matrix, an element of the Lie algebra g, while the bispinor field should be placed now in the fundamental representation of G and endowed with the extra group index, Φα → Φaα . The Dirac operator acquires the extra term: D / = γ M (∂M + iAM ) .

(13.5)

a

As was explained in Chap. 1.3, (D / Φ) transforms under gauge rotations in the same way as Φa . Finally, let us define the Dirac operator on a curved manifold in a nontrivial gauge background as   1 B C M A (13.6) D / = eA γ ∂M + ωBC,M γ γ + iAM . 4 With this definition, (D / Φ)aα transforms in the same way as Φaα under both gauge rotations and orthogonal rotations in the tangent plane with xdependent parameters, 1 1 / Φ) = θBC (x) γ B γ C D / Φ . (13.7) δΦ = θBC (x) γ B γ C Φ ⇒ δ(D 4 4 Indeed, one can be convinced that the term ∼ ∂M θBC in D / δΦ due to the variation (13.7) exactly cancels the term with the same structure coming from the variation δωBC,M in D / [see Eq. (1.51)]. Equation (13.6) is a local expression. A nontrivial question is whether the Dirac operator can be defined globally on the whole manifold. To do so, one has to define the spinor bundle, i.e. divide the manifold in several charts and define the bispinor field in each chart in such a way that, in the intersection regions, the bispinor fields from two different charts Uα and Uβ are interrelated by a transformation Ωαβ ∈ Spin(D). If non-empty overlap regions Uα ∩Uβ ∩Uγ of three different charts exist, the group-valued functions Ωαβ should satisfy there the consistency condition Ωαβ Ωβγ = Ωαγ . Consider the “purely gravitational” Dirac operator (13.6) with AM = 0 and suppose that in each chart the bispinor field represents an eigenfucntion

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of D /. Suppose further that, in the intersection regions, the sets of vielbeins eM A defined in each chart are related by the SO(D) rotations that correspond to Ωαβ . Definition 13.2. People say that the manifolds admitting spinor bundles satisfying the conditions above admit a spin structure. Not all manifolds do. For example, CP2k do not admit spin structure and the Dirac operator not including an extra gauge field AM is not well defined there. However, we will see soon that the so-called spinC structure on CP2k is admissible: if an Abelian gauge field AM of a special form is present, the Dirac operator is globally well defined. The bispinor fields in the overlapping charts are then related by Ωαβ ∈ Spin(D) × U (1). 13.1.1

Dirac operator and the supercharges

We have already encountered the flat Abelian Dirac operator (13.5) in Chaps. 5,6. We showed there that the problem of 2-dimensional motion with the Hamiltonian representing the square of the operator (13.5) is in fact supersymmetric. We will show now that it is also true in more general cases. Theorem 13.1. Consider the Dirac operator (13.6) defined on an evendimensional manifold with an Abelian or non-Abelian gauge field AM (x). Then the operators i i ˆ† = − √ ˆ = − √ D / (1 − γ D+1 ) , Q D / (1 + γ D+1 ) , (13.8) Q 2 2 2 2 where (−i)D/2 γ D+1 = εA1 ,...,AD γ A1 · · · γ AD , (13.9) D! obey the supersymmetry algebra (5.1) with the Hamiltonian 2

/ ˆ = −D H . 2

(13.10)

Proof. This follows immediately from the fact that γ D+1 anticommutes with all γ Aj and that (γ D+1 )2 = 1. Remarks. • The requirement for D to be even is essential. If D is odd, the R.H.S. of Eq. (13.9) would not give a nontrivial matrix anticommuting with γ Aj . It would be proportional to 1.

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• A collorary from (γ D+1 )2 = 1 is that γ D+1 can be diagonalized with the eigenvalues ±1. In fact, a half of these eigenvalues are +1 and a half −1. Definition 13.3. A bispinor satisfying the condition γ D+1 Ψ = Ψ is called right-handed and a bispinor satisfying the condition γ D+1 Ψ = −Ψ is called left-handed. As (13.9) commutes with the Hamiltonian, the eigenstates of the latter can be divided into the two classes: right-handed and left-handed ones. In the rest of this and in the two following sections (13.2 and 13.3) we will concentrate on Abelian gauge fields, postponing the discussion of the non-Abelian case to Chaps. 13.4, 13.5. The Hilbert space where the supercharges (13.8) and the Hamiltonian (13.10) act involves the bispinors Φα (x). Let us now try to treat this system in the same way as we did for other supersymmetric systems considered in this book: to define a Hilbert space involving the functions depending on the coordinates and also on holomorphic Grassmann variables and to map ˆ Q ˆ†, H ˆ to the operators acting there. We also would like to the triple Q, derive the expressions for the classical supercharges and the Hamiltonian. At first sight, this seems to be feasible. The γ-matrices that satisfy the √ ˆA 2 Clifford algebra (13.1) can be mapped to the quantum operators Ψ satisfying the same algebra. √ Their classical counterparts are the nilpotent Grassmann numbers ΨA 2, as in Eq. (9.61).3 Well, as far as the Dirac operator is concerned, there is no problem. It has indeed a classical counterpart—a classical real supercharge. E.g. in the flat case and in the absence of the gauge field, the latter reads Qcl = ΨM PM . (13.11) With the natural definition of the Grassmann Poisson bracket, {ΨM , ΨN }P = iδ M N [cf. the first line in (4.58)], we derive the N = 1 classical supersymmetry algebra {Qcl , Qcl }P = 2iH cl as in (5.44). It is difficult, however, to find a classical counterpart of the nonlinear operator (13.9) and it is difficult to find a √ classical counterpart of ˆ∗ = D the second Hermitian supercharge Q / γ D+1 / 2. For a generic evendimensional manifold, this is impossible [56]. We have a classical supersymmetry anomaly here [55]: N = 2 supersymmetry that our quantum system enjoys cannot be preserved at the classical level! 3 As was mentioned on p. 95, “Grassmann” becomes “Clifford” after quantization. And that means that “Clifford” goes over to “Grassmann” when we go down from quantum theory to classical one.

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The situation is better when our manifold is K¨ ahler. In this case, one can define a quantum supersymmetric system including the Dirac operator as one of two supercharges that have classical counterparts. To see that, √ A ˆ and represent the Dirac operator as we map γ A → 2 Ψ   √ M A 1 B ˆC ˆ ˆ D / = 2 eA Ψ ∂M + ωBC,M Ψ Ψ + iAM . (13.12) 2 Now we distinguish the holomorphic and antiholomorphic fermion compoˆ A ≡ (ψ a , ψˆa¯ ) with ψˆa¯ = ∂/∂ψ a . Plugging this into (13.12), we nents, Ψ obtain an operator which acts on the wave functions Φ(xM , ψ a ). The latter have 2D/2 components—the same number as the number ND of the components of the bispinor Φα (x), and there is a bijective correspondence between the two Hilbert spaces. Theorem 13.2. One can define this bijective mapping in such a way that the right-handed spinors go over to the bosonic states and the left-handed spinors to the fermionic states. Then ˆ

γ D+1 ↔ (−1)F .

(13.13)

Proof. This immediately follows from the mentioned fact that a half of the eigenvalues of γ D+1 are +1 and another half are −1. Capitalizing on the fact that, for the K¨ahler manifolds, the components ωbc,M and ω¯b¯c,M vanish [see Eq. (2.31)], we can represent D / = D / Hol − (D / Hol )† ,

(13.14)

where

(D / Hol )†

√

  1 a ˆ¯¯b ψ c − ψ c ψˆ¯b ) + iA 2 em ψ ∂ + ω ( ψ ¯ m m , a 2 bc,m   √ m 1 bˆ ¯ ˆ a ¯ c¯ c¯ b ˆ ¯ ¯ ¯ = − 2 ea¯ ψ ∂m , ¯ + ωb¯ c,m ¯ (ψ ψ − ψ ψ ) + iAm ¯ 2

D / Hol =

(13.15) (13.16)

Theorem 13.3. For the K¨ ahler manifolds, the holomorphic and antiholomorphic parts of the Dirac operator (13.15), (13.16) with Am = Am ¯ = 0 √ coincide up to an extra factor i 2 with the covariant quantum supercharges (9.11) for the system (9.1) with W = 0. ˆ in Eq. (9.11), we send the operator ψˆ¯¯b in Proof. To compare D / Hol with Q the expression (13.15) to the right, which gives us the extra term ∝ ω¯bb,m . For K¨ ahler manifolds, when ω¯bc,m = e¯nb¯ ∂m ecn¯¯ ,

n c ωb¯c,m ¯ = eb ∂m en

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[see Eq. (2.31)], we derive ω¯bb,m = ∂m ln det e¯ .

(13.17)

Hol †

Performing a similar transformation for (D / ) gives us a term ∝ ωb¯b,m ¯ = ∂m ln det e. These extra contributions exactly coincide with the structures ¯ displayed in (9.11) (we remind that the expression (9.11) was written under √ a simplifying assumption det e = det e¯ = det h). Remark. The operators D / Hol and (D / Hol )† can be downgraded to the ˆ ˆ∗ = D classical phase space functions (9.7). But the operator Q / (−1)F , which does not coincide with D / Hol + (D / Hol )† , still cannot! Suppose now that an Abelian gauge field AM is present. In the particular case when it has the form (9.15): AM = (i∂m W, −i∂m ¯ W),

(13.18)

the √ extra contributions in (13.15) and (13.16) coincide up to the extra factor i 2 with the shift (9.13) of the supercharges (9.11). In Chap. 9 (see Theorem 9.2), we established that the supercharges † (9.11) with the shift (9.13) map to the operators ∂W and ∂W of the twisted Dolbeaux complex with the twisted exterior holomorphic derivative   1 (13.19) ∂W = ∂ − ∂ W − ln det h ∧ . 4 † Alternatively, they can be mapped to the operators ∂¯W and ∂¯W of the twisted anti-Dolbeaux complex with   1 ¯ ¯ ¯ (13.20) ∂W = ∂ + ∂ W + ln det h ∧ . 4 Bearing all this in mind, we arrive at the theorem:

Theorem 13.4. For the K¨ ahler manifolds, the Dirac operator (13.6) with the gauge field (13.18) can be decomposed into the holomorphic and antiholomorphic parts as in (13.14). The holomorphic part maps to the exterior holomorphic derivative (13.19) of the twisted Dolbeault complex. The antiholomorphic part maps to the exterior antiholomorphic derivative (13.20). The fact that the Dolbeault complex maps to the Dirac complex and ˆ ˆW + Q ¯ † is isomorphic to the Dirac operator was mentioned that the sum Q W before (see the pagagraph after Theorem 9.2). In Chap. 9, we did not prove this fact. We promised to do so in Chap. 13, and we have now fulfilled this promise. However, the formulation of Theorem 13.4 involves an important restriction: the equivalence has been proven only for the K¨ahler manifolds.

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A natural question is what happens for generic complex, not necessarily K¨ ahler manifolds. We will now show that a more general version of such equivalence can also be established in this case. The proof above relied on the decomposition (13.14). It does not hold in a generic complex case for the ordinary Dirac operator because of the presence of the components ωab,m ¯ and ωa ¯¯ b,m in the spin connection. We can define, however, a deformed Dirac operator that involves not only the spin connection, but also the torsion:   1 1 B C K P tors M A D / = eA γ ∂M + ωBC,M γ γ − CM KP γ γ + iAM , (13.21) 4 24 where the torsion CM KP is given by (2.18). This is nothing but the quantum version of the supercharge Q0 in (9.61), where we also added the gauge field term. As was noted in the paragraph after Eq. (9.61), the role of the torsion term is to cancel the troublesome terms ∝ ωab,m ¯ and ∝ ωa ¯¯ b,m in the expression for D / tors . The operator D / tors admits the decomposition (13.14) and we can claim that Theorem 13.5. For a generic Dolbeault complex involving the twisted ex† † terior holomorphic derivative (13.19), the operator ∂W + ∂W , where ∂W is the Hermitian conjugate of ∂W , maps to the deformed torsionful Dirac operator (13.21) with the gauge field (13.18). Thus, for a generic complex manifold, one can define at least two different quantum supersymmetric problems: √ (1) The Dirac problem including the Hermitian supercharges −iD // 2 and √ D / γ D+1 / 2. (2) The Dolbeault problem including the Hermitian √ supercharges, one of which is the deformed Dirac operator −iD / tors / 2 and another one is the quantum version of the operator Q in (9.61) (translated if you will into the spinor language). Generically, the Hamiltonians of these two quantum problems do not coincide, though for the K¨ ahler manifolds (and, obviously, in flat space), they do. We note finally that the Dirac problem can be defined for any evendimensional manifold, while the Dolbeault problem is usually defined only for the complex manifolds.4 4 We say “usually” because, as we have seen in Chap. 9.3, there are manifolds like S 4 which are not complex, but can be made complex by removing a point. And the

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13.2

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Magnetic Field on S 2

We are going to treat in much detail the simplest nontrivial case—an Abelian gauge field on S 2 . In the mathematical language, we are dealing with a particular case of the principal fiber bundle—a line bundle. In Chap. 1.3 we defined the principal fiber bundle locally, in a particular chart, and did not discuss what happens in the overlap region. Neither did we so in the preceding section. But now we will. We are interested in the bundles equipped with principal G-connections (gauge fields). In each chart, such a connection represents a vector function AM (x) whose components are the elements of the corresponding Lie algebra represented by Hermitian matrices. In the overlap region between the two charts, we can require A0M = AM , but this requirement is too rigid. We can relax it by requiring for A0M and AM to be related by a gauge transformation (1.40). For Abelian line bundles, when ω(x) = eiχ(x) and AM (x) are ordinary functions, this boils down to A0M = AM − ∂M χ(x) .

