Noncommutative Geometry and Physics 2005: Proceedings of the International Sendai-Beijing Joint Workshop 9789812704696, 981-270-469-8

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noncommutative geometry and physics 2005 Proceedings of the International Sendai–Beijing Joint Workshop

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Ursula Carow-Watamura Satoshi Watamura Tohoku University, Japan Yoshiaki Maeda Hitoshi Moriyoshi Keio University, Japan Zhangju Liu Peking University, China Ke Wu Capital Normal University, China

noncommutative geometry and physics 2005 Proceedings of the International Sendai–Beijing Joint Workshop Sendai, Japan 1 – 4 November 2005 Beijing, China 7 – 10 November 2005

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TA I P E I

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CHENNAI

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NONCOMMUTATIVE GEOMETRY AND PHYSICS 2005 Proceedings of the International Sendai-Beijing Joint Workshop Copyright © 2007 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.

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ISBN-13 978-981-270-469-6 ISBN-10 981-270-469-8

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Preface The workshop “Noncommutative Geometry and Physics 2005” was organized by mathematicians and physicists from Keio Universty, Tohoku University and Beijing University in cooperation. The first part of this workshop was held at Tohoku University in Sendai on November 1-4, and the second part was held at Beijing University in Beijing on November 7-11. This workshop was the fifth in a series of joint workshops for mathematicians and physicists working in noncommutative geometry, deformation quantization and related topics, with the aim to stimulate discussions and the exchange of new ideas between both disciplines. Since the subject of noncommutative geometry has undergone rapid developments in the past few years, one of the important functions of our meetings is to elucidate these recent advances and the current status of research projects from mathematical point of view as well as from the physics’ side. In physical applications of noncommutative geometry, many key results have emerged on solutions of field theory on noncommutative spaces. Therefore, this was naturally one of the main subjects of the workshop. This volume includes disscussions of solitons such as monopoles and instantons in noncommutative spaces as well as in nonanticommutative superspaces. All contributions in this volume were submitted by conference speakers and participants, and were duly refereed. The articles contain presentations of new results which have not appeared yet in professional journals, or comprehensive reviews including an original part of the present developments in those topics. Effort was to provide comprehensive introductions to each subject such that the volume becomes accessible to researchers and graduate students interested in mathematical areas related to noncommutative geometry and its impact on modern theoretical physics. The workshop was held in the framework and with the support of the 21st century Center of Excellence (COE) program at Keio University, “Integrative Mathematical Sciences: Progress in Mathematics v

vi

Preface

motivated by Natural and Social Phenomena”, and it was also supported by a Grant-in-Aid for Scientific Research (No.13135202) of the Japanese Ministry of Education, Culture, Sports, Science and Technology at Tohoku University. We are also grateful to the Tohoku University, Japan, and to the Beijing University and the Academia Sinica, China, for providing the lecture rooms and their facilities which made a smooth performance of the meeting possible. The World Scientific Publishing company has been very helpful in the production of this volume, and we would like to thank Ms. Zhang Ji for her editorial guidance. In this place we wish to express our special thanks to all authors for their continuous effort in preparing these articles and the referees for their valuable comments and suggestions. Mathematical section: Zhangju Liu (Beijing Univ.) Yoshiaki Maeda (Keio Univ.) Hitoshi Moriyoshi (Keio Univ.) Physical section: Ursula Carow-Watamura (Tohoku Univ.) Satoshi Watamura (Tohoku Univ.) Ke Wu (CNU, Beijing)

Contents

I

Preface

v

DEFORMATIONS AND NONCOMMUTATIVITY

1

1 Expressions of Algebra Elements and Transcendental Noncommutative Calculus Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, Akira Yoshioka 3 2 Quasi-Hamiltonian Quotients as Disjoint Unions of Symplectic Manifolds Florent Schaffhauser 31 3 Representations of Gauge Transformation Groups of Higher Abelian Gerbes Kiyonori Gomi 55 4 Algebroids Associated with Pre-Poisson Structures Kentaro Mikami, Tadayoshi Mizutani

71

5 Examples of Groupoid Naoya Miyazaki

97

6 The Cohomology of Transitive Lie Algebroids Z. Chen, Z.-J. Liu

109

7 Differential Equations and Schwarzian Derivatives Hajime Sato, Tetsuya Ozawa, Hiroshi Suzuki

129

8 Deformation of Batalin-Vilkovsky Structure Noriaki Ikeda

151

vii

viii

II

Contents

DEFORMED FIELD THEORY AND SOLUTIONS

173

9 Noncommutative Solitons Olaf Lechtenfeld

175

10 Non-anti-commutative Deformation of Complex Geometry Sergei V. Ketov

201

11 Seiberg-Witten Monopole and Young Diagrams Akifumi Sako

219

12 Instanton Counting, Two Dimensional Yang-Mills Theory and Topological Strings Kazutoshi Ohta 239 13 Instantons in Non(anti)commutative Gauge Theory via Deformed ADHM Construction Takeo Araki, Tatsuhiko Takashima, Satoshi Watamur 253 14 Noncommuative Deformation and Drinfel’d Twisted Symmetry Yoshishige Kobayashi 261 c (2) k and Twisted Conformal Field 15 Affine Lie Superalgebra gl(2|2) Theory Xiang-Mao Ding, Gui-Dong Wang, Shi-Kun Wang 273 16 A Solution of Yang-Mills Equation on BdS Spacetime Xin’an Ren, Shikun Wang

289

17 Solitonic Information Transmission in General Relativity Yu Shang, Guidong Wang, Xiaoning Wu, Shikun Wang and Y.K.Lau

297

18 Difference Discrete Geometry on Lattice Ke Wu, Wei-Zhong Zhao, Han-Ying Guo

301

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Part I

DEFORMATIONS AND NONCOMMUTATIVITY

1

Expressions of algebra elements and transcendental noncommutative calculus Hideki Omori1 , Yoshiaki Maeda2 , Naoya Miyazaki3 , and Akira Yoshioka4 1

2

3

4

Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, 278-8510, Japan Department of Mathematics, Faculty of Science and Technology, Keio University, Hiyoshi, Yokohama, 223-8825, Japan Department of Mathematics, Faculty of Economics, Keio University, Hiyoshi, Yokohama, 223-8521, Japan Department of Mathematics, Faculty of Science, Tokyo University of Science, Kagurazaka, Tokyo, 102-8601, Japan [email protected], [email protected], [email protected], [email protected] 2 Partially supported by Grant-in-Aid for Scientific Research (#18204006.), Ministry of Education, Science and Culture, Japan. 3 Partially supported by Grant-in-Aid for Scientific Research (#18540093.), Ministry of Education, Science and Culture, Japan. 4 Partially supported by Grant-in-Aid for Scientific Research (#17540096.), Ministry of Education, Science and Culture, Japan.

Abstract Ideas from deformation quantization are applied to deform the expression of elements of an algebra. Extending these ideas to certain transcendental elements implies that 1 i~ uv in the Weyl algebra is naturally viewed as an indeterminate living in a discrete set N+ 21 or −(N+ 21 ) . This may yield a more mathematical understanding of Dirac’s positron theory. A.M.S Classification (2000): Primary 53D55, 53D10; Secondary 46L65

1 Introduction Quantum theory is treated algebraically by Weyl algebras, derived from differential calculus via the correspondence principle. However, since the algebra is noncommutative, the so-called ordering problem appears. Orderings are treated in the physics literature of quantum mechanics (cf. [1]) as the rules of association from classical observables to quantum observables, which are supposed to be self-adjoint operators on a Hilbert space.

3

4

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

Typical orderings are, the normal (standard) ordering, the anti-normal (antistandard) ordering, the Weyl ordering, and the Wick ordering in the case of complex variables. However, from the mathematical viewpoint, it is better to go back to the original understanding of Weyl, which says that orderings are procedures of realization of the Weyl algebra W~ . Since the Weyl algebra is the quotient algebra of the universal enveloping algebra of the Heisenberg Lie algebra, the Poincar´e-Birkhoff-Witt theorem shows that this algebra can be viewed as an algebra defined on a space of polynomials. As we show in §1, this indeed gives product formulas on the space of polynomials which produce algebras isomorphic to W~ . This gives the unique way of expressions of elements, and as a result one can treat transcendental elements such as exponential functions, which are necessary to solve differential equations (cf. §2.2). However, we encounter several anomalous phenomena, such as elements with two different inverses (cf. §4) and elements which must be treated as double valued (cf. [16],[17]). 1 uv should be In this note, we treat the phenomenon which shows that i~ 1 1 viewed as an indeterminate living in the set N+ 2 or −(N+ 2 ). We reach this interpretation in two different ways, by analytic continuation of inverses of 1 uv, and by defining star gamma functions using various ordered expresz+ i~ sions. We emphasise in this paper, that our approach to show the discrete picture 1 uv is not to use operator representation at all, but for the element z + i~ to express it in various orderings instead, under a leading principle that a physical object should be free from the choice of orderings(the ordering free principle), just as classical, geometric objects are expressed independent of the choice of local coordinates. Since similar discrete pictures of elements is familiar in quantum observables, treated as a self-adjoint operator, our observation gives for their justification for the operator theoretic formalism of quantum theory. However, in this note we restrict our ordering to a particular subset to avoid the multi-valued expressions. In some cases, we should be more careful about the convergence of integrals and the continuity of the product, so the detailed computations and the proof of continuity of the products will appear elsewhere.