(13.22)

The Abelian field density FM N = ∂M AN − ∂N AM (the curvature of our line bundle) stays invariant under this transformation. S 2 can be covered by two charts—the northern and the southern hemispheres overlapping at the equator, where the condition (13.22) is imposed. An important remark is that the symbol χ(x) entering (13.22) is not necessarily a uniquely defined function. The element of the U (1) gauge group ω(x) = eiχ(x) is, but χ(x) is not. We can allow for the phase χ(x) to be shifted by 2πq with an integer q when we go around the equator S 1 . This integer is a topological invariant the mapping U (1) → S 1 . R characterizing R Consider now the integral S 2 F = S 2 dA with A = AM dxM . It is the integral of a total derivative and, by Stokes’ theorem, it seems to vanish, because S 2 has no borders. However, one should take into account the fact that AM in the northern and southern hemispheres may not be the same, being related as in (13.22). Calculating separately the integrals for the northern and southern hemispheres, using Stokes’ theorem and (13.22), we derive Z I F = ∂M χ(x) dxM = 2πq . (13.23) S2

equator

Dolbeault Hamiltonian problem on S 4 \{·} with square-integrable wave functions makes sense and is supersymmetric.

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We arrive at the theorem: Theorem 13.6. For the line bundle on S 2 , the flux of the magnetic field R Φ = F is quantized as in (13.23). A very similar statement was derived in Chap. 6: the Witten index (6.65) for the supersymmetric system (6.63) that described the 2-dimensional motion in a magnetic field (and, by definition, the Witten index is an integer) is equal to Φ/2π. However, there is a certain difference. In Chaps. 5,6, we discussed the motion in flat 2-dimensional space, while now we are dealing with the motion over a curved 2-sphere. If one looks for a physical interpretation, we are dealing, for the “round” sphere with a homogeneous field density FM N εM N = const, with the motion in the field of a magnetic monopole placed at the center of the sphere. In this case, the radial component of the 3-dimensional vector potential vanishes, an electron moving in the radial direction does not feel the presence of the field, and only the angular motion, the motion over S 2 , is nontrivial. The quantization of the magnetic flux boils down then to the quantization of the magnetic charge m, q~c , (13.24) 2 where we restored the physical constants e, c, ~ present in the physical Pauli Hamiltonian (5.14). The quantization condition (13.24) was derived long time ago by Dirac [123]. Let us find the spectrum of the Hamiltonian H = −D / 2 /2 describing the angular motion of a charged particle in the field of the magnetic monopole. S 2 is a K¨ ahler manifold, and the Dirac problem is equivalent to the Dolbeault one. It is convenient, instead of decomposing S 2 into two hemispheres and imposing the matching condition (2.1) at the equator, to choose as one of the charts the whole sphere except the north pole. The second chart would in this case be a small neighbourhood of the north pole. For the first chart, we choose the complex stereographic coordinates em =

x + iy θ z = tan e−iφ = √ , 2 2

θ x − iy z¯ = tan eiφ = √ , 2 2

(13.25)

where θ is the polar angle on S 2 counted from the south to the north, and write the metric as in Eq. (2.6): ds2 =

2dzd¯ z . (1 + z z¯)2

(13.26)

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This metric is singular at z = ∞ (the north pole), but the singularity is integrable. In particular, the area of such sphere is Z dzd¯ z = 2π , (13.27) A = (1 + z z¯)2 √

corresponding to the radius R = 22 . Consider the SQM system with the action   ¯¯ Z DZDZ ¯ ¯ . + W (Z, Z) S = dθdθdt ¯ 2 4(1 + ZZ)

(13.28)

According to Theorem 13.5, this problem is isomorphic to the Dirac problem ¯ on S 2 with the gauge field (13.18). A particularly clever choice for W (Z, Z) is ¯ = − q ln(1 + ZZ) ¯ . W (Z, Z) (13.29) 2 In this case, the form iq dz ∧ d¯ z ¯ dz ∧ d¯ F = dA = −2i ∂ ∂W z = (13.30) 2 (1 + z¯z) √ is proportional to the area 2-form ih dz∧d¯ z = g dx∧dy with the coefficient q, which is nothing but the magnetic field density B—one and the same at all points. Integrating F over S 2 , bearing in mind (13.27) and recalling Theorem 13.6, we derive that q should be an integer. We can arrive at the same conclusion if we express the gauge potential (13.18) in the Cartesian H coordinates (x, y). One can be easily convinced that the circulation A · dx over a small contour embracing the north pole is ±2πB.5 This can be handled with the matching condition (13.22) between the two charts, specified above, with a unique gauge transition function ω = eiχ . In the physical literature, one can meet another interpretation. Physicists (starting from Dirac) did not cover the sphere by two different charts, but considered only one chart with the singularity of the metric at the H north pole. The gauge field is also singular there. Nonzero circulation A · dx along an infinitesimal contour around the pole may be interpreted as an infinitesimally narrow flux line (alias, the Dirac string) carrying the magnetic flux, an integer multiple of 2π, which exactly compensates the flux due to homogeneous magnetic field smeared over the sphere. The point is that the Dirac string with such a quantized flux is not physically observable. One can place it at any point on the sphere by choosing an appropriate gauge. 5 The

sign depends on the direction of the circulation, which we are too lazy to fix.

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The problem (13.28), (13.29) is so simple that the spectrum of its Hamiltonian can be explicitly found. The quantum supercharges given by the sum of (9.11) and (9.13), read   ˆ = −i(1 + z¯z)ψ ∂ − z¯(1 − q) , Q 2(1 + z¯z)   ˆ ¯ = −i(1 + z¯z)ψˆ¯ ∂¯ − z(1 + q) . Q (13.31) 2(1 + z¯z) They act upon the two-component wave functions6 Ψcov = ΨF =0 (¯ z , z) + ψΨF =1 (¯ z , z) normalized with the covariant measure √ z dz = dµ = g d¯

d¯ z dz . (1 + z¯z)2

(13.32)

(13.33)

ˆ ˆ and Q ¯ are Hermitially conjugate to one another with the The operators Q ˆ¯ In the sector F = 0, it ˆ = {Q, ˆ Q}. measure (13.33). The Hamiltonian is H reads ˆ F =0 = −(1 + z¯z)2 ∂ ∂¯ − γ(1 + z z¯)(¯ H z ∂¯ − z∂) + γ 2 z¯z − γ , (13.34) where γ =

q−1 . 2

(13.35)

ˆ F =1 is similar, one has only to interchange z and z¯ The expression for H and inverse the sign of q. Let us first find the vacuum zero-energy states. They should be annihiˆ ˆ and Q. ¯ Let q be nonzero and positive. Then the vacuum lated by both Q ˆ vanishes idenstates must have zero fermion charge. Indeed, if F = 1, QΨ ˆ ¯ tically, while the condition QΨ = 0 gives the equation   z(1 + q) (0) ¯ z , z) = 0 , (13.36) ∂− ΨF =1 (¯ 2(1 + z¯z) which does not have normalizable solutions. ˆ¯ The action of The states in the sector F = 0 are all annihilated by Q. ˆ is nontrivial and leads to the equation Q   γ z¯ (0) ∂+ ΨF =0 (¯ z , z) = 0 . (13.37) 1 + z¯z 6 Here

F is not the field density form, but the fermion charge—the eigenvalue of the ¯=ψ ∂ . operator ψ ψˆ ∂ψ

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And this equation has exactly q independent normalizable solutions: (0)

ΨF =0 = (1 + z¯z)−γ , z¯(1 + z¯z)−γ , . . . , z¯q−1 (1 + z¯z)−γ

(13.38)

(we do not bother about the normalization coefficients). The Witten index of this system is equal to q. Consider now a nonzero negative integer q. All the reasoning is repeated with the only difference that in this case there are no zero-energy states in the sector F = 0, but there are |q| independent vacuum states in the sector F = 1. The Witten index is equal to −q. Finally, if q = 0, there are no zero-energy states.7 The wave functions (13.38) (and also the wave functions of the excited states to be discussed a bit later) are defined in the chart covering the main body of S 2 and not including the north pole. In fact, for the functions (1 + z¯z)−γ , . . . , z¯q−2 (1 + z¯z)−γ , one can include the point z = ∞ in the domain of their definition—these functions vanish there. But the function z¯q−1 (1 + z¯z)−γ does not and, if q 6= 1, it does not have a definite value at z = ∞. This can be remedied if we recall that we are dealing here with the fiber bundle construction. The north pole belongs to the second chart. The wave functions in the different charts may be different, but they have to be related by a gauge transformation, Ψ0 = eiχ Ψ, with the same χ(x) that enters the transformation law (13.22) for the gauge potential. Then the covariant derivatives (∂M + iAM )Ψ in the two charts are related by the same phase factor eiχ as Ψ is. One can be convinced that in our case the wave functions in the two charts are related as  z γ Ψmain body (¯ z , z) , (13.39) Ψnorth pole (¯ z , z) = z¯ and the left-hand side is regular at z = ∞ for all the functions in the spectrum. The full spectrum was found in [124]. Consider first the sector F = 0. The Hamiltonian (13.34) commutes with the angular momentum operator m ˆ = z¯∂¯−z∂ and the eigenstates are characterized by the principal quantum number n = 0, 1, . . . and the integer angular momentum eigenvalues: m = −n, . . . , n + 2γ

7 Incidentally,

if

q ≥ 1,

m = −n + 2γ, . . . , n if

q ≤ 1.

this also follows from the so-called Lichnerowicz formula.

(13.40)

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The energies do not depend on m:8 En(q) F =0

En(q) F =0 = n(n + q)

if

q ≥ 1,

= n(n + 2 − q) + 1 − q

if

q ≤ 1.

(13.41)

As an illustration, we represent in Fig. 13.1 the spectrum of the Hamiltonian (13.34) for q = 3. E 10

n=2

4

n=1

n=0

−2

−1

0

1

2

3

4

m

Fig. 13.1 The spectrum of the Hamiltonian for the magnetic charge q = 3 in the sector F = 0. It involves 3 zero-energy ground states.

Note the relation En2−q = Enq + q − 1 ,

(13.42)

which follows naturally from (13.34) (perform there the inversion γ → −γ, z ↔ z¯). The eigenfunctions (called monopole harmonics [125, 126]) have the form9 F =0 Ψ(q≥1) = eimφ (1 − t)m/2 (1 + t)γ−m/2 Pnm,2γ−m (t) , mn F =0 Ψ(q≤1) = eimφ (1 − t)−m/2 (1 + t)m/2−γ Pn−m,m−2γ (t), (13.44) mn

where t = cos θ =

1 − z¯z , 1 + z¯z

(13.45)

8 This degeneracy is a corollary of spherical symmetry—cf. the familiar degeneracy of the states with a given l and different m for a scalar spherically symmetric potential. 9 In fact, the first line in (13.44) describes also the states with q < 1, as one can be convinced using the identities [126]

n!(n + α + β)! α,β P (t) , (n + α)!(n + β)! n n!(n + α + β)! α,β α,−β Pn+β (t) = 2−β (t + 1)β P (t) , (n + α)!(n + β)! n −α,β Pn+α (t) = 2−α (t − 1)α

(13.43)

which are valid for integer α, β. We have preferred to give two formulas, for q ≥ 1 and for q < 1, to ensure that the proper range of n is n = 0, 1, . . . in both cases.

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eiφ =

p

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z¯/z, and Pnα,β (t) are the Jacobi polynomials:

Pnα,β (t) = n

X Γ(α + n + 1) n!Γ(α + β + n + 1)

k=0

   k n Γ(α + β + n + k + 1) t − 1 . (13.46) k Γ(α + k + 1) 2

The eigenfunctions and the eigenvalues in the sector F = 1 are given by the same expressions with the change q → −q, m → −m, z ↔ z¯ (the eigenfunctions involve, of course, the extra Grassmann factor ψ). When E 6= 0, they can be obtained from the eigenfunctions (13.44) by the action ˆ Inversely, the eigenfunctions of the excited states in of the supercharge Q. the sector F = 0 can be obtained from those in the sector F = 1 by the ˆ ¯ action of Q. If one chooses q = 1 in the sector F = 0 (or q = −1 in the sector F = 1), the expressions are simplified. In this case, γ = 0 and the Hamiltonian reduces to the ordinary Laplacian. The eigenfunctions represent then the ordinary spherical harmonics. The spectrum (13.41) reduces to E = n(n + 1) as it should. 13.2.1

Fractional magnetic charge

For a fiber bundle on S 2 , the magnetic charge must be an integer. The same is true for an ordinary Schr¨odinger problem with the Pauli Hamiltonian (13.10) including the field of a magnetic monopole: if q is not an integer, the wave function becomes singular at one point at least. This is so even if we allow for the wave functions not to coincide in the different charts, but be related by a gauge transformation with a uniquely defined ω(x) = eiχ(x) . Using a still different physical language: if q is not an integer, the Dirac string associated with the singularity of AM (x) becomes observable! This looks weird, and that is why fractional charges are not usually considered, but we want to emphasize: all this does not mean yet that a spectral problem cannot be defined in this case. Indeed, we have seen in Chap. 9.3 that though S 4 is not a complex manifold, one can well define the Dolbeaux complex on S 4 \{·}, and the corresponding spectral problem is supersymmetric. Is the same true for the Dirac problem on S 2 ? Can one define it on S 2 \{·} and to keep the Hamiltonian supersymmetric? The answers to two parts of this question are different [127]: Theorem 13.7. 1. A meaningful spectral problem can be defined even if the magnetic charge is fractional. 2. This problem is not supersymmetric.

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Proof. To define a spectral problem, we have to define the Hilbert space. As a first try, we do the same as we did in Chap. 9.3 for S 4 and include in the Hilbert space all square-integrable functions with the measure (13.33). With this condition, consider the spectrum of the Hamiltonian (13.34), where we do not assume q to be an integer anymore. We obtain then two families of formal solutions to the Schr¨odinger equation: Ψnm = z¯m (1 + t)γ Pnm,2γ−m (t) with

En = n(n + 1 + 2γ)

(13.47)

and ˜ nm = z¯m (1 + t)m−γ Pnm,m−2γ (t) Ψ with En = (n + m + 1)(n + m − 2γ) . (13.48) The integers n, m take the values n = 0, 1, . . . ; m = 0, ±1, . . ., but we have to pick up only the normalizable solutions. For an integer q, the normalizable wave functions in (13.48) are the same as in (13.47) [coinciding also with the functions (13.44)]. For example, three zero energy states for q = 3 represented in Fig. (13.1), z¯ z¯2 1 , , , (13.49) Ψ1,2,3 = 1 + z¯z 1 + z¯z 1 + z¯z are described by (13.47), if choosing n = 0, m = 0, 1, 2, and also by (13.48), if choosing there n = 2, m = 0; n = m = 1; n = 0, m = 2 [the identities (13.43) are at work!]. But for a fractional charge, we encounter a challenge. Take e.g. q = 1 ⇒ γ = − 41 and n = m = 0. Equations (13.47) and (13.48) give two 2 different eigenfunctions: |1i = (1 + t)−1/4 = (1 + z¯z)1/4 with E = 0 1 1 with E = (13.50) |2i = (1 + t)1/4 = 1/4 2 (1 + z¯z) The function |1i is singular at the north pole t = −1, but it is squareintegrable. The energies of the states (13.50) are different, but these states are not mutually orthogonal. And that means that the Hamiltonian is not ˆ Hermitian! One can also see it if one compares the matrix elements h1|H|2i ˆ and h2|H|1i. They differ by a total derivative which does not vanish in this case:  Z dzd¯ z 1 ¯ ˆ ˆ h1|H|2i − h2|H|1i ∼ (1 + z¯z)1/4 (1 + z¯z)2 ∂∂ (1 + z¯z)2 (1 + z¯z)1/4  1 2¯ 1/4 − (1 + z ¯ z) ∂∂(1 + z ¯ z) (1 + z¯z)1/4   Z  Z  Z z¯ d¯ z dz z dzd¯ z + ∂¯ dzd¯ z∼ ∼ ∂ 6= 0 . (13.51) 1 + z¯z 1 + z¯z (1 + z¯z)2

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If one looks at the main reason for this mismatch, it is due to the fact that we included in the Hilbert space the functions that are singular on S 2 . This suggests a remedy: let us redefine the Hilbert space in our problem and include there only the wave functions that are regular on S 2 . Then the state |1i is not admissible. We have no problem with hermiticity in the reduced Hilbert space and the spectral problem is well defined there. It is well defined in the sector F = 0 and also in the sector F = 1. However, this spectral problem is not supersymmetric. To see this, ˆ in (13.31) on the consider the result of the action of the supercharge Q regular function |2i. We obtain   1 ψ¯ z z¯ ˆ ∼ . (13.52) Q|2i ∼ ψ(1 + z¯z) ∂ − 4(1 + z¯z) (1 + z¯z)1/4 (1 + z¯z)1/4 This function is not regular as z = ∞. In other words, the state |2i is admissible and benign as far as the Hamiltonian spectral problem is conˆ the action of Q ˆ brings us cerned, but it does not belong to the domain of Q: outside of the Hilbert space where the Hamiltonian is defined. As a result, certain states do not have superpartners and supersymmetry is lost. A similar phenomenon (the loss of apparent supersymmetry due to the fact that the superpartners of some states lie outside the Hilbert space and thus do not belong to the spectrum of the Hamiltonian) is known to show up for some other systems. The simplest example [128] is probably Witten’s supersymmetric Hamiltonian (5.28), 0 2 ˆ2 ˆ¯ . ˆ = P + [W (x)] + 1 W 00 (x)(ψ ψˆ¯ − ψψ) H 2 2

(13.53)

with the superpotential W (x) = ωx2 /2 − ln x, considered at the half-line x ≥ 0. In the bosonic sector F = 0, the Hamiltonian is 2 2 ˆ2 ˆ B = P + ω x − 3ω . H 2 2 2

(13.54)

The ground state Ψ0 ∝ exp{−ω 2 x2 /2} has the negative energy E = −ω, which is obviously non-consistent with supersymmetry. The reason for the trouble is that the function Ψ0 does not belong to the domain of the ˆ 0 behaves as ∝ 1/x near x = 0 ˆ ∝ Pˆ − iW 0 : the function QΨ supercharge Q and is not square-integrable. To make the spectrum supersymmetric, we should in this case restrict the Hilbert space and consider only the functions that vanish at the origin, Ψ(0) = 0. With this restriction, the ground state ˜ 0 (x) = xΨ0 (x) has zero energy. Ψ

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On the other hand, for a magnetic field with noninteger flux, there is no way to make the Pauli Hamiltonian supersymmetric. We have just shown that the Hilbert space of regular functions on S 2 does not constitute the ˆ but this is also true for any other restricted or enhanced domain of Q, Hilbert space. We address the reader to Ref. [127] for further details. 13.3

Line Bundles on CPn

Another name for 2-sphere is CP1 . In the previous section, we showed that the model (13.28), (13.29) maintains its apparent supersymmetry under quantization iff the topological charge q is an integer. The same concerns the case when the sphere is “crumbled” and the magnetic field is not homogeneous. Indeed, we know from Theorem 9.2 that, for K¨ahler manifolds (and CP1 is K¨ ahler), the Dirac complex with the gauge field (13.18) maps to the twistedDolbeault complex with gauge field  a “shifted”   1 1 . (13.55) W − ln det h A˜M = i∂m W − ln det h , −i∂m ¯ 4 4 For a complex manifold, a “pure” Dolbeault Hamiltonian H Dolb = {∂, ∂ † } is well-defined and Hermitian. The same concerns the twisted Dolbeault Hamiltonian with the gauge field (13.55) provided that AM represents a fiber bundle and one can safely travel between the charts. Then, according R to Theorem 13.6, the flux S 2 F is quantized. And the same is true for Z Z ˜ F = F − 2π . (13.56) S2

S2

Let us now go to CPn>1 . Each such manifold can be glued up from n + 1 disks of complex dimension n (see p. 48). In each chart, the metric can be chosen in the Fubini-Study form (2.80):   ¯ 1 z p z¯m δmp¯ − , k = 1, . . . , n . (13.57) hmp¯ = 1 + z¯k z k 1 + z¯k z k Introduce now the gauge field with the prepotential q W = ln det h . (13.58) 2(n + 1) We will show that the parameter q is not arbitrary, but quantized. In this section, we will prove it by pure geometric reasoning, and in the next chapter we will put on our glasses and derive the quantization of q using the supersymmetric methods. Theorem 13.8. For the field (13.55) with W given by (13.58) to represent a benign line bundle so that the twisted Dolbeault complex is well defined, q should be an integer if n is odd and half-integer if n is even.

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Proof. Equation (13.57) implies det h =

1 (1 +

z¯k z k )n+1

.

(13.59)

Then Eq. (13.55) gives  n+1 q z¯m (13.60) − 4 2 1 + z¯k z k and the complex-conjugated expression for A˜m ¯ . This field is defined in the chart C0 including the origin z k = 0, but not including the points of CPn where one of z k hits infinity. To describe the whole manifold, one needs n extra charts with the coordinates C1 : (u1 = 1/z 1 , u2,...,n = z 2,...,n /z1 ), C2 : (v 2 = 1/z 2 , v 1,3,...,n = z 1,3,...,n /z 2 ), etc. We want to prolongate A˜M to these charts by performing a gauge transformation (13.22) in the overlap regions in such a way that the transformed field would not be singular. When z 1 → ∞, the gauge field (13.60) behaves as iα¯ z2 iα¯ z3 iα A˜2 (z) ∼ 1 1 , A˜3 (z) ∼ 1 1 , . . . , (13.61) A˜1 (z) ∼ 1 , z z¯ z z¯ z where α = (n + 1)/4 − q/2. At first sight, it is not singular at z 1 → ∞ and an extra gauge transformation is not needed, but one should recall that A˜M is a covariant vector, whose holomorphic component transforms under a variable change as ∂z n ˜ A˜m (u) = An (z) . (13.62) ∂um k 1 k We then see that A˜2,3,... (u ) are regular at u = 0, but A˜1 (u ) ∼ −iα/u1 is singular and a gauge transformation (13.22) is required. We should perform it in the overlap region C0 ∩ C1 . The topology of the latter is S 1 × Cn−1 . The function ω(x) = eiχ(x) defines a mapping S 1 × Cn−1 → U (1), which has the same distinct topological classes as the mapping S 1 → U (1) that we had for CP1 . Only some discrete values of α are admissible. To find them, recall our convention: x + iy A˜x − iA˜y √ u1 = √ , A˜1 = . (13.63) 2 2 Then the singular part of A˜ is 2αy 2αx A˜x = − 2 , A˜y = 2 . (13.64) x + y2 x + y2 Formally, one can get rid of it by applying the gauge transformation (13.22) with χ(x, y) = 2α arctan(y/x). To assure for eiχ(x,y) to be uniquely defined, 2α should be integer. In other words, n+1 q− = integer , (13.65) 2 from which the statement of the theorem follows. A˜m = i



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Remark. A corollary of this theorem is the fact mentioned above: CPn with even n do not admit spin structure: the “purely gravitational” Dirac operator (13.6) with AM = 0 (implying q = 0 ) is not well defined—there is a topological obstacle for that. For odd n, the value q = 0 is admissible.

13.4 13.4.1

Non-Abelian Fields



Instantons on S 4

Non-Abelian bundles

In Chaps. 9,12 and up to now in this chapter, we have mostly discussed Abelian gauge fields, the line bundles in the mathematical language. But the general construction of the principal fiber bundles outlined in Chap. 1.3 applies also to non-Abelian groups. The connection AM (x) belongs then to the Lie algebra of any simple or semi-simple gauge group G. The connections in the different charts Cj are related as in (1.40) with the group-valued transition functions ω(x) that are uniquely defined in the overlap regions Cj ∩ Ck . An interesting mathematics (and physics) emerges when the mappings Cj ∩ Ck → G are not trivial. For simple non-Abelian G, this happens when the dimension of the manifold is D = 4 or more. If D = 4, the overlap regions can have the topology of S 3 . Topologically nontrivial mappings correspond then to nontrivial elements of π3 (G). Note that there may also be another source for nontrivial topology. The elements of the center of G do not act faithfully on the gauge fields belonging to the adjoint representation. That means that the relevant gauge group is G/ZG rather than just G. The fundamental group of G/ZG may be nontrivial. For example, for the unitary groups, π1 [SU (N )/ZN ] = ZN . If the fundamental group of the 4-dimensional manifold we are interested in is also nontrivial, this can bring about the configurations with fractional Pontryagin number (13.70). These configurations were first discovered by ’t Hooft and are called torons [129]. But these complications are beyond the scope of our book. Here we will consider only the simplest non-Abelian bundle that has the gauge group G = SU (2) and is defined on S 4 , which is simply connected and involves no torons. S 4 can be covered by two charts—the northern and the southern hemispheres. The overlap region has the topology of S 3 and only π3 (G) is relevant. Topologically distinct bundles are characterized by an integer number— an element of π3 [SU (2)] = π3 (S 3 ) = Z. Let us first derive an algebraic

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expression for this integer, an analog of the expression Z i ω −1 dω, ω ∈ U (1) n = − 2π S 1 for the mapping S 1 → U (1) [cf. (13.23)].

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(13.66)

Theorem 13.9. The topological class of the mapping S 3 → SU (2) is given by the expression Z  1 q= Tr ω −1 dω ∧ ω −1 dω ∧ ω −1 dω , ω ∈ SU (2). (13.67) 2 24π S 3 Proof. Consider first the similar integral over the proper SU (2) group: Z  1 Tr ω −1 dω ∧ ω −1 dω ∧ ω −1 dω . (13.68) qSU (2) = 2 24π SU (2) The integrand in (13.68) is a 3-form dµ that is invariant under a group multiplication, dµ(ωα) = dµ(αω) = dµ(ω) ,

(13.69)

where α is an arbitrary constant SU (2) matrix. Up to a coefficient, there is unique 3-form enjoying this property — the invariant volume form. The coefficient 1/24π 2 has been chosen in such a way that the integral (13.68) is equal to 1. A dedicated reader can check that qSU (2) = 1, indeed, by an explicit calculation. As the integral does not depend on the parameterization, one may use any convenient one. One can e.g. choose ω(ξ) = ξ4 − iσj ξj with the condition ξ42 + p ξ 2 = 1. (It is convenient to integrate only over a half of S 3 by posing ξ4 = 1 − ξ 2 and multiply the result by 2.) Going back to the integral (13.67), we make there a local variable change from the coordinates x parameterizing S 3 to the coordinates ξ parameterizing SU (2). The integrand acquires the Jacobian factor |∂x/∂ξ|. Then the integral acquires an extra integer factor indicating the number of times that the sphere S 3 covers the group. This factor can also be negative (for example, if x = −ξ) or zero. Now we can prove the following important theorem: Theorem 13.10. For a principal fiber bundle on S 4 with the SU (2) gauge group, the integral Z 1 q = Tr{F ∧ F } , (13.70) 8π 2 S 4 where F = dA + iA ∧ A is a field density 2-form representing a Hermitian 2 × 2 matrix [see (1.45)], is an integer.