2 K-ordered expressions for algebra elements We introcuce a method to realize the Weyl algebra via a family of expressions. This leads to a transcendental calculus in the Weyl algebra. 2.1 Fundamental product formulas and intertwiners Let SC (n) and AC (n) be the spaces of complex symmetric matrices and skewsymmetric matrices respectively, and MC (n)=SC (n)⊕AC (n). For an arbitrary

2 K-ordered expressions for algebra elements

5

fixed n×n-complex matrix Λ∈MC (n), we define a product ∗Λ on the space of u] by the formula polynomials C[u i~

f ∗Λ g = f e 2 (

P ←− ij − − → ∂ui Λ ∂uj )

g=

X (i~)k k!2k

k

Λi1 j1· · ·Λik jk ∂ui1· · ·∂uik f ∂uj1· · ·∂ujk g.

(1) u], ∗Λ ) is an associative algebra. It is known and not hard to prove that (C[u u ], ∗Λ ) is determined by the skew-symmetric (a) The algebraic structure of (C[u part of Λ (in fact, by its conjugacy class A → t GAG). u ], ∗Λ ) is isomorphic to the (b) In particular, if Λ is a symmetric matrix, (C[u usual polynomial algebra. Set Λ=K+J, K∈SC (n), J∈AC (n). Changing K for a fixed J will be called a deformation of expression of elements, as the algebra remains in the same isomorphism class. Example of computations: ui ∗Λ uj =ui uj +

i~ ij Λ , 2

ui ∗Λ uj ∗Λ uk =ui uj uk +

i~ ij (Λ uk +Λik uj +Λjk ui ). 2

By computing the ∗Λ -product using the product formula (1), every element of the algebra has a unique expression as a standard polynomial. We view these expressions of algebra elements as analogous to the “local coordinate expression” of a function on a manifold. Thus, changing K corresponds to a local coordinate transformation on a manifold. In this context, we call the product formula (1) the K-ordered expression by ignoring the fixed skew part J. Following a familiar notion in quantum we  call the K-ordered  mechanics,  0 Im 0 −Im expression for the particular K=0, , , the Weyl ordering, Im 0 −Im 0 normal ordering, anti-normal ordering, respectively. The intertwiner between a K-ordered expression and a K ′ -ordered expression, which we view as a local coordinate transformation, is given in a concrete form : Proposition 2.1 For symmetric matrices K, K ′ ∈ SC (n), the intertwiner is given by K′

IK (f ) = exp

 i~ X 4

i,j

(K

′

ij

 K K′ −K ij )∂ui ∂uj f (= I0 (I0 )−1 (f )),

(2)

K′

u ]; ∗K+J ) → (C[u u ]; ∗K ′ +J ) between algebras. giving an isomorphism IK : (C[u u] : Namely, for any f, g ∈ C[u K′

K′

K′

IK (f ∗K+J g) = IK (f ) ∗K ′ +J IK (g).

(3)

6

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka K

u ], the set {I0 (f ); K∈SC (n)} forms a family of eleThus, for every f ∈C[u ments which are mutually intertwined. We denote this family by f∗ viewing K as an algebraic object, and we use often  the notation :f∗ :K+J =I0 (f ). 0 −Im u ], ∗Λ ) is called the Weyl algebra, In the case n=2m and J= , (C[u Im 0 with isomorphism class denoted by W2m . In fact, if J is non-singular, then u ], ∗Λ ) is isomorphic to the Weyl algebra. (C[u 1 uk ) t(z+s i~

2.2 The star exponential function e∗

Using the various ordered expression of elements of algebra, we can treat elementary transcendental functions. For an object H∗ , the ∗-exponential func∗ ∗ tion etH is defined as the family :etH ∗ ∗ :Λ of real analytic solutions in t of the evolution equations d ft =:H∗ :Λ ∗Λ ft , f0 =1. (4) dt For instance, for every z∈C, we have 1 z+s i~ uk

:e∗

s

1

:Λ =ez :e∗ i~

uk

2

:Λ =ez es

kk 1 4i~ K

1

es i~ uk .

(5)

When we fix the skew part J of Λ, we often abbreviate the notation to : :K , ∗K for : :K+J , ∗K+J respectively. Since the exponential law 1 (z+w)+(s+t) i~ uk

:e∗

1 z+s i~ uk

:K =:e∗

1 w+t i~ uk

:K ∗K :e∗

:K

holds for every K, it is better to write 1 uk (z+w)+(s+t) i~

e∗ 1 z+s i~ uk

by viewing :e∗

1 z+s i~ uk

=e∗

1 w+t i~ uk

∗e∗

:K as the K-ordered expression of the (ordering free) ex-

z+s

1

u

ponential element e∗ i~ k . Under this convention, one may write for instance ij :ui ∗uj :K =ui uj + i~ 2 (K+J) . s

1

u

We remark that even for the simplest exponential function e∗ i~ k , formula (5) gives the following (cf. [12]). 1 P uk 2n i~ Proposition 2.2 If Im K kk − , 2

(19)

1 . 2

(20)

1

1 1 e 2 tz i~ 2uv tanh 2 t dt, 1 e cosh 2 t

Re z
− 12 , which is also the holomorphic 1 domain for (z+ i~ uv)−1 −∗ . All of these results are easily proved for the Weyl ordering. However, if t 1 uv κ∈C−{κ≥1} ∪{κ≤−1}, then :e∗i~ :κ is rapidly decreasing in t, and the same computation gives the following:

4 Inverses and their analytic continuation

15

1 Proposition 4.3 For every z with Re z>− 21 , the two inverses (z+ i~ uv)−1 +∗ 1 uv)−1 and (z− i~ −∗ are defined in the κ-ordered expression for κ∈C−{κ≥1} ∪ {κ≤−1}.

−1 1 1 Note that (z+ i~ uv)−1 +∗ ∗(−z− i~ uv)−∗ diverges for any ordered expression. However, the standard resolvent formula gives the following:

Proposition 4.4 If z+w6=0, then  1 1  1 −1 (z+ uv)−1 +(w− uv) +∗ −∗ z+w i~ i~ 1 1 is an inverse of (z+ i~ uv)∗(w− i~ uv). In particular, for every positive integer n, and for every complex number z such that Re z> − (n+ 12 ), the κ-ordered
−1 1 1 1 1 uv))−1 expression of 2n (1+ n1 (z+ i~ +∗ +(1− n (z+ i~ uv))−∗ gives an inverse of 1 1− n12 (z+ i~ uv)2∗ for κ∈C−{κ≥1}∪{κ≤−1}.

4.1 Analytic continuation of inverses 1 1 Recall that (z± i~ uv)−1 ±∗ are holomorphic on the domain Re z > − 2 . It is −1 −1 1 1 natural to expect that (z± i~ uv)±∗ =C(C(z± i~ uv))±∗ for any non-zero constant C. To confirm this, we set C=eiθ and consider the θ-derivative Z 0 1 eiθ t(z± i~ uv) eiθ e∗ dt. −∞

In the (κ, τ )-ordered expression, the phase part of the integrand is bounded in t and the amplitude is given by (1−κ)e

2eiθ tz , + (1+κ)e−eiθ t/2

eiθ t/2

κ6=1.

Hence, the integral converges whenever Re eiθ (z± 21 ) > 0, and by integration 1 uv)−1 by parts this convergence does not depend on θ. It follows that (z± i~ ±∗ 1 are holomorphic on the domain C−{t; −∞ − 21 uv) t(z− i~ dt= . (z− uv)∗e∗ 1−̟00 z=− 21 i~ −∞

As suggested by these formulas, we extend the definition of the ∗-product as s 1 uv follows: For every polynomial p(u, v) or p(u, v)=e∗ i~ , Z 0 1 t(z± 1 uv) e∗ i~ dt. p(u, v)∗(z± uv)−1 (25) = lim p(u, v) ∗ +∗ N →∞ i~ −N Hence we have the formula  1 1 1 Re z > − 12 = . (z+ uv)∗(z+ uv)−1 +∗ 1−̟00 z=− 12 i~ i~

(26)

Considering 1 1 1 1 ◦ n n n uv)∗(z+ uv)−1 uv)∗v n ∗(v ◦ )n ∗(z+ uv)−1 +∗ ∗v =(v ) ∗(z+ +∗ ∗v i~ i~ i~ i~ and using the formula (23), we have the following:

(v ◦ )n ∗(z+

Theorem 4.2 If we use definition (25) for the ∗-product, then  1 1 1 uv)∗(z+ uv)−1 = 1 1 +∗ 1− n! ( i~ u)n ∗̟00 ∗v n i~ i~  1 1 1 (z− uv)∗(z− uv)−1 = 1 1 −∗ 1− n! ( i~ v)n ∗̟00 ∗un i~ i~ (z+

z6∈−(N+ 12 ) , z=−(n+ 12 )

(27)

z6∈−(N+ 21 ) . z=−(n+ 12 )

(28)

18

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

Although z=−(n+ 12 ) are all removable singularities for (27) and (28) as a function of z, it is better to retain these singular points. These formulas give in particular for every fixed positive integer m  1 1 1 1 1 z6∈−(N+m+ 12 ) −1 (1+ (z+ uv))∗(1+ (z+ uv))+∗ = 1 1 k k 1− k! ( i~ u) ∗̟00 ∗v z=−(k+m+ 12 ) m i~ m i~ (29) for arbitrary k ∈ N. We state the following identity for later use: 1 1 1 ̟00 ∗v n ∗(−n− + uv)=̟00 ∗( u∗v)∗v n =0. 2 i~ i~

(30)

5 An infinite product formula Recall the classical formula sin πx=πx ∞ Y

k=1

(1−

1 x2 )= k 2 2i

Z

x2 k=1 (1− k2 ).

Q∞

χ[−π,π] (t)eitx dt= lim

n→∞

Z Y n

Rewrite this as follows:

(1+

k=1

1 2 ∂ )δ(t)eitx dt, k2 t

where χ[−π,π] (t) is the characteristic function of the interval [−π, π]. It follows that χ[−π,π] (t)=2i lim

n→∞

n Y

k=1

(1+

1 2 ∂ )δ(t) k2 t

in the space of distributions. 1 it i~ uv :κ is not singular on t ∈ R, we For κ uch that | κ+1 κ−1 |6=1 , so that :e∗ compute as follows: Z Z 1 uv) ±it 1 uv) it(z± i~ :κ dt= χ[−π,π] (t)eitz :e∗ i~ :κ dt. χ[−π,π] (t):e∗ Fixing a cut-off function ψ(t) of compact support such that ψ=1 on [−π, π], we see that Z Z Y n 1 ±it 1 uv t(z± 1 uv) χ[−π,π] (t):e∗ i~ :κ dt=2i lim (1+ 2 ∂t2 )δ(t)ψ(t)etz :e∗ i~ :κ dt. n→∞ k k=1

Integration by parts gives Z n n Y Y 1 1 ±it 1 uv t(z± 1 uv) :(1+ 2 ∂t2 )e∗ i~ :κ . lim δ(t) (1+ 2 ∂t2 )ψ(t)etz :e∗ i~ :κ dt= lim n→∞ n→∞ k k k=1

k=1

Hence we have in the κ-ordered expression that Z n Y 1 1 1 it(z± i~ uv) χ[−π,π] (t)e∗ dt=2i lim (1− 2 (z± uv)2 )∗ . n→∞ k i~ k=1

5 An infinite product formula

19

Noting that sin∗ π(z±

1 1 uv)=π(z± uv)∗ i~ i~

Z

1 uv) it(z± i~

χ[−π,π] (t)e∗

dt ∈ Hol(C2 ),

we have n

sin∗ π(z±

Y 1 1 1 1 uv)=π(z± uv)∗ lim ∗ (1− 2 (z± uv)2 )∗ n→∞ i~ i~ k i~

(31)

k=1

in Hol(C2 ). In particular, we have κ+1 Proposition 5.1 In the κ-ordered expression with | κ−1 |6=1, we have n

sin∗ π(z+

Y 1 1 1 1 uv)=π(z+ uv)∗ lim ∗(1− 2 (z+ uv)2 ). n→∞ i~ i~ k i~ k=1

This is identically zero on the set z∈Z+ 12 . The formula in Proposition 5.1 may be rewritten as n

sin∗ π(z+

Y 1 1 1 1 1 1 (z+ i~ uv) uv)=π(z+ uv)∗ lim ∗(1− (z+ uv))∗e∗k n→∞ i~ i~ k i~ k=1 n Y

∗

k=1

1 1 1 − 1 (z+ i~ uv) ∗(1+ (z+ uv))∗e∗ k . k i~

In §6, we will define a star gamma function via the two different inverses mentioned previously and give an infinite product formula for the star gamma function.
−1 1 1 1 uv)−1 5.1 The product with (z+ i~ +∗ and with 1+ m (z+ i~ uv) +∗

1 1 First we consider the product (z+ i~ uv)−1 ±∗ ∗ sin∗ π(z+ i~ uv) in two different ways. One way is by defining:

(z+

1 1 uv)−1 uv) ±∗ ∗ sin∗ π(z+ i~ i~ n   (32) Y 1 1 1 1 2 = lim (z+ uv)−1 ∗(1− (z+ ∗ (z+ uv)∗ uv) ) . ±∗ n→∞ i~ i~ k2 i~ k=1

Qn

1 uv)∗ Since (z+ i~ (27) and (30) give

k=1

1 ∗(1− k12 (z+ i~ uv)2 ) is a polynomial, Proposition 3.3, ∞

Y 1 1 1 1 uv)= ∗(1− 2 (z± uv)2 ). (z+ uv)−1 ±∗ ∗ sin∗ π(z± i~ i~ k i~ k=1

(33)

20

Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

The second way is by defining 1 1 (z+ uv)−1 uv)= lim ±∗ ∗ sin∗ π(z+ N →∞ i~ i~

Z

0

1 uv) t(z+ i~

e∗

−N

∗ sin∗ π(z+

1 uv). (34) i~

This may be written as the complex integral 1 2i

Z

0+πi −∞+πi

t(z+ 1 uv) e∗ i~ dt−

1 2i

Z

0−πi

−∞−πi

1 uv) t(z+ i~

e∗

dt.

Rπ 1 Adding − 12 −π eit(z+ i~ uv) dt to this expression gives the clockwise contour integral along the boundary of the domain D={z∈C; Re z0, then | κ−1 positive real part.

Proposition 5.2 Suppose Re κ>0 and κ∈C−{κ≥1}∪{κ≤−1}. Then in the κ-ordered expression, we have Z Z 0 1 1 1 π it(z+ i~ 1 uv) uv) t(z+ i~ ∗ sin∗ π(z+ uv)= dt. e∗ lim e∗ N →∞ −N i~ 2 −π According to (31) the right hand side gives the same result as (33), that is, Q∞ 1 1 2 ∗(1− 1 k2 (z+ i~ uv) ).

Proposition 5.3 Suppose Re κ>0 and κ∈C−{κ≥1}∪{κ≤−1}. Then in the 1 1 uv)∗(z+ i~ uv)−1 κ-ordered expression, the product sin∗ π(z+ i~ +∗ is an entire −1 1 function of z. Namely, all singularities of (z+ i~ uv)+∗ at −(N+ 21 ) are cancelled out in formulas (29) and (30). By a proof similar to that of Proposition 5.3, we obtain Proposition 5.4 Suppose Re κ>0, and κ∈C−{κ≥1}∪{κ≤−1}. Then in the κ-ordered expression,

6 Star gamma functions

sin∗ π(z−

21

1 1 uv)∗(z− uv)−1 −∗ i~ i~

is a well defined entire function of z. 1 1 uv)∗(z 2 −( i~ uv)2 )−1 In particular, sin∗ π(z+ i~ ±∗ is a holomorphic function of z in C. 1 1 1 (z+ i~ uv))−1 Consider next the product (1+ m +∗ ∗ sin∗ π(z+ i~ uv). Since

(1+

1 1 1 (z+ uv))−1 uv)−1 +∗ =m(m+z+ +∗ , m i~ i~

1 1 and sin∗ π(z+m+ i~ uv)=(−1)m sin∗ π(z+ i~ uv) by the exponential law, the product formula is essentially the same as above. Hence we see the following:

Proposition 5.5 Suppose Re κ>0, and κ∈C−{κ≥1}∪{κ≤−1}. Then in the 1 1 1 κ-ordered expression, the product sin∗ π(z+ i~ uv)∗(1+ m (z+ i~ uv))−1 +∗ is an entire function of z with no removable singularity. Remark 2 Suppose Re κ − 12 , the right hand side of (36) converges and is holomorphic with respect to z . However, Γ∗ (− 12 ± uv ~i ) is singular. Throughout this section, ordered expressions are always restricted to κ∈C−{κ≥1}∪{κ≤−1}. 6.1 Analytic continuation of Γ∗ (z ±

uv ) ~i

As with the usual gamma function, integration by parts gives the identity Γ∗ (z+1 ±

uv uv uv )=(z ± )∗Γ∗ (z ± ). ~i ~i ~i

(37)

Using

uv uv −1 uv )=(z ± ) ∗Γ∗ (z+1 ± ), ~i ~i ±∗ ~i and careful treating continuity inssues, we have Γ∗ (z ±

Proposition 6.2 Γ∗ (z ± C−{−(N+ 21 )}. τ (z± uv )

uv ~i )

extends to a holomorphic function on z ∈

~i ∗̟00 =(z ± 21 )−1 ̟00 , we see the following remarkable feature Since e∗ of these star functions: Z N τ τ (z± uv ) uv ~i e−e e∗ Γ∗ (z ± dτ ∗̟00 = Γ(z ± 21 )̟00 , )∗̟00 ≡ lim N →∞ −N ~i Z N uv τ (z± uv ~i ) , y)∗̟00 ≡ lim e∗ B∗ (z ± (1−eτ )y−1 dτ ∗̟00 = B(z ± 21 , y)̟00 . N →∞ −N ~i (38)

6 Star gamma functions

23

6.2 An infinite product formula We see in the same notation as above Z 0 uv
−1 uv τ (z± uv ~i ) dτ = z+ B∗ (z± , 1) = e∗ , ~i i~ ∗± −∞

1 Re z > − . 2

(39)

We now compute uv )Γ(y) = Γ∗ (z ± ~i

ZZ

τ σ τ (z± uv ~i ) σy −(e +e )

R2

e∗

e e

dτ dσ.

We change variables by setting eσ = et (1−es ),

τ = t+s,

where − ∞ < t < ∞, −∞ < s < 0.