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The invariant (13.70) is called the Pontryagin number, the integral of the second Chern class Tr{F ∧ F }/8π 2 (and the 2-form F/2π proportional to the magnetic field density, the integral of which over a compact 2-manifold is an integer, is called the first Chern class). Proof. Note the identity 2i A ∧ A ∧ A . (13.71) 3 (The term ∼ A∧A∧A∧A in F ∧F vanishes due to the property t[a tb tc td] = 0: the group SU (2) has only three generators.) R By Stokes’ theorem, the4integral (13.70) is proportional to the integral Tr{K} over the border. S has no borders, and integral seems to be zero. We have to recall, however, that, to define a fiber bundle, the manifold should be represented as a union of charts. In our case, it is sufficient to divide S 4 into two charts — the northern and the southern hemispheres. They overlap at the equator S 3 . The gauge fields in the two charts are different, being related in the overlap region by a gauge transformation F ∧ F = dK,

where

K = A ∧ dA +

A0 = (ωA + idω)ω −1 .

(13.72)

Then the integral (13.70) over the whole S 4 is equal to the difference of two integrals (Chern-Simons invariants) Z Z 1 1 0 Tr{K } − Tr{K} . (13.73) q = 8π 2 S 3 8π 2 S 3 A short calculation gives   Z 1 1 −1 −1 −1 −1 q= Tr −id(dω∧Aω ) − dω · ω ∧ dω · ω ∧ dω · ω . (13.74) 8π 2 S 3 3 The first term is the integral over a total derivative, which vanishes (we are now in one of the charts, where A is uniquely defined), and the second term coincides up to a sign with the invariant (13.67). It is an integer.

13.4.2

Self-dual fields



Semi-dymanical variables

An especially interesting class of topologically nontrivial gauge fields on S 4 are self-dual or anti-self-dual fields satisfying the condition 1√ 1 FM N = ±F˜M N = ± EM N P Q F P Q = ± g εM N P Q F P Q (13.75) 2 2 They are interesting both from the physical and mathematical viewpoints. The physical interest comes from the following noteworthy fact:

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Theorem 13.11. (Anti-)self-dual fields realize the minimum of the action functional 10 Z 1 √ Tr{FM N F M N } g d4 x S = (13.76) 2 in a given topological class. Proof. Consider the functional Z √ A = Tr{(FM N − F˜M N )(F M N − F˜ M N )} g d4 x .

(13.77)

A is positive definite and hence Z 1 √ S ≥ Tr{FM N F˜ M N } g d4 x . (13.78) 2 The integral on the R.H.S. is nothing but the Pontryagin number q in (13.70) multiplied by 8π 2 . Obviously, when FM N = F˜M N , q is positive, the inequality in (13.78) becomes an equality, and the minimum of the action S = 8π 2 q is realized. If q < 0, one should consider the functional Z √ B = Tr{(FM N + F˜M N )(F M N + F˜ M N )} g d4 x . (13.79) and derive S ≥ 8π 2 |q|. The minimum is realized for anti-self-dual fields F = −F˜ . If a field configuration realizes the minimum of the action, it represents a solution to the classical equations of motion. And such solutions are quite obviously relevant for physics. The mathematical interest of self-dual confugurations is that the corresponding SQM models enjoy the extended N = 4 supersymmetry. Indeed, if we remove one point from S 4 , we obtain a conformally flat 4-dimensional manifold, which is HKT (see Chap. 9.3), and we know from Theorem 12.11 that adding a self-dual gauge field does not destroy the N = 4 supersymmetry of the HKT sigma model. An attentive reader can protest at this point: Theorem 12.11 was proven only for Abelian fields, and we are now interested in the non-Abelian case! 10 In Chap. 4, we discussed only the action functionals like (4.15) that appear in classical mechanics. The functional (13.76) describes the field-theory action of a pure 4dimensional non-Abelian gauge field. And such fields and their actions are indispensible to describe the Nature—the dynamics of strong and electroweak interactions. The factor 1/2 in (13.76) (which is, of course, irrelevant for the theorem) is a common physical convention and was included not to confuse a reader-physicist. And a reader-mathematician can simply ignore this footnote and this theorem.

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However, a generalization of this theorem to the non-Abelian case exists. In particular, one can make the following observation [122]: Theorem 13.12. Consider a conformally flat 4-dimensional manifold with the metric11 ds2 =

dxM dxM , f 2 (x)

(13.80)

Consider the matrix quantum supercharges √ a ˆ = ψj (σ † )ja [(PˆM + AM )f + i∂M f ψk ψˆ¯k ] , 2Q M √ ˆ ¯ = [f (Pˆ + A ) − i∂ f ψ ψˆ¯k ](σ ) ψˆ¯j , 2Q a

M

M

M

k

M aj

(13.81)

where PˆM = −i∂M , AM are Hermitian matrices belonging to any semisim† ja ple Lie algebra g, (σM )aj = (σ, i)aj and (σM ) = (σ, −i)ja , as in Eqs. (3.9), (3.11). The statement is that these supercharges satisfy the extended N = 4 supersymmetry algebra iff the field density FM N = ∂M AN −∂N AM + i[AM , AN ] is self-dual. A similar N = 4 supersymmetric system includes an anti-self-dual FM N also. The supercharges of such system differ from (13.81) by the inter† change of σM and σM . This is equivalent to the interchange of two spinor representations of SO(4). Proof. This observation can be verified by an explicit calculation,12 but it represents in fact a particular case of a more general Theorem 13.13 to be proven at the end of the chapter. The following remarks are in order. 11 Following bad habits acquired in Chap. 9.3, we do not distinguish here between the covariant and contravariant indices. 12 The following identities are handy for this purpose: † σ(M σN = δM N , ) † (σ[M )ja (σN ] )bk =

† p σ[M σN = i¯ ηM N σp , ]

† p σ[M σN ] = iηM N σp ,

1 1 † † (σ σ )j k δba + (σ[N σM ) a δkj , ] b 2 [M N ] 2

(13.82)

p p where ηM ¯M N and η N are the ’t Hooft symbols, which we have already seen in Chap. 3: p p ηm4 = −η4m = δmp , p η¯M N

p ηM N

p ηmn = εmnp .

(13.83)

p and differs from by the sign in the first relation in (13.83). The tensors ηM N p are self-dual and η¯M N are anti-self-dual.

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• The Hamiltonian corresponding to the supercharges (13.81) is13 ˆ = 1 f (PˆM + AM )2 f − i f 2 FM N ψσ † σN ψˆ¯ H M 2 4   1 † 2 j j 2 ˆ ˆ ˆ ¯ ¯ ¯ ˆ −if ∂M f (PN + AN ) ψσ[M σN ] ψ + f ∂ f ψj ψ − (ψj ψ ) . (13.84) 2 • If f = 1+x2M /2, the metric (13.80) coincides up to a factor with (13.26) and describes S 4 . If the gauge field is absent, the supercharges (13.81) are equivalent to the quantum version of the supercharges (9.78), (9.79). • When AM = 0, the Hamiltonian (13.84) coincides up to a factor with the square of a torsionful Dirac operator—see the discussion in Sect. 1.1. • If AM = 0 and f (xM ) depends only on three out of four coordinates, one can perform a Hamiltonian reduction in the spirit of Chap. 11 to obtain the Hamiltonian (10.50) of the symplectic sigma model. When the gauge field is Abelian, the Lagrangian of the model can be derived from the superspace formalism of Chap. 10 or by Legendre transformation of the classical Hamiltonian. The field-dependent part of such Lagrangian reads i † ¯ ψ. (13.85) LA = −x˙ M AM + f 2 FM N ψσM σN 4 An interesting question is how to derive the Lagrangian in the non-Abelian case. The conventional Legendre transformation does not work: the Hamiltonian (13.84) belongs to g and has a matrix nature, while the action and the Lagrangian must be scalars. Still, the Lagrangian of the model can be written by invoking the so-called semi-dynamical variables. The main idea is to introduce the extra variables ϕα and the complex conjugated ϕ¯α belonging to the fundamental and antifundamental representations of the gauge group and to replace the Hermitian generators tp ∈ g in (13.84) (AM ≡ ApM tp ) by the structure T p = ϕα (tp )α β ϕ¯β .

(13.86)

The variables ϕα can have the ordinary or the Grassmann nature. Grassmann variables are somewhat more popular in this context; the idea comes back to the classical paper [46], where a triplet of Grassmann variables was introduced to describe spin dynamics. But the same idea can be realized 13 Attention: the factor 1/2 in the first term of the second line in Eq. (20) of Ref. [122] was redundant. Note also the different conventions for the sign of AM .

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with ordinary bosonic variables [130]. In the context of N = 4 supersymmetric quantum mechanics, such extra bosonic variables were discussed in [131]. Thus, we introduce the ordinary complex variables ϕα and write instead of (13.85) the following expression [132]: Lnon−Ab = −iϕ¯α ϕ˙α + B(ϕα ϕ¯α − 1) A i p † ¯ p (13.87) −x˙ M ApM T p + f 2 FM N T ψσM σN ψ 4 with T p defined in (13.86). The field B is a non-dynamical auxiliary variable, alias a Lagrange multiplier. The variation over B gives the constraint ϕα ϕ¯α = 1 .

(13.88)

The variables ϕα enter the Lagrangian with a time derivative, but the Lagrangian involves only the linear term in the generalized velocities ϕ˙ α , not the quadratic term, as is common for the conventional bosonic dynamical variables like xM . That is why we called this kind of variables “semi-dynamical”.14 The variables ϕα can be interpreted as generalised holomorphic coordinates. Then −iϕ¯α are their canonical momenta. Under quantization, ϕ¯α → ∂/∂ϕα . The classical constraint (13.88) gives the following differential constraint to be imposed on the wave functions: ϕα

∂ Φ(ϕα , xM , ψj ) = Φ(ϕα , xM , ψj ) . ∂ϕα

(13.90)

The solution to (13.90) is simple: Φ(ϕα , xM , ψj ) = Aα (xM , ψj ) ϕα .

(13.91)

This nicely maps to the matrix description where the wave functions depend only on xM and ψ j , but belong to the fundamental representation of the gauge group. The quantum version of the structure (13.86) is the operator ∂ . Tˆp = ϕα (tp )α β ∂ϕβ

(13.92)

14 I could not resist mentioning here that similar semi-dynamical variables appear in some popular field theory models studied by physicists. I am talking about 3-dimensional Chern-Simons theory with the action Z S ∝ εijk d3 x Tr{Ai ∂j Ak } . (13.89)

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It is easy to see that Tˆp satisfy the same commutation relations as the generators tp . And the operator Tˆp Tˆp boils down [bearing in mind (13.91)] to the multiplication by the Casimir invariant in the fundamental representation cF = tp tp . The full Lagrangian involving the term (13.87) and the other fieldindependent terms is invariant under N = 4 supersymmetry transformations, whose explicit form can be found in Ref. [132]. For the gauge group SU (2), the N = 4 invariance of the action can also be derived using the harmonic superspace formalism by introducing semi-dynamical harmonic superfields. Again, we address an interested reader to Ref. [132]. 13.4.3

Explicit solutions

Yang monopole In Sect. 2 we discussed at length the topologically nontrivial line bundle on S 2 with constant field density—the Dirac monopole field. A direct nonAbelian analog of this field confuguration is the Yang monopole [133]. The Yang monopole is a self-dual configuration with the same value of the invariant Tr{FM N F M N } (the same action density) at all the points of S 4 . The metric of S 4 of radius R is expressed in the stereographic coordinates as 16R4 (dxM )2 . (13.93) ds2 = (x2 + 4R2 )2 In a certain conveniently chosen gauge, the Yang solution has a simple form, ApM =

p N 2ηM Nx , x2 + 4R2

p 16R2 ηM p N ˜p FM = F = − , N MN (x2 + 4R2 )2

p where ηM N is the ’t Hooft symbol (13.83). Then 1 3 Tr{FM N F M N } = 4 2 R and the action (13.76) is

(13.94)

(13.95)

3 3 8π 2 R4 VS 4 = 4 = 8π 2 , (13.96) 4 R R 3 as it should be for a self-dual configuration with unit topological charge. S =

Instantons There are also self-dual topologically nontrivial configurations where Tr{FM N F M N } is not constant. The configurations where the characteristic support of the action integral (13.76) is much smaller than the radius R

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of the sphere represent a particular physical interest. In this case, we are actually dealing with the self-dual configurations on R4 whose field strength vanishes at infinity. These configurations were discovered in [134] and are called instantons.15 They look similar to (13.94), ApM =

p N 2ηM Nx , 2 2 x +ρ

p ˜p FM N = FM N = −

p 4ρ2 ηM N , (x2 + ρ2 )2

(13.97)

but the parameter ρ has now the meaning of the instanton size, which is small compared to R. When x2  ρ2 , the field density falls down as ∝ 1/x4 , while the gauge potential tends to a pure gauge configuration AˆM = i(∂M ω)ω −1

with

ω =

x0 − ixj σj √ . x2

(13.98)

The configuration (13.97) carries unit topological charge. It is also easy to write the anti-self-dual antiinstanton configuration with topological charge −1: ApM =

p N 2¯ ηM Nx . 2 2 x +ρ

(13.99)

Generic (anti-)self-dual configurations of an arbitrary topological charge q have more complicated form. They were constructed in [135] (see also [136]). 13.5

KLW Theorem and its HKT Generalization

We proved in Theorem 12.11 that a generic HKT geometry admits the inclusion of an Abelian gauge field with the tensor FM N (or rather its tangent space projection) belonging to sp(n) subalgebra of the so(4n) algebra, so that N = 4 supersymmetry of the model is left intact. In Theorem 13.12, we have shown that this result can be generalized to the non-Abelian case, but only for a limited class of Obata-flat 4-dimensional HKT manifolds with a conformally flat metric. A natural question is whether a non-Abelian generalization exists also for the generic HKT geometry. The answer to this question is positive. Theorem 13.13. The extended N = 4 supersymmetry of an HKT SQM model is left intact also in the presence of a non-Abelian gauge field iff N eM A eB FM N belongs to the subalgebra sp(n) of so(4n) (and to the gauge algebra g, of course). 15 This name may look strange, but to explain its meaning, one would have to plunge deeply into physics of non-Abelian fields, which is not our intention here.

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In the hyper-K¨ ahler case, a direct operator proof of this theorem was given by Kirschberg, L¨ ange and Wipf in Ref. [137]. Another variant of such proof [which is similar to the proof of Theorem 10.5 that the (1, 2, 1) SQM models living on hyper-K¨ ahler manifolds enjoy N = 8 supersymmetry] and its generalization for the HKT manifolds was suggested in Ref. [93]. However, in the HKT case, this method works well only for 4-dimensional manifolds. In higher dimensions, it meets certain difficulties. Fortunately, we can prove this theorem in a rather simple way, capitalizing on the result of Theorem 12.11, which was proven in the preceding chapter. Proof. In the absence of the gauge field, the classical and also quantum HKT supercharges were found to have the form (10.78). It is not difficult to include the field there: one only has to substitute PˆM → PˆM + AM . The commutators of the supercharges involve many different terms and most of them have exactly the same form in the non-Abelian case as in the Abelian one. The only specifics of the non-Abelian system is the presence of the terms involving the commutators [AM , AN ]. These commutator terms come from the structures [PˆM + AM , PˆN + AN ] = −i(∂M AN − ∂N AM ) + [AM , AN ] = −iFM N ;

(13.100)

they are added to the derivatives ∼ ∂M AN to form the non-Abelian field densities. Thus, the only new feature of the non-Abelian system, compared to the Abelian one is the presence of the commutator term in the field density (13.100). And if FM N commutes with all three complex structures [so that FAB belongs to sp(n)], the N = 4 superalgebra holds by the same token as it holds (as we know from Theorem 12.11) in the Abelian case. The Hamiltonian of the system is obtained from the Hamiltonian of the system without the field by substituting PˆM → PˆM + AM and adding the extra term ˆMΨ ˆN . ˆ = − i FM N Ψ (13.101) ∆F H 2 To give the reader more feeling on how it actually works, compare the ˆ 0 and Q ˆ 1 . We field-dependent terms in the squares of the supercharges Q present the latter in the form ˆ0 = Ψ ˆ M (PˆM + AM ) + irrelevant bits, Q ˆ1 = Ψ ˆ N IN R (PˆR + AR ) + Q

irrelevant bits.

(13.102)

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ˆ 1 )2 is The field-dependent part in (Q 1 ˆM ˆN S R ˆ Ψ Ψ IM IN [PS + AS , PˆR + AR ] 2 i ˆM ˆN S R = − Ψ Ψ IM IN FSR , (13.103) 2 where we neglected the gauge-non-invariant terms involving the derivatives of the complex structure: they have the same form as in the Abelian case to be eventually cancelled out. Using the fact that F and I commute, we can bring (13.103) in the form (13.101)—the structure, which also shows ˆ 2 . We conclude that Q ˆ2 = Q ˆ2. up in Q 0 0 1 ˆ ˆ 1 }, we obtain the structure ∼ In the anticommutator {Q0 , Q M N R ˆ Ψ ˆ I[N FRM ] , which vanishes due to skew symmetry of IN R and FRM Ψ ˆ1, Q ˆ 2 } gives the and the property [F, I] = 0. The anticommutator {Q structure ∼ IF J + JF I, which vanishes due to IJ = −JI = K and [F, I] = [F, J] = [F, K] = 0. An immediate corollary of this theorem is the fact that, in the instanton background, the eigenstates with λ 6= 0 in the spectrum of the Dirac operator D / are double degenerate, and hence the spectrum of the excited states in the Hamiltonian −D / 2 /2 is four-fold degenerate, in accord with the old observation made in Ref. [138].

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Chapter 14

Atiyah-Singer Theorem

14.1

Generalities

The Atiyah-Singer theorem is a very general statement concerning the properties of certain elliptic differential operators. Some special cases of this statement (e.g. the Riemann-Roch theorem) were known to mathematicians since the 19th century, but the general statement was first conjectured by Gelfand in 1960 [139]. It was announced as a theorem by Atiyah and Singer in 1963 [140], and the complete proof was published by them in a series of papers in 1968 [59]. We are going to show in this chapter that the formulation and the proof of this theorem have in fact a very transparent and concise supersymmetric interpretation. But let us explain first what is this theorem about. To please our reader-mathematician, we will use a moderately fancy mathematical language—the same as in the original papers, but a reader-physicist need not worry—we will immediately give a clarifying example. Let X be a compact closed manifold.1 Let E and F be two vector bundles over X, let Γ(E) and Γ(F ) be the spaces of smooth sections of ˆ be an elliptic differential operator acting on the these bundles and let D sections in Γ(E) in such a way that the results of this action belong to Γ(F ) ˆ on any section in Γ(F ) gives zero. Correspondingly, the and the action of D ˆ † acts on the sections in Γ(F ) to produce the sections conjugate operator D ˆ and D ˆ † are nilpotent. in Γ(E) and annihilates the latter. Obviously, both D ˆ is the difference between the number Definition 14.1. The index of D ˆ = 0 and the number of of the independent solutions of the equation Df † ˆ the independent solutions of the equation D g = 0 with f ∈ Γ(E) and g ∈ Γ(F ). 1A

generalization to manifolds with boundaries also exists [141]. 305

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In mathematical notation, ˆ − dim ker D ˆ† . I = dim ker D

(14.1)

The AS theorem says that this index is determined by the topology of the manifold and of its structures. What particular topology and what ˆ particular structures—depends on D. ˆ is the even-dimensional Dirac The promised important example of D ˆ is the operaoperator D / acting on the bispinors. To be more precise, D D+1 tor D / (1 + γ ) that acts nontrivially on the right-handed bispinors ΨR satisfying γ D+1 ΨR = ΨR (see Definition 13.3) to produce the left-handed ˆ† = D spinors, which it annihilates. The conjugate operator D / (1 − γ D+1 ) acts on the left-handed bispinors to produce the right-handed ones. Thus, ˆ and D ˆ † act in complementary directions, as is required by the conditions D of the theorem. The Atiyah-Singer index coincides in this case with (0)

(0)

ind(D /) = n+ − n− ,

(14.2) (0)

(0)

/ with positive chirality and n− where n+ is the number of zero modes of D is the number of zero modes of D / with negative chirality. Without any doubt, our reader has already recognized in ˆ = D D / (1 + γ D+1 ) ,

ˆ† = D D / (1 − γ D+1 )

(14.3)

ˆ † and D ˆ † to Q. ˆ Also in a ˆ corresponds to2 Q the supercharges (13.8): D ˆ and D ˆ † can be interpreted as supercharges. general case the operators D Indeed, being nilpotent, they satisfy the algebra (5.1), with the operator ˆ D ˆ † } playing the role of the Hamiltonian. And then the index (14.1), {D, (14.2) is, up to the sign flip, none other than the Witten index (6.53)! In other words, the Atiyah-Singer theorem is in fact a statement about the ground (vacuum) states of the supersymmetric Hamiltonian known to us from Chap. 6. We noticed there that the Witten index stays invariant under smooth deformations of the parameters of the model. But this is tantamout to saying that the index has topological nature. Similar to Mr. 2 The reason for the different dagger attributions here can be traced back to Benjamin Franklin who defined the sign of the electric charge two and a half centuries ago in such a way that the charge of the electron (of the existence of which Franklin had no idea) turned out to be negative. As a result, the ground states of the Landau Hamiltonian for the electron in a magnetic field with a positive projection on the third axis have negative chiralities. And it was convenient for us when we were giving the supersymmetric interpretation to the Landau problem in Chap. 5, to ascribe to these ground states a ˆ the operator that increases the fermion charge by 1. zero fermion charge and call Q

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AS theorem

Supersymmetry

Γ(E)

bosonic (or fermionic) states

Γ(F )

fermionic (or bosonic) states

ˆ D ˆ† D,

supercharges

ˆ D ˆ †} {D,

Hamiltonian

AS index

Witten index

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Jourdain back in 1670 and like Dr. Landau in 1930, Prof. Atiyah and Prof. Singer spoke in 1963 supersymmetric prose without realizing this! To make the last statement absolutely clear, we gave in Table 14.1 a small dictionary of correspondences. A nontrivial question is how to evaluate the index. In Chap. 6, we described a general method to do so which capitalizes on the fact that all the excited states of a supersymmetric Hamiltonian are doubly degenerate, and hence one can replace the index (6.53) by the supersymmetric partition function (6.49). The latter can be presented as (6.48), where the Euclidean evolution operator K(qj , ψα ; qj , ψ¯α ; −iβ) is expressed as the path integral (6.46). The sum (6.49) does not depend on β and it is convenient to assume β to be small, in which case the path integral can be evaluated! Following Refs. [65], we did so by substituting for the complicated integral (6.46) its small β approximation (6.45). After that, (6.48) was reduced to the finitedimensional integral (6.54), which can be evaluated for all SQM systems we are familiar with. In Chap. 6, we calculated the Witten index for the simplest supersymmetric systems which described one-dimensional motion (Witten’s quantum mechanics) and two-dimensional motion in a magnetic field. And now we are interested in more complicated systems studied in the previous chapters and related to geometry of manifolds. How to evaluate Witten indices there? An example of such evaluation was already presented in Chap. 8. We calculated there the Witten index for the supersymmetric system (8.6) de-

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scribing the de Rham complex and showed that the index is given by the integral (8.38) and coincides with the Euler characteristic of the manifold. However, there are problems where the calculation of the index is not so easy. We will discuss them in the rest of the chapter. 14.2

Hirzebruch Signature

Like the Euler characteristic, the Hirzebruch signature is a topological invariant associated with the de Rham complex. Consider the de Rham complex on a manifold of dimension D = 4k. Consider the Hodge duality operator (1.24). According to (1.25), the square of this operator is sometimes +1 and sometimes −1, which is not convenient for our purposes. But a remarkable fact is that, if D = 4k, one can define the signature operator, the composition3 τˆ = (−1)p(p+1)/2 ◦ ?

(14.5) † ˆ which squares to 1 and anticommutes with the operator R = d + d (hence it commutes with the Laplace-de Rham operator 4 = −dd† − d† d). It follows that the eigenvalues of τ are either +1 or −1. Thus, the whole set of p-forms can be divided in this case into two subsets: the subset S+ of the forms with positive signature for which τˆα = α and the subset S− of ˆ τˆ} = 0 implies that the forms with negative signature. The property {R, ˆ the action of the operator R on a positive-signature state gives a state with negative signature and vice versa. ˆ plays in this setting the same role as the Dirac opIn other words, R erator plays for the Dirac complex, while τˆ plays the role of γ D+1 . In the ˆ ± τˆ) are nilpotent, Hermitially full analogy with (14.3), the operators R(1 conjugate to one another and play the role of the supercharges. Note that these Hirzebruch supercharges do not coincide at all with the de Rham supercharges d and d† . However, their anticommutator gives the same Hamiltonian as in the de Rham complex. Note also that the extended N = 4 supersymmetry does not hold here: the anticommutators ˆ ± τ ), d} and {R(1 ˆ ± τ ), d† } do not vanish. {R(1 Definition 14.2. The Hirzebruch signature of a 4k–dimensional manifold is defined as 3 This

sign = b+ − b− ,

(14.6)

τˆαp = (−1)(4k−p)(4k−p+1)/2 ? αp = (−1)p(p−1)/2 ? αp .