Since eτ +eσ = et , this gives a diffeomorphism of R×R− onto R2 . The Jacobian 1 is given by dτ dσ = 1−e s dtds. Hence we have the fundamental relation between the gamma function and the beta function Z ∞Z 0 t uv t(y+z± uv s(z± uv ~i ) −e ~i ) e ∗e∗ (1−es )y−1 dtds )Γ(y) = e∗ Γ∗ (z ± ~i −∞ −∞ (40) uv uv =Γ∗ (y+z ± )∗B∗ (z ± , y). ~i ~i Integration by parts gives (z ±

uv uv uv )∗B∗ (z ± , y+1) = yB∗ (1+z ± , y+1). ~i ~i ~i

To prove this, note that uv d τ (z± uv τ (z± uv ~i ) ~i ) e∗ = (z ± )∗e∗ , dτ ~i τ

lim e−e

τ →±∞

uv +zτ ±τ ~i e∗

τ d −eτ e = −eτ e−e , dτ

= 0 for

1 Re z > − . 2

uv uv Since B∗ (z ± uv ~i , y+1) = B∗ (z ± ~i , y)−B(1+z ± ~i , y), we have the functional equation y+z ± uv uv uv ~i B∗ (z ± , y) = ∗B∗ (z ± , y+1). (41) ~i y ~i

Iterate (41) to obtain B∗ (z ±

(y+z ± uv , y) = ~i

uv ~i )∗(y+1+z

y(y+1)

±

uv ~i )

∗B∗ (z ±

uv , y+2). ~i

Using the notation (a)n = a(a+1) · · · (a+n−1),

{A}∗n = A∗(A+1)∗ · · · ∗(A+n−1),

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Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

we have B∗ (z ±

{y+z ± uv uv uv ~i }∗n ∗B∗ (z ± , y) = , y+n). ~i (y)n ~i

(42)

Similarly, integration by parts gives the formula Γ∗ (1+z ±

uv uv uv ) = (z ± )∗Γ∗ (z ± ), ~i ~i ~i

1 Re z > − . 2

for

(43)

Iterate (43) to obtain uv uv  uv . ) = Γ∗ (z ± )∗ z ± ~i ~i ~i ∗n Qn uv −1 Lemma 6.1 B∗ (z ± uv k=0 ∗(k+z ± ~i )±∗ . ~i , n+1) = n! Γ∗ (n+1+z ±

(44)

Proof. The right hand side of the above equality will be denoted by

n! . (±) {z± uv ~i }∗n+1

The case n = 0 is given by (39). Suppose the formula holds for n. For the case n+1, we see that uv B∗ (z ± , n+2) ~i Z 0 τ (z± uv ~i ) = e∗ (1−eτ )(1−eτ )n dτ = −∞

It follows that B∗ (z ±

n! {z ±

uv (±) ~i }∗n+1

−

n! {1+z ±

uv (±) ~i }∗n+1

.

(n+1)! uv . , n+2) = (±) ~i } {z ± uv ∗n+2 ~i

In this subsection, we give an infinite product formula for the ∗-gamma function. By Lemma 6.1, we see that Z 0 1 n! τ (z± uv ~i ) e∗ , Re z > − . (1 − eτ )n dτ = uv (±) 2 −∞ {z ± ~i }∗n+1 Replacing eτ by

1 τ′ ne ,

namely setting τ = τ ′ − log n in the left hand side, and (log n)(z± uv ~i )

multiplying both side by e∗ Z

log n

−∞

τ ′ (z± uv ~i )

e∗

, we have

1 ′ n! (log n)(z± uv ~i ) ∗e∗ (1− eτ )n dτ ′ = . (±) uv n {z ± ~i }∗n+1

(45)

Lemma 6.2 The Weyl ordering of the left hand side of (45) converges when R ∞ τ ′ (z± uv τ′ ~i ) −e n→∞ to −∞ e∗ e dτ ′ in Hol(C2 ).

6 Star gamma functions ′

25

τ′

Proof. Obviously, limn→∞ (1− n1 eτ )n =e−e uniformly on each compact subset as a function of τ ′ . In the Weyl ordering, it is easy to show that Z log n ′ Z ∞ ′ τ′ τ′ τ (z± uv τ (z± uv ′ ~i ) −e ~i ) −e e∗ lim e∗ e dτ = e dτ ′ n→∞

−∞

−∞

2

in Hol(C ). Thus it is enough to show that Z log n ′ τ′ 1 ′ τ (z± uv ~i ) lim (e−e −(1− eτ )n )dτ ′ =0 e∗ n→∞ −∞ n in Hol(C2 ). This is easy in the Weyl ordering. Applying the intertwiner gives the desired result. The right hand side of (45) equals for Re z > − 12 1 (log n−(1+ 21 +···+ n ))(z± uv ~i )

e∗

∗(z ±

n z± uv 
~i z ± uv uv −1 Y  ~i −1 k ∗e )∗± ∗ 1+ . ∗ ±∗ ~i k k=1

(log n−(1+ 1 +···+ 1 ))(z± uv )

2 n ~i = The left hand side of (45) converges, and limn→∞ e∗ −γ(z± uv ~i ) obviously, where γ is Euler’s constant. By the continuity of the ∗e∗ s uv multiplication e∗ ~i ∗, we have the convergence in Hol(C2 ) of

n   Y 1 uv
−1 k1 (z± uv ~i ) lim ∗ 1+ (z ± . ) ±∗ ∗e∗ n→∞ k ~i k=1

Hence we have the convergence in Hol(C2 ) of the infinite product formula Γ∗ (z+

∞   Y 1
−1 k1 (z+ i~ 1 1 uv uv −γ(z+ uv uv) ~i ) ) = e∗ ∗ ∗ (z+ uv) ∗e 1+ ∗(z+ )−1 ∗ +∗ ~i ~i ∗+ k i~ k=1 (46) −

1

(z+ uv )

1 m ~i to both side of (46) and (z+ uv Fix m∈N. Multiplying (1+ m ~i )e∗ using the abbreviated notation  Y Y  1
−1 k1 (a+ i~ uv 1 1 uv) (z=a) = (z+ )−1 (a+ uv) ∗e ∗ 1+ ∗ +∗ ~i +∗ k i~ k6=m

k6=m

we have uv uv 1 − 1 (z+ uv ~i ) ∗Γ∗ (z+ ) (1+ (z+ ))∗e∗ m m i~ ~i Q  1 (z=z) k6=m
z6∈ − (N+m+ 21 ) (47) = Q 1 1 1 n n z= − (n+m+ 2 ) k6=m (z=−n−m− 2 )∗ 1− n! ( i~ u) ∗̟00 ∗v

−1 1 1 where n∈N. As opposited to the case that (1+ m (z+ uv i~ ))+∗ ∗ sin∗ π(z+ i~ uv) is entire function (cf. Proposition 5.5), there are removable singularities with respect to z.

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Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki, and Akira Yoshioka

Multiplying (47), we have


− k1 (z+ uv ~i ) to both sides of (46) and using 1+ k1 (z+ uv ~i ) e∗

Q∞

k=1

N  Y
− 1 (z+ i~1 uv)  1 1 1 ∗ 1+ (z+ uv) ∗e∗ k ∗Γ∗ (z+ uv) N →∞ k i~ i~ k=1  z6∈ − (N+ 21 ) Pn 1 11 k = k 1− k=0 k! ( i~ u) ∗̟00 ∗v , z= − (n+ 12 ),

lim

in Hol(C2 ), where n∈N.

1 uv) 7 Products with sin∗ π(z+ i~ 1 1 In this section we show that sin∗ π(z+ i~ uv)∗Γ∗ (z+ i~ uv) is well defined as an entire function of z. By recalling Euler’s reflection formula, this product may 1 . We define the product by the integral be understood as Γ (1−(z+ 1 uv)) ∗

i~

1 1 uv)∗Γ∗ (z+ uv) i~ i~ Z T′ 1 1 τ τ (z+ uv ) uv) −πi(z+ i~ uv) πi(z+ i~ i~ = lim −e∗ )∗e−e e∗ dτ (e∗ ′ T,T →∞ −T Z ∞ τ (τ −πi)(z+ uv (τ +πi)(z+ uv i~ ) i~ ) −e∗ )dτ. = e−e (e∗

2i sin∗ π(z+

−∞

(48)

The κ-ordered expression of (48) is given as follows: :(48):κ =

Z

∞+πi

τ −πi

e−e

−∞+πi

τ −πi

By using e−e

τ +πi

=e−e

Z (

τ (z+ uv i~ )

e∗

dτ −

Z

∞−πi

τ +πi

e−e

−∞−πi

τ (z+ uv i~ )

e∗

dτ.

, this is given by the integral

∞+πi

−∞+πi

−

Z

∞−πi

−∞−πi

τ

τ (z+ uv i~ )

)ee e∗

dτ.

Note this is not a contour integral, but is defined for κ∈C−{κ≥1}∪{κ≤−1}. The following is our main result: 1 1 uv)∗Γ∗ (z+ i~ uv) is defined as an entire function Theorem 7.1 sin∗ π(z+ i~ 1 of z, vanishing at z∈N+ 2 in any κ-ordered expression such that Re κ 0 is not formulated as a σ-model. It may be better to think of our theory as a certain gauge theory in which higher-order differential forms provide “gauge fields”.

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61

3.2 Equation of motion To derive the equation of motion from the functional, we begin with some comments on Lie group structures on G(W ) and G(W )C : we can make G(W ) into an infinite dimensional Lie group whose Lie algebra g(W ) is the vector space g(W ) = A2k (W )/d(A2k−1 (W )) with the trivial Lie bracket. The exponential map exp : g(W ) → G(W ) is given by the following composition of homomorphisms: ι

A2k (W )/d(A2k−1 (W )) → A2k (W )/A2k (W )Z → G(W ). In a similar way, we can also make G(W )C into a complex Lie group. As the notation indicates, G(W )C gives rise to a complexification of G(W ). Since the tangent space of G(W )C is seen, we compute the equation of motion for the action functional EW to obtain: Lemma 3.2 A Deligne cohomology class f ∈ G(W )C such that δ(f )|∂W = 0 is a critical point of EW if and only if f satisfies: d∗ δ(f ) = 0, where d∗ = − ∗ d∗ is the formal adjoint of d : A2k+1 (W, C) → A2k+1 (W, C). d E (f + tα) for α ∈ g(W )C . Because δ(α) = dα, Proof. We compute dt t=0 W Stokes’ theorem leads to: Z Z d α ∧ d ∗ δ(f ), α ∧ ∗δ(f ) − 2ℓπ EW (f + tα) = 2ℓπ dt t=0

∂W

W

which implies the lemma. ⊔ ⊓

We note that f ∈ G(W )C always satisfies the equation dδ(f ) = 0. Thus, in the case where W has no boundary, f is a solution to the equation of motion if and only if δ(f ) is a harmonic form. So Lemma 2.2 allows us to identify the space of solutions with H 2k+1 (W, Z) × (H 2k (W, R)/H 2k (W, Z)). We also note that f ∈ G(W )C such that ∗δ(f ) = iδ(f ) or ∗δ(f ) = −iδ(f ) also gives rise to a solution to the equation of motion. This motivates us to introduce the following subgroups in G(W )C :

Definition 3.3 We define the chiral subgroup G(W )+ C and the anti-chiral subgroup G(W )− to be the following subgroups in G(W ) C: C ∓ G(W )± C = Ker δ = {f ∈ G(W )C | δ(f ) ∓ i ∗ δ(f ) = 0}.

In the case of k = 0 and W is a Riemann surface, G(W )+ C is isomorphic to the group of holomorphic functions f : W → C/Z, and G(W )− C the group of anti-holomorphic functions f : W → C/Z. A 2k-form α ∈ A2k (W, C) satisfying the “self-dual” condition i ∗ δ(f ) = δ(f ) is called a chiral 2k-form (see [15, 20], for example). By means of the homomorphism ι in Lemma 2.2 (b), a chiral 2k-form induces an element in + G(W )+ C . This is the reason that G(W )C is named the chiral subgroup.

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Kiyonori Gomi

3.3 Analogy of the Polyakov-Wiegmann formula The action functionals in the Wess-Zumino-Witten models obey the so-called Polyakov-Wiegmann formula. There is a similar formula for the functional EW . We define ΓW : G(W )C × G(W )C → C by Z δ − (f ) ∧ δ + (g). ΓW (f, g) = 4ℓπi W

Lemma 3.4 Suppose that ∂W = ∅. For f, g ∈ G(W )C we have: eEW (f ) · eEW (g) = (exp ΓW (f, g)) eEW (f +g) . Proof. A straight computation gives: EW (f + g) − EW (f ) − EW (g) + ΓW (f, g) = 2ℓπi

Z

W

δ(f ) ∧ δ(g).

Recall Rthat δ(f ), δ(g) ∈ A2k+1 (W, C)Z . Since ℓ is taken to be an integer, we ⊓ have ℓ W δ(f ) ∧ δ(g) ≡ 0 in C/Z under the present assumption on W . ⊔

The point R in the above proof is that W has no boundary: if W has a boundary, then W δ(f ) ∧ δ(g) is not necessarily an integer. To take into account contributions of the boundary, we introduce a complex line bundle. Definition 3.5 Let M be a compact oriented (4k + 1)-dimensional smooth manifold (without boundary). (a) We define the line bundle LM over G(M )C by LM = G(M )C × C. (b) We define the product structure LM × LM → LM by (f, z) · (g, w) = (f + g, zw exp 2ℓπiSM,C (f, g)),

where SM,C : G(M )C × G(M )C → C/Z is defined to be Z f ∪g SM,C (f, g) = M

by using the cup product and the integration for smooth Deligne cohomology. Lemma 3.6 Suppose that ∂W 6= ∅. For f ∈ G(W )C we define an element eEW (f ) ∈ L∂W to be eEW (f ) = (f |W , exp EW (f )). Then we have: eEW (f ) · eEW (g) = (exp ΓW (f, g)) eEW (f +g) . Proof. By means of Lemma 2.4 (b), we have Z Z (f ∪ g)|∂W = S∂W,C (f |∂W , g|∂W ) = ∂W

W

δ(f ∪ g) =

Z

W

δ(f ) ∧ δ(g).

Now this lemma follows from the formula in the proof of Lemma 3.4. ⊔ ⊓

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63

3.4 Central extension The product structure on LM induces a group multiplication on the complement LM \{0} of the image of the zero section. We denote the group by ˜ )C . Obviously, G(M ˜ )C is a central extension of G(M )C : G(M ˜ )C −→ G(M )C −→ 1. 1 −→ C∗ −→ G(M ˜ ) of G(M ): By restriction, we also obtain the central extension G(M ˜ ) −→ G(M ) −→ 1. 1 −→ U (1) −→ G(M ˜ ) = G(M ) × U (1) as We can express the group multiplication in G(M (f, u) · (g, v) = (f + g, uv exp 2ℓπiSM (f, g)), R where SM (f, g) ∈ R/Z is defied to be SM (f, g) = M f ∪ g by using again the cup product and the integration for smooth Deligne cohomology. Proposition 3.7 Let M be a compact oriented smooth (4k + 1)-dimensional ˜ ) is non-trivial as a central extension. manifold. If ℓ 6= 0, then G(M In [12], the proof in the case of ℓ = 1 is given. We can easily generalize the ˜ )C is also non-trivial, proof to the case of ℓ 6= 0. The central extension G(M ˜ since it is a complexification of G(M ). As an example, we consider the case of k = 0 and M = S 1 . In this ˜ 1) ∼ b (1)/Z2 , where case, G(S 1 ) ∼ = LU (1) as mentioned, and we have G(S = LU b LU (1)/Z2 is the universal central extension of LU (1), ([19]). 3.5 Toward the quantum theory

To approach the quantum theory, we appeal to a method by using path integrals formally. In the case of ∂W = ∅, we may describe the partition function of our theory as: Z ZW =

F ∈G(M)C

eEW (F ) DF,

where DF is a formal invariant measure on G(M )C . In the case of ∂W 6= ∅, the probability amplitude eEW (F ) is formulated as an element in L∂W . Hence the formal path integral gives the section ZW ∈ Γ (L∂W ) by Z ZW (f ) = eEW (F ) DF. F ∈G(W )C ,F |∂W =f

On a formal level, we can reconstruct the partition function for a (4k + 2)dimensional manifold W without boundary from the above sections by cutting W along a submanifold M of dimension 4k + 1. This suggests that the space

64

Kiyonori Gomi

of sections Γ (LM ) contains the quantum Hilbert space of our theory, as is in the Wess-Zumino-Witten models [9, 10]. By construction, the central extension G(M )C acts on the line bundle LM ˜ )C covering the action of G(M )C on itself. Though G(M )C is abelian, G(M ˜ )C is not. Hence the space of sections Γ (LM ) gives rise to a two sided G(M module. Compared to the Wess-Zumino-Witten models, one may expect that the quantum Hilbert space in Γ (LM ) is of the form “⊕λL ,λR HλL ⊗ HλR ”, ˜ )C , where HλL and HλR are certain left and right irreducible modules of G(M ˜ respectively. This motivates us to study representations of G(M )C , which is the subject of the next section.

4 Representations of smooth Deligne cohomology In this section, we deal with representations of smooth Deligne cohomology groups, culling out some results from [13]. After the statement of the classification, we explain a relationship between the representations and the quantum Hilbert space of a chiral 2-form due to Henningson [15]. We also consider an analogy of the space of conformal blocks in the Wess-Zumino-Witten model. 4.1 Representations of smooth Deligne cohomology First of all, we make a general remark: for a compact oriented (4k + 1)˜ ) is constructed by using dimensional manifold M , the central extension G(M 2ℓπiS(·,·) the group 2-cocycle e : G(M ) × G(M ) → U (1). Hence a representation ˜ ) such that the center U (1) acts as the scalar multiplication (˜ ρ, H) of G(M corresponds bijectively to a projective representation (ρ, H) of G(M ) with its ˜ )C . We cocycle e2ℓπiSM . There is a similar correspondence in the case of G(M use the correspondences freely in the following. For the smooth Deligne cohomology group G(M ), admissible representations of level ℓ are certain projective unitary representations on Hilbert spaces with their cocycle e2ℓπiSM . They are characterized by the representations of the subgroup A2k (M )/A2k (M )Z ⊂ G(M ) obtained by restriction. Generalizing straightly the proof of Theorem 1.1 given in [13], we can obtain the following classification of admissible representations: Theorem 4.1 Let M be a compact oriented (4k+1)-dimensional Riemannian manifold. For a positive integer ℓ, admissible representations of G(M ) of level ℓ have the following properties: (a) An admissible representation is equivalent to a finite direct sum of irreducible admissible representations. (b) The number of the equivalence classes of irreducible admissible representations is (2ℓ)b r, where b = b2k (M ) = b2k+1 (M ) is the Betti number, and r is the number of elements in the set {t ∈ H 2k+1 (M, Z)| 2ℓ · t = 0}.

Representations of gauge transformation groups of higher abelian gerbes

65

If H 2k+1 (M, Z) is torsion free, then the number of the equivalence classes of irreducible admissible representations of level ℓ is (2ℓ)b . The outline of constructing these irreducible representations is as follows: 1. We let G 0 (M ) be the subgroup A2k (M )/A2k (M )Z in G(M ). Using the Riemannian metric on M , we decompose the subgroup G 0 (M ) as follows:
G 0 (M ) = H2k (M )/H2k (M )Z × d∗ (A2k+1 (M )), where H2k (M ) is the group of harmonic 2k-forms on M , H2k (M )Z = H2k (M ) ∩ A2k (M )Z the subgroup of harmonic 2k-forms with integral periods, and d∗ : A2k+1 (M ) → A2k (M ) the formal adjoint of d given by d∗ = − ∗ d∗. Using the Riemannian metric again, we define an inner product ( , ) on d∗ (A2k+1 (M )) and a compatible complex structure J on the completion V of d∗ (A2k+1 (M )) such that: (ν, Jν ′ ) = ℓSM (ν, ν ′ ),

ν, ν ′ ∈ d∗ (A2k+1 (M )) ⊂ V.