(14.4)

actually means

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where b± are the numbers of the closed but not exact forms, belonging to S+ and S− . Remarks • The terminology is a little awkward here, but we hope that the reader will not confuse two different notions: the signature of a form (the eigenvalue of τˆ) and the Hirzebruch signature of a manifold. • Do not mix up the Hirzebruch signature with the Euler characteristic (8.33), which also can be presented in the form (14.6), but the numbers b+ and b− would have in that case a completely different meaning, being the sums of the Betti numbers bp with even and odd p. • In fact, only the forms of degree p = D/2 = 2k contribute to (14.6). Indeed, the duality operator transforms p-forms to (D − p)-forms. Hence, for any form αp6=D/2 , the form αp + τˆαp belongs to S+ , while the form αp − τˆαp belongs to S− . The first form contributes to b+ , while the second one to b− . The difference (14.6) is not sensitive for that. • The Hirzebruch signature is a particular case of the Atiyah-Singer index and is a topological invariant. On the supersymmetric side, we are dealing with the system with the Hamiltonian (8.28). Theorem 14.1. The operator (14.5) has in this language a nice representation  4k  Y ∂ − ψA . (14.7) τˆ = ∂ψA A=1

Proof. The operator (14.7) can be represented as a product of k blocks:   4  8  Y Y ∂ ∂ − ψA × − ψA × · · · . τˆ = (14.8) ∂ψA ∂ψA A=1

A=5

Each block commutes with “foreign” fermion variables (for example, the first block commutes with ψ A=5,...,4k ). We leave it for the reader to check, using the definitions (1.24) and (1.48), that the action of, for example, the first block on the variables ψ M =1,2,3,4 and their different products is the same as is prescribed by (14.4). The Hirzebruch signature is n o ˆ sign = Tr∗ τˆe−β H ,

(14.9)

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where Tr∗ stands for the operator trace in Hilbert space. In Chap. n 6 owe discussed the ordinary partition function ˆ Z = Tr∗ e−β H and the supersymmetric partition function Z˜ = o n ˆ ˆ Tr∗ (−1)F e−β H . We showed there that both Z and Z˜ admit a path integral representation. The same is true for the expression (14.9). The corresponding path integral can be calculated. We will not do so here and only quote the result. Theorem 14.2. The signature is given by the integral " # Z R 1/2 2π sign = det , R tan 2π

(14.10)

where R is the matrix 2-form associated with the Riemann curvature tensor: 1 RAB = RAB M N dxM ∧ dxN . (14.11) 2 Proof. It consists in a not so simple calculation of the appropriate path integral. The reader can find it in the original papers [60]. The integrand in (14.10) should be understood as a Taylor series, " # R 1 1/2 2π Tr{R ∧ R} + higher-order terms, (14.12) det = 1− R 24π 2 tan 2π D/2

z }| { where only the term ∼ R ∧ . . . ∧ R proportional to the volume form is relevant. For 4-dimensional manifolds, Z 1 sign = − Tr{R ∧ R} . (14.13) 24π 2 The simplest nontrivial example is CP2 . This manifold is 4-dimensional. The nontrivial Betti numbers are b0 = b2 = b4 = 1. The closed but not exact 2-form α2 happens to have the eigenvalue +1 with respect to τˆ and the Hirzebruch signature is equal to 1. The integral (14.13) also gives this value. 14.3 14.3.1

Dirac Index K¨ ahler manifolds

Consider now the index of the Dirac operator defined as n o D 2 /2 ID = Tr∗ γ D+1 eβ/ ,

(14.14)

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We start our discussion with a more simple case when geometry is K¨ ahler and the gauge field is Abelian. As we know from Theorem 13.4, the Dirac complex is equivalent in this case to the twisted Dolbeault complex, which is described by the SQM model (9.1). Let us try to calculate the Witten index in this model using the recipe (6.54). To this end, we need to know the classical Hamiltonian of the model. It was written in (9.17). In the K¨ ahler case, the 4-fermion term in the Hamiltonian diappears and the Bismut spin connection (9.10) coincides with the ordinary Levi-Civita one. The Hamiltonian acquires the form    ¯ ¯ Kahl c d¯ HW = hkj Pj − iωa¯b,j ψ a ψ b + i∂j W P¯k¯ − iωcd, ¯k ¯ ψ ψ − i∂k ¯W ¯ − 2 ∂j ∂k¯ W ψ j ψ¯k . (14.15) The corresponding Lagrangian is i ¯ ¯ ¯ LKahl = hj k¯ z˙ j z¯˙ k + (ψ a ψ¯˙ a¯ − ψ˙ a ψ¯a¯ ) + i(z¯˙ k ωa¯b,k¯ + z˙ j ωa¯b,j )ψ a ψ¯b W 2 ¯ ¯ (14.16) + i(z¯˙ k ∂k¯ W − z˙ j ∂j W ) + 2 ∂j ∂k¯ W ψ j ψ¯k . Let us do the calculation for CPn . Choose W in the form q q W = ln det h = − ln(1 + z¯z) , (14.17) 2(n + 1) 2 as in (13.58) (we use the shorthand z¯z ≡ z¯k z k ), substitute it in (14.15) and do the integral Z Y n n dPk dP¯k dz k d¯ z k Y a ¯a Kahl dψ dψ exp{−βHW } . (14.18) IW = (2π)2 a=1 k=1

We derive: IW =

 q n Z Y qn 1 dz k d¯ zk = . n+1 2π (1 + z¯z) n!

(14.19)

k

We have found in Chap. 13 that, for the gauge field to represent a benign line bundle, q should be integer for odd n and half-integer for even n. For n = 1, when we are dealing with CP1 ≡ S 2 , the expression (14.19) boils down to IW = q, which is integer. But for all n ≥ 2, we obtain a fractional result for the index, which makes no sense. That means that our calculation was wrong. At some point, we should have made an error. We did, indeed, but it was not a trivial error. The point is that, no matter how small β is, we are not allowed in this particular case to replace the functional integral (6.46) for the Euclidean evolution operator by its infinitesimal expression and assume that the variables z k (τ ), ψ a (τ ), etc. in the functional integral (6.50) do not depend on τ .4 Instead, we should 4 Hopefully, there will be no confusion between the Euclidean time τ and the signature operator τˆ.

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• Impose the periodic boundary conditions z k (β) = z k (0),

ψ a (β) = ψ a (0),

ψ¯a (β) = ψ¯a (0) ,

• Expand all the variables in Fourier series, X k k + z(m) e2πimτ /β , z k (τ ) = z(0)

etc.

(14.20)

(14.21)

m6=0

• Substitute this into (6.50). • Do the integral over the nonzero Fourier modes in the Gaussian approximation. We will show later that the Gaussian approximation is sufficient here—one need not take into account the quartic terms and terms of still higher power. Using the physical slang, it is not sufficient in this case to calculate the functional integral in the tree approximation, but the one-loop approximation is sufficient and gives the correct answer. The actual calculation of the path integral is not so simple. In the original papers devoted to the supersymmetric proof of the Atiyah-Singer theorem [60], the details of this calculation were not given. Some more details (but not all the details) were displayed in Refs. [142]. In the next section, we will present this calculation in the most explicit terms. But before coming to grips with the technicalities, we shall present the result and discuss it. The Witten index of the model (9.1) for a K¨ahler manifold reads # " Z R 4π , (14.22) IW = eF /2π det1/2 R sin 4π where F is the field strength 2-form, F=

1 ¯ FM N dxM ∧ dxN = −2i ∂j ∂k¯ W dz j ∧ d¯ zk , 2

(14.23)

and R was defined in (14.11). The symbol eF /2π is deciphered as eF /2π = 1 +

F F ∧F + ... + 2π 8π 2

and the determinant factor as " # det 1/2

R 4π R sin 4π

= 1+

1 192π 2

(14.24)

Z Tr{R ∧ R} + . . . ,

(14.25)

The factor eF /2π in the integrand is the tree contribution coming from the integral (14.18), while the determinant factor comes from nonzero Fourier

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modes. Only those terms of the expansion of eF /2π and of 4π sin(R/4π)/R that are combined to give a 2n-form in the integrand contribute. Consider first the simplest case of CP1 . In this case, the determinant factor is absent, there is only one term in the expansion (14.24) and the Witten index is simply equal to the magnetic flux q. This agrees with the results of Chap. 13.2. For CP2 , the index represents a sum of two terms, Z Z 1 1 F ∧ F + Tr{R ∧ R} . (14.26) IW (CP2 ) = 8π 2 192π 2 In the first term in the R.H.S. of (14.26), we recognize the Abelian version of the Pontryagin number (13.70) and the second term coincides with the Hirzebruch signature of CP2 given in Eq. (14.13) and taken with the coefficient −1/8. For higher n, the index is a sum of many different topological invariants. The estimate (14.19) for the index would be reproduced if one ignores this curvature-dependent determinant factor. We include now this factor and do the calculation for CP2 first. Bearing in mind that sign(CP2 ) = 1, we derive from (14.26) and (14.13): q2 1 − . 2 8 This expression is integer for half-integer q: IW (CP2 ) =

(14.27)

IW (q = ±1/2) = 0, IW (q = ±3/2) = 1, IW (q = ±5/2) = 3, etc. (14.28) To express the result for higher n, introduce s = |q| − n+1 2 . It follows from (13.65) that s must be an integer. If s ≥ 0, the index is given by a binomial coefficient   n+s IW = (14.29) n if q is positive and involves an additional factor (−1)n if q is negative. And when s < 0, i.e. when |q| < (n + 1)/2, the index vanishes.5 The final result (14.29) for the Witten index in CPn can in principle be derived from (14.22) in the same way as for CP2 by summing up all the relevant topological invariants. But it is complicated, and we even do not know whether anybody did so. There is a much simpler and direct way that 5 Note in passing that the index (14.29) is closely related to the Witten index in the 3dimensional supersymmetric Yang-Mills-Chern-Simons field theory [143]. See Ref. [144] for the detailed discussion.

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consists in counting the number of independent ground states [137,145], i.e. the number of the normalized solutions to the equations ˆ¯ cov Φ = 0 , ˆ cov Φ0 = Q Q 0

(14.30)

where the covariant supercharges were defined in Eqs. (9.11),(9.13)):     z¯k aˆ ¯¯b ω ¯ ˆ cov = −iψ k ∂ + 1 q − n + 1 ψ Q + ψ ab,k , ∂z k 2 2 1 + z¯z     1 n+1 zk ∂ ¯ ¯ cov aˆ k b ˆ ˆ ¯ ¯ ¯ − Q = −iψ q+ + ψ ψ ωa¯b,k¯ . (14.31) ∂ z¯k 2 2 1 + z¯z By “normalized” we mean normalized with the covariant measure: Z Y Y dz k d¯ zk dψ a dψ¯a k

a

b ¯b 1 ¯ 0 (z k , z¯k , ψ¯a )Φ0 (z k , z¯k , ψ a ) = 1. e−ψ ψ Φ n+1 (1 + z¯z)

(14.32)

An analysis shows that the solutions to (14.30) may exist only in the sector of zero fermion charge or of fermion charge n. Let first q be positive. In that case, the normalized solutions to (14.30) may exist only in the sector with zero fermion charge. Indeed, for the states of fermion charge n, the equation ˆ¯ cov Φ = 0 gives ˆ cov Φ0 = 0 is satisfied automatically, while the equation Q Q 0 ∂k¯ Φ0 = c

zk Φ0 1 + z¯z

with c > 0. This equation has no normalized solutions. Consider the states of zero fermion charge. In this case the equation ˆ ¯ cov Φ0 = 0 is satisfied automatically, while the equation Q ˆ cov Φ0 = 0 Q implies ∂k Φ0 = −

s¯ zk Φ0 . 2(1 + z¯z)

(14.33)

The normalized solutions to (14.33) exist only for positive s. They read Φ0 = (1 + z¯z)−s/2 P (¯ z) ,

(14.34)

where P (¯ z ) is a polynomial of z¯k of degree not higher than s. The dimension of the vector space of such polynomials exactly coincides with (14.29). For negative q, the vacuum states are present only in the sector of fermion charge F = n. Their number is also given by (14.29). For odd n the vacuum states are fermionic, hence the factor (−1)n in the index.

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An important remark is in order. Equation (14.22) is the result for the Witten index of the model (9.1) with the Hamiltonian (9.17). This index does not exactly coincide with the Dirac index defined in (14.14)—IW and ID may differ by a sign (see the footnote on p. 306). Indeed, we have seen that the ground state in the Landau problem is bosonic, so that the Witten index is positive, but in the matrix language, this ground state has negative chirality, so that the Dirac index is negative. This mismatch is not so surprising: the signs of both indices depend on convention. For the Witten index, the sign depends on what state is assumed to be bosonic and what state fermionic, it can be reversed by ¯ For the Dirac index, it depends on the changing the convention ψ ↔ ψ. definition of what is left and what is right. We will see a little bit later that the Dirac index (14.14) of an even-dimensional manifold (K¨ahler or not) is given by almost the same formula as (14.22), but with a negative sign before F in the exponent: # " Z R 4π . (14.35) ID = e−F /2π det1/2 R sin 4π In other words, ID = IW for 4k-dimensional manifolds and ID = −IW for 4k + 2-dimensional manifolds. 14.3.2

Non-K¨ ahler manifolds

Consider now a complex, but not necessarily K¨ahler manifold. As we know from the previous chapter, one can define in this case two different complexes: (i) the Dolbeault complex, which is described by the SQM model (9.1) and which, according to Theorem 13.5, maps to a deformed torsional Dirac operator and (ii) the standard Dirac complex. Theorem 14.3. Torsions do not affect the value of the index. Proof. This simply follows from the fact that the Dirac operator in an even-dimensional manifold admitting spin structure enjoys supersymmetry associated with the pair of Hermitian supercharges −i/ D and −i/ Dγ D+1 , irrespectively of whether the torsions are present or not. This supersymmetry holds under an arbitrary continuous deformation of the tensor CM N L . Thus, the torsions can be unwound all the way to zero so that the index is not changed.6 6 See Ref. [146] for a more detailed discussion. Note the essential difference with the gauge field. We have seen in Sec. 13.2 that, to keep supersymmetry, AM should be a