2. We construct the projective representation (ρ, H) of d∗ (A2k+1 (M )). The representation is realized as the representation of the Heisenberg group associated to the symplectic form (·, J·) : V × V → R. The representation space H is a completion of the symmetric algebra S(W ), where W is the eigenspace in V ⊗ C of J with its eigenvalue i. 3. Let X (M ) denote the set of homomorphisms λ : H2k (M )/H2k (M )Z → R/Z. For λ ∈ X (M ), we construct the projective representation (ρλ , Hλ ) of G 0 (M ). The representation space is Hλ = H. The action of (η, ν) ∈ (H2k (M )/H2k (M )Z ) × d∗ (A2k+1 (M )) is ρλ (η, ν) = e2πiλ(η) ρ(ν). 4. We construct the projective representation (ρλ , Hλ ) as the representation induced from the representation (ρλ , Hλ ) of the subgroup G 0 (M ) ⊂ G(M ). The projective representations (ρλ , Hλ ) and (ρλ′ , Hλ′ ) are equivalent if and only if there is ξ ∈ H 2k+1 (M, Z) such that λ′ = λ + 2ℓs(ξ). Here the homomorphism s : H 2k+1 (M, Z) → X (M ) is defined by Z η ∧ ξR mod Z, s(ξ)(η) = M

where ξR is a de Rham representative of the real image of ξ. If H 2k+1 (M, Z) is torsion free, then s is an isomorphism by the Poincar´e duality. Thus, among the representations (ρλ , Hλ ), we have (2ℓ)b inequivalent representations. For example, we again consider the case of k = 0 and M = S 1 . Then we have the isomorphism G(S 1 ) ∼ = LU (1), and an admissible representation of G(S 1 ) of level ℓ gives rise to a positive energy representation of LU (1) of level 2ℓ, and vice verse. This is because the construction of irreducible admissible representations outlined above coincides with that of irreducible positive energy representations given in [19]. The number of the equivalence classes of irreducible positive energy representations of LU (1) of level 2ℓ is 2ℓ, which is consistent with Theorem 4.1.

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Kiyonori Gomi

4.2 Relationship to Henningson’s work The irreducible admissible representations in the case of k = 1 have certain relationship to the quantum Hilbert space of a chiral 2-form studied in a work of Henningson [15]. To explain the relationship, some notations in [15] will be used here without change. Let M be a compact oriented 5-dimensional Riemannian manifold such that H 2 (M, Z) and H 3 (M, Z) are torsion free. Then, by means of Theorem 4.1, the number of the equivalence classes of irreducible admissible representations of G(M ) of level ℓ = 1 is 2b . These irreducible representations are parameterized by the set Coker{2s : H 3 (M, Z) → X (M )} ∼ = (Z/2Z)b . In [15], Henningson studied the quantum Hilbert space V+ of a chiral 2form on R × M . The Hilbert space V+ can be expressed as a Hilbert space 0 0 tensor product: V+ = V ′ ⊗ V+ . The Hilbert space V+ admits a further decom0 0 position: V+ = ⊕a+ ∈H 3 (M,Z2 ) Va+ . Accordingly, we have the following decomposition into different version of the chiral theory: M V ′ ⊗ Va0+ . V+ = a+ ∈H 3 (M,Z2 )

Under the present assumption on M , the homomorphism s : H 3 (M, Z) → X (M ) induces the natural isomorphism: H 3 (M, Z2 ) ∼ = Coker{2s : H 3 (M, Z) → X (M )}. Hence we can naturally identify the parameterization space of the Hilbert spaces V ′ ⊗ Va0+ with that of the equivalence classes of irreducible admissible representations. If we take fixed lifting a+ ∈ H 3 (M, Z) of elements a+ ∈ H 3 (M, Z2 ), then (ρs(a+ ) , Hs(a+ ) ) represents the equivalence class of irreducible admissible representations corresponding to V ′ ⊗ Va0+ . In addition, we can find a natural isomorphism between V ′ ⊗ Va0+ and Hs(a+ ) . On the one hand, the construction in [15] implies that the Hilbert space V ′ is the completion of the symmetric algebra S(W ), so that V ′ = H. The Hilbert space Va0+ is spanned by |k+ , a+ i, (k+ ∈ H 3 (M, Z)). Thus, we have the following expression by using a Hilbert space direct sum: M d V ′ ⊗ Va0+ = H ⊗ C|k+ , a+ i. 3 k+ ∈H (M,Z)

On the other hand, it is shown in [13] that, for each λ ∈ X (M ), the representation Hλ |G 0 (M) of G 0 (M ) obtained by the restriction of Hλ is expressed as the following Hilbert space direct sum: M d Hλ |G 0 (M) = Hλ+2s(ξ) . 3 ξ∈H (M,Z)

Now the natural isomorphism V ′ ⊗ Va0+ → Hs(a+ ) follows from the obvious identification H ⊗ C|k+ , a+ i ∼ = Hs(a+ +2k+ ) .

Representations of gauge transformation groups of higher abelian gerbes

67

4.3 An analogy of the space of conformal blocks Projective representations of G(M )C , rather than G(M ), concern the field theory in Section 3. It is known [19] that positive energy representations of LU (1) extend to that of LC∗ in a certain way. The following proposition generalizes the fact. Proposition 4.2 Let M be a compact oriented (4k + 1)-dimensional Riemannian manifold, and (ρ, H) an admissible representation of G(M ) of level ℓ. Then there is an invariant dense subspace E ⊂ H, and (ρ, E) extends to a projective representation of G(M )C . We can prove the proposition above by a straight generalization of the proof in the case of ℓ = 1 described in [13]. By means of the representations of G(M )C , we consider below an analogy of the space of conformal blocks in higher dimensions. Before the consideration, we notice that Lemma 3.6 and Theorem 4.1 lead to: (i) For a compact oriented (4k+2)-dimensional Riemannian manifold W with boundary, the following map gives rise to a homomorphism: ˜ r+ : G(W )+ C → G(∂W )C ,

f 7→ (f |∂W , exp EW (f )).

(ii) For a compact oriented (4k + 1)-dimensional Riemannian manifold M , we can parameterize the equivalence classes of irreducible admissible representations of G(M ) of level ℓ by a finite set Λℓ (M ). Now, for W and λ ∈ Λℓ (M ), we define CB(W, λ) to be the vector space consisting of continuous linear maps ψ : Eλ → C such that ψ(˜ ρλ (r+ (f ))v) = + ψ(v) for all v ∈ Eλ and f ∈ G(M )C : +

CB(W, λ) = Hom(Eλ , C)G(W )C . As the simplest example, we let W = D 4k+2 be the standard (4k + 2)dimensional disk whose boundary is S 4k+1 . In a direct way, we can compute CB(W, λ) to obtain a finite dimensional vector space: in the case of k = 0, the parameterization set Λℓ (S 1 ) is identified with Z/ℓZ, and we obtain:  C, (λ = 0) CB(D2 , λ) ∼ = {0}. (λ 6= 0) In the case of k > 0, the parameterization set Λℓ (S 4k+1 ) = {0} consists of a single element, and we obtain: CB(D4k+2 , 0) ∼ = C. At present, the only example available is the above one. Computations of other examples as well as a proof that CB(W, λ) is a finite dimensional vector space still remain as problems.

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Acknowledgement. Thanks are due to organizers and audiences during International Workshop on Noncommutative Geometry and Physics 2005, November 1–4, 2005, at Tohoku University. The author’s research was supported by Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists.

References 1. L. Breen, On the classification of 2-gerbes and 2-stacks. Asterisque No. 225 (1994), 160 pp. 2. J-L. Brylinski, Loop spaces, Characteristic Classes and Geometric Quantization. Birkh¨ auser Boston, Inc., Boston, MA, 1993. 3. J. Cheeger and J. Simons, Differential characters and geometric invariants. Lecture Notes in Math. 1167(1985), Springer Verlag, 50-80. 4. A. L. Carey, M. K. Murray and B. L. Wang, Higher bundle gerbes and cohomology classes in gauge theories. J. Geom. Phys. 21 (1997), no. 2, 183–197. 5. P. Deligne and D. S. Freed, Classical field theory. Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), 137–225, Amer. Math. Soc., Providence, RI, 1999. 6. J. L. Dupont and F. W. Kamber, Gerbes, simplicial forms and invariants for families of foliated bundles. Comm. Math. Phys. 253 (2005), no. 2, 253–282. 7. H. Esnault and E. Viehweg, Deligne-Be˘ılinson cohomology. Be˘ılinson’s conjectures on special values of L-functions, 43–91, Perspect. Math., 4, Academic Press, Boston, MA, 1988. 8. P. Gajer, Geometry of Deligne cohomology. Invent. Math. 127 (1997), no. 1, 155–207. 9. K. Gaw¸edzki, Conformal field theory: a case study. Conformal field theory (Istanbul, 1998), 55 pp., Front. Phys., 102, Adv. Book Program, Perseus Publ., Cambridge, MA, 2000. 10. K. Gaw¸edzki, Topological actions in two-dimensional quantum field theories. Nonperturbative quantum field theory (Carg`ese, 1987), 101–141, NATO Adv. Sci. Inst. Ser. B Phys., 185, Plenum, New York, 1988. 11. J. Giraud, Cohomologie non-ab´elienne. Grundl. 179, Springer Verlag (1971). 12. K. Gomi, Central extensions of gauge transformation groups of higher abelian gerbes. J. Geom. Phys. to appear. hep-th/0504075. 13. K. Gomi, Projective unitary representations of smooth Deligne cohomology groups. math.RT/0510187. 14. K. Gomi and Y. Terashima, Higher-dimensional parallel transports. Math. Res. Lett. 8 (2001), no. 1-2, 25–33. 15. M. Henningson, The quantum Hilbert space of a chiral two-form in d = 5 + 1 dimensions. J. High Energy Phys. 2002, no. 3, No. 21, 15 pp. 16. N. Hitchin, Lectures on special Lagrangian submanifolds. Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), 151–182, AMS/IP Stud. Adv. Math., 23, Amer. Math. Soc., Providence, RI, 2001. 17. M. K. Murray, Bundle gerbes. J. London Math. Soc. (2) 54 (1996), no.2, 403416. 18. R. Picken, A cohomological description of abelian bundles and gerbes. Twenty years of Bialowieza: a mathematical anthology, 217–228, World Sci. Monogr. Ser. Math., 8, World Sci. Publ., Hackensack, NJ, 2005.

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19. A. Pressley and G. Segal, Loop groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1986. 20. E. Witten, Five-brane effective action in M -theory. J. Geom. Phys. 22 (1997), no. 2, 103–133.

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Algebroids associated with pre-Poisson structures Kentaro Mikami1 and Tadayoshi Mizutani2 1

2

Dep of Computer engineering, Akita University, Japan [email protected] Dep of Mathematics, Saitama University, Japan [email protected]

1 Introduction There are several ways to generalize Poisson structures. A Jacobi structure (or a local Lie algebra structure), in which we do not require the Leibniz identity for the bracket, and a Nambu-Poisson structure, where the brackets are not binary but n-ary operations satisfying a generalized Leibniz rule called fundamental identity, are well-known examples. Also, a Dirac structure is a natural generalization of a Poisson structure. As another direction of studying Poisson geometry, we would like to do some trial or attempt to generalize the concepts, ideas, or theories of Poisson geometry into some area where the Poisson condition is not fulfilled. In the first half of this note, we show briefly our trials in this context, namely in almost Poisson geometry. As we will see in short, a Poisson structure gives a Lie algebroid. It is natural to handle a Leibniz algebroid as generalization of a Lie algebroid. Thus, it is meaningful to study the fundamental properties of Leibniz algebra or super Leibniz algebra. In the second half of this note, after we recall some properties of Leibniz modules, we define super Leibniz algebras and super Leibniz modules keeping the exterior algebra bundle of the tangent bundle with Schouten bracket as a prototype of a super Lie algebra (and so a super Leibniz algebra). We will show that an abelian extension is controlled by the second super cohomology group. The notion of super Leibniz bundles is clear, but unfortunately we do not have the proper notion of anchor, so far. In near future, we hope we could find concrete examples of super Leibniz bundles tightly connected to the properties of Poisson geometry, and could understand what the anchor should be.

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2 Review of elements of Poisson geometry Definition 1. By a Poisson structure or a Poisson bracket on a manifold M , we mean a binary operation on the function space C ∞ (M ) of M , C ∞ (M ) × C ∞ (M ) ∋ (f, g) 7→ {f, g} ∈ C ∞ (M ) satisfying 1. R-bilinearity 2. skew-symmetry 3. Jacobi identity 4. Leibniz rule

{λf + µg, h} = λ{f, h} + µ{g, h} (λ, µ ∈ R) {g, f } = −{f, g} {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0 {f, gh} = {f, g}h + g{f, h}.

We call a manifold with a specified Poisson structure as a Poisson manifold. Definition 2. On a Poisson manifold, the Hamiltonian vector field Xf of f ∈ C ∞ (M ) is given by hXf , dgi := {f, g} Proposition 1. X{f,g} = [Xf , Xg ] holds for f, g ∈ C ∞ (M ). This comes from Jacobi identity of the Poisson bracket. Example 1. Every manifold has the trivial Poisson structure {f, g} := 0. Example 2. For a given symplectic structure ω, the Poisson bracket is defined by {f, g} := hXf , dgi where ω ♭ (Xf ) := −df or Xf := −ω ♯ (df ) .

It is well-known that the cotangent bundle T∗ (Q) of a manifold Q has a canonical symplectic structure. The Poisson bracket satisfies {τ ∗ f, τ ∗ g} = 0,

{X, τ ∗ g} = τ ∗ hX, dgi,

{X, Y } = [X, Y ]

where τ : T∗ (Q) → Q is the bundle projection, and f, g ∈ C ∞ (Q), X, Y ∈ Γ (T(M )) and considered as linear functions along the fibres of T∗ (M ), and [X, Y ] is the usual Lie bracket of X and Y . Example 3. Let g be a Lie algebra of finite dimension. Consider the dual space g∗ as the underlying manifold. Since an element of g is a linear function on g∗ , g is a subspace of C ∞ (g∗ ). forF, H ∈ C ∞ (g∗ ), andµ ∈ g∗ , the Poisson bracket is defined by δF δH , ], µi {F, H}(µ) := h[ δµ δµ d δF δF (ν) := F (µ + tν)|t=0 (ν ∈ g∗ ) and ∈ g∗ ∗ ∼ where = g. In fact, δµ X dt δµ cijk zi holds where (zi ) is a basis of g whose structure constants {zj , zk } = i

are (cijk ). This bracket is called Lie-Poisson bracket on the dual space of a Lie algebra.

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Definition 3. For a given Poisson bracket {·, ·}, there exists a unique π ∈ Γ (Λ2 T(M )) satisfying π(df, dg) = hπ, df ∧ dgi = {f, g} . π is called the Poisson 2-vector field (or the Poisson tensor field) of the Poisson bracket. Proposition 2. Conversely, for a given 2-vector field π, define a new bracket by {f, g} := π(df, dg) = hπ, df ∧ dgi . This bracket is Poisson, namely satisfies the Jacobi identity if and only if the Schouten bracket [π, π]S vanishes. We recall here the definition and the related properties of the Schouten bracket . Now on, we abbreviate Γ (Λ• T(M )) to Λ• T(M ), and so on. Definition 4. The Schouten bracket [·, ·]S is the homogeneous bi-derivadimM X tion on Λ• (T(M )) of degree −1 •=0

Λp T(M ) × Λq T(M ) ∋ (P, Q) 7→ [P, Q]S ∈ Λp+q−1 (T(M )) uniquely defined by the following five conditions. Property (6) is called super Jacobi identity. 1. [f, g]S = 0 f, g ∈ Λ0 (T(M )) = C ∞ (M ) 2. [X, f ]S = hX, df i = Xf X ∈ Λ1 (T(M )), f ∈ Λ0 (T(M )) 3. [X, Y ]S = [X, Y ]Lie bracket X, Y ∈ Λ1 (T(M )) 4. [P, Q]S = −(−1)(p−1)(q−1) [Q, P ]S 5. [P, Q ∧ R]S = [P, Q]S ∧ R + (−1)(p−1)q Q ∧ [P, R]S 6. (−1)(p−1)(r−1) [[P, Q]S , R]S + (−1)(q−1)(p−1) [[Q, R]S , P ]S + (−1)(r−1)(q−1) [[R, P ]S , Q]S = 0, where the small letter p means the ordinary degree of the capital letter P , i.e., P ∈ Λp (T(M )). Remark 1. The Schouten bracket on the decomposable elements is given by [X1 ∧ · · · Xp ,Y1 ∧ · · · Yq ]S =

q p X X i

j

ci · · · Y1 ∧ · · · c (−1)i+j [Xi , Yj ] ∧ X1 ∧ · · · X Yj · · · ∧ Yq

for Xi , Yj ∈ Γ (T(M )) (p, q ≥ 1). We often abbreviate [·, ·]S to [·, ·]. The sign convention of the Schouten bracket here is different from that in Vaisman’s book [9].

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Given a 2-vector field π on M , we have a bundle map from T∗ (M ) to T(M ) defined by β 7→ π(β, ·) = ιβ π = β π . This map is denoted by π ♯ , π ˜ , or often π itself if there is no danger of confusion. The Hamilton vector field Xf is then written as π ♯ (df ).

3 Algebroids related with Poisson structures 3.1 Lie algebroids We start this section by a famous result by B. Fuchssteiner [2]. Theorem 1 ([2]). Given a Poisson 2-vector field π on M , define a bracket by {α, β}π := Lπ♯ (α) β − Lπ♯ (β) α − d(π(α, β))

for each α, β ∈ Γ (T∗ (M )). Then {·, ·}π yields a Lie algebra structure on Γ (T∗ (M )) and the following equality holds: {α, f β}π = hπ ♯ α, df iβ + f {α, β}π . Replacing T(M ) by a general vector bundle we obtain the notion of Lie algebroids.