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How to calculate this index? The first option that comes to mind is to study the generic SQM model (9.1) describing the Dolbeault complex and try to calculate the index there. However, this program meets serious difficulties due to the presence of the 4-fermion term in the generic Dolbeault Hamiltonian (9.8). We have seen above that the constant field approximation (6.54) does not work for the Dolbeault complex and the calculation of the path integral over nonzero modes in the Gaussian approximation is necessary. In the K¨ ahler case, one does not need to go beyond the Gaussian approximation, this will be proven in the next section devoted to explicit path integral calculations. And it is there that we will show that it does not hold for the Hamiltonian (9.8), for which all the loops must be taken into account. Obviously, we are not able to do so. We choose therefore another option and calculate the path integral for the Dirac index, for which the Gaussian approximation is sufficient, as we will see. To this end, we consider instead of (9.1) the torsionless N = 1 supersymmetric model of Sec. 9.2:   Z 1 1 gM N X˙ N − AM (X ) DX M = gM N (x) x˙ M x˙ N L = i dθ 2 2 i ˙ A + ωAB,M x˙ M ΨB ) − AM x˙ M + i FAB ΨA ΨB. (14.36) + ΨA ( Ψ 2 2 In the generic non-K¨ ahler case, it has only one classical conserved real supercharge   i C A B Q0 = eM Ψ P − ω Ψ Ψ + A . (14.37) M AB,M M C 2 The quantum counterpart of iQ0 maps to the standard Dirac operator. For a complex manifold, we may express everything in complex notation: M = {m, m}, ¯ A = {a, a ¯}. Now, if geometry is not K¨ahler, the components ωab,M and ωa¯¯b,M are different from zero. As a result, the Dirac operator ˆ 0 and the Hamiltonian H ˆ =Q ˆ 2 do not commute with the operator of the Q 0 ¯ a¯ as they did in the K¨ahler case. They commute, fermion charge Fˆ = ψ a ψˆ ˆ however, with the operator (−1)F . And that means that the mapping of the bispinor Hilbert space to the Hilbert space of the holomorphic wave functions Φ(xM , ψ a ) in the spirit of Theorem 13.2, such that γ D+1 is mapped ˆ to (−1)F , can still be defined. Then the second quantum supercharge can ˆ D+1 , and the Dirac index (14.14) is still mapped, up to be defined as −i/ Dγ n o ˆ ˆ . a possible sign flip, to the Witten index Tr∗ (−1)F exp{−β H} connection of a regular fiber bundle belonging to a distinct topological class. It cannot be continuously deformed in an arbitrary way without losing supersymmetry.

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The n R latter omay be evaluated as the functional integral of β exp − 0 LE (τ ) with the Euclidean version of the Lagrangian (14.36) and the periodic boundary conditions for both the bosonic and fermionic fields. We will see in the next section that the result of this evaluation does not depend on whether the manifold is K¨ahler or not and coincides, if the conventions are properly adjusted, with the expression (14.35). Now note that neither the Lagrangian (14.36) nor the result (14.35) depend on the assumption that the manifold is complex. We only need ˆ to assure the mapping γ D+1 → (−1)F (this is necessary to express the index as a path integral where fermion fields satisfy periodic boundary conditions). To this end, we have to define the Hilbert space of the wave functions Φ(xM , ψ a ) depending on the holomorphic fermion variables. But complex Grassmann variables in the tangent space can be introduced for all even-dimensional manifolds: locally, any such manifold is complex. The only condition that we need to derive (14.35) is that the manifold admits spin structure so that the notion of the Dirac index makes sense. 14.4

Functional Integral Calculation

The Witten index of any supersymmetric system is given by the integral (6.48). So far we calculated it in the approximation where the evolution kernel K was replaced by its infinitesimal form (6.45). As was mentioned above, this approximation does not work for the problem of interest and we have to evaluate the functional integral (6.46) for the kernel at small, but finite β. In our case, it is convenient first to do the Gaussian integral over Q τ dPM (τ ) in the Hamiltonian path integral for our system [a generalization of (6.17) involving also the integration over Grassmann variables] and represent the path integral in the Lagrangian form:7 Z Yq Y Y g[xM (τ )] dxM (τ ) dΨA (τ ) IW = C τ

( Z exp −

M

A

)

β

LE (τ )dτ

(14.38)

0 7 To be quite meticulous, we had to write first a finite-number-of-points approximation of the Hamiltonian path integral, integrate over the momenta there to obtain a finitenumber-of-points approximation of (14.38) and send the number of points to infinity at the end of the day. In this way, the exact numerical coefficient in the integral for the index can be fixed. We will not do so here, addressing an interested reader to Ref. [99] where this calculation was accurately done in the K¨ ahler case.

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with8 1 1 ˙ A + ωAB,M x˙ M ΨB ) gM N (x)x˙ M x˙ N + ΨA (Ψ 2 2 i (14.39) + iAM x˙ M − FM N ΨM ΨN . 2 Let first space be flat. In this case, the Cecotti-Girardello method works and we can reduce the path integral (14.38) for the index to the ordinary integral:  D/2 Z   Y iβ i flat dD x dΨC exp IW FAB (x)ΨA ΨB , (14.40) = 2πβ 2 LE =

C

−D/2

where the factor (2πβ) comes from the integration over momenta in the original Hamiltonian path integral [that is how we fixed the coefficient C in (14.38)] and the factor iD/2 appears due to the relation D/2

Y

dψ c dψ¯c = iD/2

c

D Y

dΨC

(14.41)

C

Q between the measure C dΨC and the canonical fermion measure expressed via the holomorphic variables ψ c , ψ¯c that we used in Chap. 7 and in Eq. (14.18).9 Only the term of order D/2 in the expansion of the exponential in (14.40) works. Using ! Z D Y dΨC ΨA1 · · · ΨAD = (−1)D/2 εA1 ...AD , (14.42) B=C

we derive flat IW =

1 (4π)D/2 (D/2)!

Z εA1 ...AD

dD x FA1 A2 · · · FAD−1 AD . (14.43)

R The integral (14.43) corresponds to the integral eF /2π in the language of forms.10 The result (14.22) for the index is thus reproduced in the case when the manifold is flat and the second factor in the integrand is absent. We can as well reproduce in this way the result (14.35) for the canonically defined Dirac index, but to this end, we have to modify somewhat our conventions. Namely, we now suppose that √ √ ψ 1 = (Ψ1 − iΨ2 )/ 2, etc. and dψ 1 = (dΨ1 + idΨ2 )/ 2, etc. (14.44) 8 It is worthwhile to note that, in contrast to the Minkowski Lagrangian (14.16), the Lagrangian (14.39) is not real. It need not be. On the other hand, the Euclidean path integral for the index is, of course, real. √ √ 9 We remind our standard convention: ψ 1 = (Ψ1 + iΨ2 )/ 2, ψ 2 = (Ψ3 + iΨ4 )/ 2, etc., √ which dictates dψ 1 = (dΨ1 − idΨ2 )/ 2 etc. 10 Do not forget about the factor 1/2 in the definition (14.23).

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This gives the factor (−i)D/2 rather than iD/2 in (14.41), which brings about the factor (−1)D/2 in (14.43), in agreement with (14.35). For an arbitrary curved manifold, the determinant factor is non-trivial. As was mentioned above, to calculate it, we must go beyond the recipe (6.54), expand the fields in Fourier series and evaluate the path integral (14.38) in the Gaussian approximation. Substitute the expansion xM (τ ) = xM (0) +

0 X

iΩm τ xM , (m) e

M xM (−m) = x(m) ,

m

ΨA (τ ) = ΨA (0) +

0 X

A eiΩm τ , ψ(m)

A ΨA (−m) = Ψ(m) ,

(14.45)

m

P0 P with m ≡ m6=0 and Ωm = 2πm/β, into (14.38). To calculate the functional integral in the Gaussian approximation (we will justify the validity A of this approximation later), we keep only quadratic in xM (m) , Ψ(m) terms and do the τ integral. The quadratic part of the relevant terms in the Lagrangian [we can now disregard the second line in (14.39) involving gauge fields: the latter have already been successfully handled] gives Z β 0 0 iβ X β X 2 (0) M (2) A Ωm gM N x(−m) xN + Ωm Ψ A LE dτ = (m) (−m) Ψ(m) 2 2 0 m m 0

+

X iβ A B  (0) (0) N Ψ(0) Ψ(0) ∂M ωAB,N − ∂N ωAB,M Ωm x M (−m) x(m) 4 m (0)

−iβ ΨA (0) ωAB,M

0 X

B Ωm xM (−m) ψ(m) .

(14.46)

m

Now we diagonalize the sum in (14.46) by the substitution (0)

A B R ΨA (m) ⇒ Ψ(m) − ωAB,R Ψ(0) x(m) .

(14.47)

The variable change (14.47) brings (14.46) in the simple form Z β 0 0 X X (2) B N LE dτ = AmAB ΨA Ψ + DmM N xM (−m) (m) (−m) x(m) , (14.48) 0

m

m

where iβ Ωm δAB , 2  β i (0) (0) B = Ω2m gM N + Ωm RM N,AB ΨA Ψ (0) (0) 2 2

AmAB = DmM N

(14.49)

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and RM N,AB is the Riemann tensor (1.58). Note that the matrix of the partial derivatives corresponding to the variable change (14.47) is triangle and has a unit superdeterminant. The functional integral over the non-zero modes is then given by a product of a large (in the continuous limit, infinite) number of finite-dimensional determinants, which can be symbolically written as grav. factor ∝

0 Y

det1/2 kAm k · det−1/2 kDm k .

(14.50)

m

Fixing the coefficient by the requirement for the determinant factor to reduce to unity in the flat case and the requirement of general covariance, we deduce

0 p Y

iβ −1/2 (0)

(14.51) g det grav. factor =

δAB + 2πm RAB , m (0)

D where RAB = RAB,CD ΨC (0) Ψ(0) /2. The infinite m-product in (14.51) can be done by writing the determinant as the product of its eigenvalues. All the eigenvalues of the real skew-symmetric matrix iRAB are imaginary. They split into D/2 pairs {iλ1 , −iλ1 } , {iλ2 , −iλ2 }, etc. By orthogonal rotations, we can bring iRAB to the canonical form:      0 λ1 0 λ2 iRAB = diag , ,... . (14.52) −λ1 0 −λ2 0

Each block associated with a given λα brings about the following factor in the determinant in Eq. (14.51): ∞ Y m=1

βλα /2 (2πm)2 = . (2πm)2 + (βλα )2 sinh(βλα /2)

(14.53)

We now use the following elementary mathematical fact: Let Q be a real skew-symmetric matrix brought to the canonical form (14.52) and let f (Q) be an even function of Q. Then Y det1/2 [f (Q)] = f (iλα ) (14.54) α

This finally gives grav. factor =

p

g (0) det1/2

βR/2

. sinh(βR/2)

(14.55)

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To obtain the index, the factor (14.55) in the integrand should be mulA B tiplied by exp{iβFAB ψ(0) ψ(0) /2}, by the factor (−i/2πβ)D/2 and then inQ Q A tegrated over the zero modes M dxM A dψ(0) . Incorporating the factor (0) −i/(2πβ) in the argument of f (R) (we are capitalizing on the fact that QD only the term ∼ A=1 ΨA in the expansion of the integrand contributes in the integral), and translating the result into the geometric language of forms [not forgetting the sign factor in (14.42)], we finally derive the result (14.35). It is now clear why we had to insert in this particular case the 1-loop gravitational factor in the tree-level integral (14.18) for the index. Formally, the factor (14.51) tends to 1 for small β and, naively, the corrections involving β and its higher powers can be neglected. We see, however, that each factor of β in the expansion is multiplied by a bi-fermion structure, as is also the case for the expansion of the integrand in (14.40). For the fermion integral not to vanish, we have to pick up the terms ∼ (βψ 2 )D/2 in A B the expansion of both the factor exp{iβFAB ψ(0) ψ(0) /2} and of the 1-loop factor (14.55)—they come on equal footing. On the other hand, if one tries to evaluate the path integral more precisely, taking into account cubic and other higher-order terms in xM (m) and Rβ A ψ(m) in the mode expansion for 0 LE dτ , the corresponding corrections would involve more powers of β than the powers of ψ 2 , to be suppressed in the small β limit. This justifies neglecting two-loop and higher-loop effects in the functional integral. Now we can also explain why we abandonned the idea to calculate the index for the Dolbeault Hamiltonian (9.8) involving a 4-fermion term. The latter would give large contributions ∼ (βψ 4 )k ∼ β −k to the integral (6.54). They even seem to diverge in the limit β → 0. Of course, this does not actually happen—for example, for a manifold of real dimension 4, we obtain the contribution Z ∼ d4 x εM N P Q ∂M CN P Q , which is an integral of a total derivative and vanishes. But for higher dimensions, the large terms involving the expansion of e−βH may multiply some (normally, small) structures coming from higher loops and the vanishing of all these contributions is not at all obvious. 14.4.1

Non-Abelian Dirac index

The Euclidean supersymmetric Lagrangian (14.39) involves an Abelian gauge field, and the calculation above was performed for an Abelian gauge

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field. But the Dirac index can also be defined and calculated for a nonAbelian gauge background. To write the corresponding supersymmetric Lagrangian, one has to introduce on top of ΨA also extra bosonic or fermionic variables in the fundamental representation of the gauge group (see the discussion on p. 299). But if one is only interested in the index, it is more convenient to work in the Hamiltonian approach. To this end, we define the supersymmetric problem (13.8) – (13.10) where the Dirac operator has the form (13.5), but with a non-Abelian matrix-valued gauge field Anon−Ab = AaM ta . M For simplicity, we consider now only flat geometry—the “gravitational” (nontrivial curvature) factor in the integrand of the index integral does not depend on whether the extra gauge field is Abelian or non-Abelian and was calculated before. As was also the case for Abelian AM , the pure gauge index can be calculated in the tree approximation. The integral [we are now using the representation (6.59)] reads Z Y n o dPM dxM ˆ flat Tr γ D+1 e−β H Inon−Ab = 2π ZM n o 1 D D+1 iβFAB γ A γ B /4 = d x Tr γ e (2πβ)D/2 Z D/2 (−1) ε = dD x Tr{FA1 A2 · · · FAD−1 AD } (14.56) A ...A (4π)D/2 (D/2)! 1 D —the obvious generalization of (14.43) [where the Witten → Dirac transitional factor (−1)D/2 should be inserted]. The index depends on the representation to which ta belong. For a localized SU (2) gauge field in R4 and if ta belong to the fundamental representation, the index coincides with the Pontryagin number (13.70). In particular, for the instantons (13.97), this index is equal to unity. That means that an instanton supports one fundamental fermion (a quark) zero mode and that this zero mode is right-handed [14].

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Bibliography

[1] E. Witten, Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982) 661. [2] Yu.A. Golfand and E.P. Likhtman, Extension of the algebra of Poincare group generators by bispinor generators, JETP Lett. 13 (1971) 323. [3] T. Eguchi, P.B. Gilkey and A.J. Hanson, Gravitation, gauge theories and differential geometry, Phys. Repts. 66 (1980) 213. [4] P. Candelas, Lectures on complex manifolds, in [Trieste 1987, Proceedings, Superstrings 87, pp. 1-88]. [5] N.E. Mavromatos, A note on the Atiyah-Singer index theorem for manifolds with totally antisymmetric H torsion, J. Phys. A21 (1988) 2279. [6] J.-M. Bismut, A local index theorem for non K¨ ahler manifolds, Math. Ann. 284 (1989) 681. [7] A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957) 391. See also L. Nirenberg, Lectures on Linear Partial Differential Equations, Amer. Math. Soc., 1973. ¨ [8] A. Clebsch, Uber die simultane Integration linearer partieller Differentialgleichungen, J. Reine. Angew. Math. 65 (1866) 257-268; ¨ G. Frobenius, Uber das Pfaffsche Problem, J. Reine. Agnew. Math. 82 (1877) 230-315. [9] A.V. Smilga, Comments on the Newlander-Nirenberg theorem, to appear in: [Varna 2019, Proceedings, Lie Theory and its Application in Physics], arXiv:1902.08549 [math-ph]. [10] A. Opfermann and G. Papadopoulos, Homogeneous HKT and QKT manifolds, arXiv:math-ph/9807026. [11] C.M. Hull, The geometry of supersymmetric quantum mechanics, arXiv:hep-th/9910028. [12] H. Hopf, Zur Topologie der komplexen Mannigfaltigkeiten, in:[Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York], p. 167. [13] E. Calabi, M´etriques k¨ ahleriennes et fibr´es holomorphes, Ann. Ecol. Norm. Sup. 12 (1979) 269. [14] G. ’t Hooft, Computation of the quantum effects due to a four-dimensional pseudoparticle, Phys. Rev. D14 (1976) 3432.

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Index

R-symmetry, 108 p-q symbol, 124 ’t Hooft symbols, 53, 58, 298, 301 (anti-)self-duality, 52, 57–60, 195, 273, 296–298, 301, 302

bracket Gr¨ onewold-Moyal, 91 for Grassmann variables, 112 Lie, 87, 88, 110, 198, 222, 256 Poisson, 81 for Grassmann variables, 94, 95, 279

action functional, 82, 297 in superspace, 137 analytic superspace and superfields, 154 anomaly chiral, 112, 209 conformal, 112 supersymmetric, 112, 132, 279 approximation Gaussian, 119, 312, 316, 317, 319 one-loop, 312 tree, 312 auxiliary variables, 143, 144, 149, 150, 158, 162, 175, 201–204, 239, 242, 249, 250, 300

canonical coordinates and momenta, 80 Cartan structure equation, 27 Chern class, 6, 296 Chevalleu basis, 70 Clifford algebra, 95, 191, 218, 256, 276, 279 complex anti-Dolbeault, 37, 182–184 de Rham, 1, 3, 4, 20 Dirac, 192, 196, 281, 292, 308, 311, 315 Dolbeault, 4, 5, 37, 181, 183, 196, 232, 316 twisted, 182, 184, 190, 311 K¨ ahler–de Rham, 200, 232 complex structure, 41 connection affine, 2, 14 Bismut, 33 Levi-Civita, 15 Obata, 27, 66, 68, 150, 213, 219, 222, 239, 264

Berezin integral, 92, 136 Betti numbers, 168–170, 309, 310 bispinors, 276 left- and right-handed, 279 bosonic and fermionic states, 100 superfield components, 136 bra and ket vectors, see Dirac notation 333

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principal, see gauge field spin, 26, 27 covariant and contravariant vectors, 12 covariant derivative harmonic, 152, 155, 247 in geometry, 14, 26 supersymmetric, 137, 138, 152 with gauge field, 23 de Rham cohomology, 22, 168 Dirac Hamiltonian, 278 matrices, 276 notation, 89 operator, 26, 112, 275–282, 294, 306, 308, 310, 315, 316, 322 deformed (by torsions), 282, 299, 315 string, 285, 289 Dolbeault Hamiltonian, 37, 38, 201, 292, 316, 321 twisted, 292 Dolbeault laplacian, see Dolbeault Hamiltonian equation for a geodesic, 79 Hamilton, 80 Lagrange, 82 for Grassmann variables, 94 Maxwell, 102 Schr¨ odinger, 88, 89, 290 Euler characteristic, 168–171, 173, 176, 308, 309 evolution operator, 98, 116–126, 307, 311 kernel of, 116 for Grassmann variables, 121–125 exterior derivative, 3, 19 fermion charge, 128, 167, 169, 174, 175, 181, 221, 223, 286, 306, 314, 316 fiber bundle, 22, 287, 289, 292, 296

line, 277, 283, 284, 292 canonical (for CPn ), 181 principal, 6, 22, 24, 26, 283, 294, 295 tangent, 26 vector, 305 field density tensor, 24 Fock-Schwinger gauge, 56 form closed, 20, 169 exact, 21, 169 harmonic, 21 invariant volume, 19 K¨ ahler, 39 Maurer-Cartan, 58 fundamental solution, 116, 121 gauge degree of freedom, 234 field, 23 field theories, 208–210, 235 transformation, 23, 241, 242, 250, 283, 287, 289, 293, 296 Gauss-Bonnet formula, 170, 171 generators of Grassmann algebra, 92 Lie algebra, 56, 70–74 supersymmetry transformations, see supercharges ghosts, 98 Hadamard formula, 74, 230, 231 Hamiltonian classical, 80 quantum, 3, 96 crypto-Hermitian, 98 Hamiltonian reduction, 5, 59, 218, 234–243, 299 Hamiltonian system, 80 harmonic bridge, 249–251, 258, 260–263, 265, 268–269 harmonic charge, 153, 155–158, 210, 246 heat kernel, 121 higher-derivative systems, 86, 164 Hodge

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decomposition, 168 diamond, 38 duality, 18 for complex manifolds, 36 holonomy group, 34 homogeneous spaces, 76 hypercomplex structure, 63, 214, 215, 233, 256 index (signature) Hirzebruch, 308–310, 313 Atiyah-Singer, 306 Dirac, 310–322 Witten, 5, 6, 118, 125–132, 169, 170, 287, 306, 316 instanton, 57, 58, 60, 195, 301, 302, 322 integral of motion, 80, 81, 84, 88, 91, 95, 234 Jacobi identity, 81, 88, 95, 110, 222 Jacobi polynomials, 289 K¨ ahler potential, 5, 35, 47, 240, 245 Kodaira identities, 223 Landau diamagnetism, 102 Landau Hamiltonian (for an electron in a magnetic field), 5, 101–105 Laplace-Beltrami operator, 3, 20, 167 Laplace-de Rham operator, 3, 20, 308 Lefschetz operators, 223 Legendre transformation, 82, 110, 243 Lichnerowicz formula, 287 Lorentz force, 102 magnetic charge, 104, 107, 130, 283–289 fractional, 6, 289–292 moment of electron, 101 monopole, 59, 209, 243, 284, 289 manifold almost complex, 39, 48 bi-HKT, 5 bi-K¨ ahler, 5

335

bi-spinorial, 5, 211 complex, 29 HKT, 4, 54 Obata-flat, 150, 213, 219, 222, 245, 302 Hopf, 47, 68 hyper-K¨ ahler, 4, 54 hypercomplex, 51 K¨ ahler, 4, 35 orientable, 12 quasicomplex, 5 Riemannian, 13 spinorial, 5, 208 metric Delduc-Vallent, 69, 270–271 Eguchi-Hanson, 270 Fubini-Study, 48, 292 Hermitian, 29 HKT—reducible and irreducible, 267, 271 Taub-NUT, 58–59, 267–270 monopole harmonics, 288 Morse function and Morse indices, 173 Morse theory, 1, 5, 171–174 Nijenhuis tensor, 42, 47, 49, 70, 71, 185, 255 Obata families, 263, 266, 267 octonions, 49 ordering ambiguity, 90 normal, 123, 132 Weyl, 90, 112, 131, 165, 179 oscillator anharmonic, 120 Grassmann, 93–96, 100, 105, 121–125 harmonic, 99, 103, 104, 117, 121, 234 in holomorphic variables, 94, 99 supersymmetric, 99–100 Pauli Hamiltonian, 101, 135, 180,

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284, 289, 292 Pauli matrices, 53, 54, 101, 207, 251 4-dimensional generalization of, 53, 298 phase space, 80 for Grassmann variables, 91, 94, 110 Planck constant, 89 Pontryagin number, 294, 296, 313, 322 prepotential for gauge fields in a Dolbeault model, 180, 231, 292 harmonic HKT, 6, 156, 212, 262, 265 harmonic hyper-K¨ ahler, 5, 265, 268 harmonic symplectic, 210 in a bi-HKT model, 241 in a bi-K¨ ahler model, 205 in a bi-symplectic model, 211 in a symplectic model, 207, 208, 226 in an HKT model, 212, 218, 240, 241, 245 principle Huygens-Fresnel, 117 least action, 83 pseudoconjugation, 151, 155, 157, 246, 247, 254 pseudoreality condition, 54, 147, 156, 157, 246–248, 252, 253, 263, 270 quasivielbeins, 261 Ricci tensor, 17, 57, 58 Riemann curvature tensor, 17 root multiplets, 218, 243 roots (of Lie algebra), 71–75 semi-dynamical variables, 299–301 sigma model N =2 de Rham, 171 sigma model N = 1, 161, 316, 318 N =2

de Rham, 162 de Rham quasicomplex, 176, 204, 238 de Rham with potential, 171 de Rham with quasitorsions, 174 Dolbeault, 177–196, 311 Dolbeault with gauge field, 191 N =4 bi-HKT, 62, 135, 217–220, 241, 242, 256 bi-K¨ ahler, 204–206, 218, 242 bi-symplectic, 211, 218 HKT, 211–216, 239–241, 243, 252–273, 302 HKT with gauge field, 272–273 K¨ ahler, 198–203, 232 K¨ ahler quasicomplex, 203–204, 239, 241 symplectic, 207–210, 228, 243, 299 symplectic with gauge field, 208–210 N =8 bi-HKT, 224 bi-K¨ ahler, 223–224, 226 hyper-K¨ ahler, 220–223, 233, 246–252, 264 symplectic, 226–228 not supersymmetric, 161 spectrum continuous, 91 discrete(pure point), 91 spin structure, 278, 294, 315, 317 spinC structure, 278 spinors, 96, 101, 275–278 spontaneous breaking of supersymmetry, 105, 107, 128, 130 stereographic coordinates, 30, 284, 301 super-Jacobian, 255, 320 supercharges, 3, 97 superfield N = 1, 136

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N N N N N N N N

=2 =4 =4 =2 =4 =4 =4 =4

chiral, 140 (3, 4, 1), 147 (4, 4, 0), 149 real, 139 chiral, 145 harmonic (V ++ ), 156 harmonic (q + linear), 155 harmonic (q + nonlinear), 156 N = 8 harmonic, 157 Grassmann, 137, 163, 202 mirror, 147, 150, 205, 211, 212, 217, 223, 224 supergroup, 134 supermanifold, 79 superpotential, 106, 107, 146, 171, 202, 203, 291 supertime, 5 supertransformations of dynamical variables, 110, 111, 136, 140, 141, 164, 184, 185, 197, 213, 238, 251, 258, 259 of superfields, 143, 185, 197, 205, 206, 213, 217, 223, 225, 226, 255 of superspace coordinates, 134, 138, 144, 152 supervariables, see superfields symplectic notation (for vectors and complex structures), 53 tangent space, 25 target space, 26

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theorem Atiyah-Singer, 6, 115, 170, 305–322 Frobenius, 43 KLW, 302–304 Newlander-Nirenberg, 5, 42, 141, 188 Noether, 84–86, 88, 110, 135, 164, 175, 178, 199, 235 Riemann-Roch, 305 Samelson, 69 Stokes, 21, 283, 296 time analytic, 154 holomorphic (left and right), 140 torons, 294 torsion, 15 holomorphic, 183–184, 191, 192, 217, 218, 232 transition functions, 12, 29, 32, 193, 293, 294 wave function, 3, 88 for Grassmann variables, 3, 18 spinorial, see spinor wedge product, 3, 18 Wess-Zumino gauge, 148 model, 109, 147 Wiener measure, 119, 120 Witten’s SQM model, 105–107, 129, 130, 143, 144, 291, 307 Yang monopole, 301

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