Definition 5. A vector bundle L over M is a Lie algebroid if and only if (a) Γ (L) is endowed with a Lie algebra bracket [·, ·] over R, i.e., [·, ·] is skewsymmetric R-bilinear and satisfies Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 (b) there exists a bundle map (called anchor ) a : L → T(M ) which induces a : Γ (L) → Γ (T(M )) is a(f x) = f a(x) (c) and satisfying [x, f y] = ha(x), df iy + f [x, y] where x, y, z ∈ Γ (L), f ∈ C ∞ (M ). Example 4. (1) T∗ (M ) with the bracket {·, ·}π defined from the Poisson 2vector field π is a Lie algebroid whose anchor is π ♯ . (2) T(M ) is a Lie algebroid with the identity map as the anchor. Assume that a 2-vector field π is not necessarily Poisson. Then we look at the space {α ∈ Γ (T∗ (M )) | α

[π, π]S = 0} =: ker[π, π]S

and ask the questions. Is ker[π, π]S closed with respect to {·, ·}π ? Does {·, ·}π satisfy Jacobi identity?

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We would like to say this trial is one of pre-Poisson attempts. The answers for the both are yes and we get a theorem. Theorem 2 (Mikami-Mizutani [7]). If rank ker[π, π]S is constant, ker[π, π]S is a Lie algebroid having bracket defined by {α, β}π := Lπ♯ (α) β − Lπ♯ (β) α − d(π(α, β)) and the anchor α 7→ π ♯ (α). 3.2 Dirac structures As stated already, a 2-vector field π is Poisson if and only if [π, π]S = 0. On the other hand, T. Courant and A. Weinstein studied Poisson condition from more geometrical point of view. They handle the bundle homomorphism π ♯ : T∗ (M ) → T(M ). They claim that “Poisson condition is equivalent to some property of the graph of π ♯ , (Dirac structure)”, and generalize their discussion from a graph to a relation. On T(M ) ⊕ T∗ (M ), Courant([1]) defined h(Y1 , β1 ), (Y2 , β2 )i+ := iY1 β2 + iY2 β1 (fibre wise)   1 [[(Y1 , β1 ), (Y2 , β2 )]] := [Y1 , Y2 ] , LY1 β2 − LY2 β1 − d (iY1 β2 − iY2 β1 ) 2 T(e1 , e2 , e3 ) := h[[e1 , e2 ]], e3 i+ where (Yj , βj ) = ej ∈ Γ (T(M ) ⊕ T∗ (M )) (j = 1, 2, 3). Remark 2. In general, T is not tensor field, and only skew-symmetric in the first two arguments. Definition 6. A sub-bundle L ⊂ T(M ) ⊕ T∗ (M ) is an almost Dirac structure if L is maximally isotropic with respect to the pairing h·, ·i+ , i.e., L is a sub-bundle of rank dimM , and the restriction of h·, ·i+ to L × L is identically zero. Proposition 3. If L is an almost Dirac structure,


T|L (e1 , e2 , e3 ) = S hβ1 , [Y2 , Y3 ]i + LY1 hβ2 , Y3 i 123

where ej = (Yj , βj ) ∈ Γ (L), (j = 1, 2, 3). Especially, T|L is tensorial, and skew-symmetric in 3 arguments. Example 5. Let π be an arbitrary 2-vector field on M . The graph of π ♯ , L = {(π ♯ (β), β) | β ∈ T∗ (M )} is an almost Dirac structure. T|L ((π ♯ (β1 ), β1 ), (π ♯ (β2 ), β2 ), (π ♯ (β3 ), β3 )) =

1 [π, π]S (β1 , β2 , β3 ) . 2

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Example 6. Let ω be a 2-form on M . The graph of ω ♭ , L = {(X, iX ω) | X ∈ T(M )} is almost Dirac. T|L ((X1 , iX1 ω), (X2 , iX2 ω), (X3 , iX3 ω)) = (dω)(X1 , X2 , X3 ) . The Courant bracket [[·, ·]] is skew-symmetric, but does not satisfy Jacobi identity. In fact, let (J1 , J2 ) be the components of Jacobiator, i.e., (J1 , J2 ) := S [[[[(Y1 , β1 ), (Y2 , β2 )]], (Y3 , β3 )]] . 123

Then, J1 = 0 holds, but J2 is complicated. Proposition 4. J2 is given explicitly, and the restriction of J2 to an almost Dirac structure L is given by J2 |L (· · · ) =


1 d T|L (· · · ) 2

Definition 7. An almost Dirac structure L is a Dirac structure if Γ (L) is closed by bracket [[·, ·]], i.e., it satisfies T|L ≡ 0. In the case of an almost Dirac structure defined by a 2-vector field, a Dirac structure gives a Poisson structure of the base manifold. In the case of a 2-form, a pre-symplectic structure. When L is almost Dirac, we consider the following “sub-bundle” ker(T|L ) : = {e ∈ L | T|L (e1 , e2 , e) = 0, e1 , e2 ∈ L} . Again we would like to say this is one of pre-Poisson trials. Theorem 3 (Mikami-Mizutani[8]). Let L be an almost Dirac structure and ker(T|L ) be of constant rank. Then ker(T|L ) is a Lie algebroid with the bracket [[·, ·]], and the anchor ρker(T|L ) , which is the restriction of the first projection ρ : T(M ) ⊕ T∗ (M ) → T(M ) to ker(T|L ). Application of Theorem 3 to Theorem 2 (Mikami-Mizutani[8]) We know that the graph of a general 2-vector field π is almost Dirac. Since 1 TL (·, (π ♯ (βj ), βj ), ·) = [π, π]S (·, βj , ·), we have ker TL := {(π ♯ (γ), γ) | γ ∈ 2 ker[π, π]S }. The bracket is computed as follows [[(π ♯ (α), α), (π ♯ (β), β)]] = ([π ♯ (α), π ♯ (β)], {α, β}π ) = (π ♯ ({α, β}π ), {α, β}π ) .

If we pick up the first components of the elements of ker TL , this computation gives another proof of Theorem 2 of Mikami-Mizutani.

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4 Leibniz algebra and Leibniz algebroid Definition 8. A vector bundle A over M is a Leibniz algebroid if (a) Γ (A) is endowed with a Leibniz algebra structure over R, i.e., there is a binary operation [·, ·] : Γ (A) × Γ (A) → Γ (A) which is non skew in general and R-bilinear, and satisfying [[x, y], z] = [x, [y, z]] − [y, [x, z]] (Leibniz identity) where x, y, z ∈ Γ (A). (b) there exists a bundle map (called anchor ) a : A → T(M ) (c) the following compatibility condition holds: [x, f y] =ha(x), df iy + f [x, y] where x, y, z ∈ Γ (A), ha(x), df i = La(x) f for f ∈ C ∞ (M ). Remark 3. (1) The condition [[x, y], z] = [x, [y, z]] − [y, [x, z]] is equivalent to [x, [y, z]] = [[x, y], z] + [y, [x, z]] and this means [x, ·] satisfies Leibniz identity or left-derivation rule. (2) If it is skew-symmetric, Jacobi identity is equivalent to Leibniz identity as we see [[x, y], z] = −[[y, z], x] − [[z, x], y] = [x, [y, z]] − [y, [x, z]] (using skew-symmetry) (3) The difference between Lie algebroids and Leibniz algebroids is just relaxing the skew-symmetric property. But, we can see some hidden “almost” skew-symmetric property for Leibniz algebroid. The reason is: [[x, y], z] = [x, [y, z]] − [y, [x, z]] (Leibniz rule) this shows RHS is skew-symmetric in x and y, thus 0 = [[x, y] + [y, x], z]

(x, y, z ∈ Γ (A)).

(4) It follows a([x, y]) = [a(x), a(y)] (x, y ∈ Γ (A)) by the following computations. Since [y, f z] = (La(y) f )z + f [y, z], we have
[x, [y, f z]] = [x, La(y) f z + f [y, z]] LHS =[[x, y], f z] + [y, [x, f z]]

=(La([x,y])f )z + f [[x, y], z] + [y, (La(x) f )z + f [x, z]] =(La[x,y] f )z + f [[x, y], z] + (La(y) La(x) f )z+ + (La(x) f )[y, z] + (La(y) f )[x, z] + f [y, [x, z]] RHS =(La(x) La(y) f )z + (La(y) f )[x, z] + (La(x) f )[y, z] + f [x, [y, z]]
Thus, we have La([x,y])f − La(x) La(y) f + La(y) La(x) f z = 0. This means we have the result.

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Example 7. Let M be a manifold. A := T(M ) ⊕ T∗ (M ). For a pair of sections of A, define a bracket by [(X, α), (Y, β)] := ([X, Y ], LX β − ιY dα) and the anchor a(X, α) := X. Then this is a Leibniz algebroid, but not a Lie algebroid. 4.1 Abelian extension of Leibniz algebra In this subsection, we shall study an algebraic property of a Leibniz algebra and related bi-module (a Leibniz bi-module), namely the second Leibniz cohomology group with coefficient in the Leibniz module. Definition 9 (Leibniz module). Let (g, [·, ·]) be a Leibniz algebra. A module A is called a g-bi module if A is a module where g acts from both, left and right, and it holds the following three conditions. (a · g) · h = a · [g, h] − g · (a · h) (g · a) · h = g · (a · h) − a · [g, h]

[g, h] · a = g · (h · a) − h · (g · a)

where g, h ∈ g and a ∈ A, and we denoted the left action of g on A by g · a, and the right action of g on A by a · g (g ∈ g, a ∈ A). When a Leibniz g-module A is given, we can construct cochain complex and cohomology groups which are called Leibniz cohomology groups (cf. J.-L. Lodays’s works, for example [4] or [5]). Definition 10. For each non-negative integer k, k-th cochain complex consists of k-multilinear maps from g × · · · × g to A and the coboundary operator | {z } k−times

is given by

(δψ)(g1 , . . . , gk+1 ) =

k X i=1

+

(−1)i−1 gi · ψ(. . . gbi . . .) + (−1)k+1 ψ(g1 , . . . , gk ) · gk+1

X i