1997 Summer School in High Energy Physics and Cosmology: ICTP, Trieste, Italy, 2 June-4 July 1997 [Volume 14] 9814528617, 9789814528610

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1997 SUMMER SCHOOL IN

HIGH ENERGY PHYSICS AND COSMOLOGY

This page is intentionally left blank

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

~

t lliill

UNITED NATIONS EDUCATIONAL, SCIENTIFIC

~ AND CULTURAL ORGANIZATION

INTERNATIONAL ATOMIC ENERGY AGENCY

I~ ,~ \

THE ICTP SERIES IN THEORETICAL PHYSICS - VOLUME 14

1997 SUMMER SCHOOL IN

HIGH ENERGY PHYSICS AND COSMOLOGY leTP, Trieste, Italy 2 June - 4 July 1997

Editors

E. GAVA INFN, Italy

A. MASIERO SISSA, Italy

K. S. NARAIN S. RANDJBAR-DAEMI G. SENJANOVIC A. SMIRNOV fCTP, Trieste, Italy

Q. SHAFI University of Delaware, USA

b World Scientific

' II

Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. POBox 128. Farrer Road. Singapore 912805 USA office: Suite lB. 1060 Main Street, River Edge. NJ 07661 UK office: 57 Shelton Street. Covent Garden. London WC2H 9HE

British Library Cataloguing-in-PubHcation Data A catalogue record for this book is available from the British Library.

1997 SUMMER SCHOOL High Energy Physics and Cosmology

Copyright @ 1998 by International Centre for Theoretical Physics

ISBN 9789814528610

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

v

Preface The Trieste Summer School in High Energy Physics and Cosmology provides an important forum whereby active physicists from the developing countries have an opportunity to interact with experienced and often leading researchers from across the globe in a stimulating environment. It also happens to be one of the most comprehensive schools covering almost all of the forefront topics. In this volume, recent developments in the nonperturbative aspects of string theory, duality in N = 1 string compactifications, orientifolds and Ftheory as well as the matrix model description of M-theory are reported. The D-brane approach to black holes and their entropy is also discussed. The phenomenological part is devoted to the main fields of present-day astroparticle physics: supersymmetry, neutrino physics and cosmology. The two most important aspects of supersymmetry are covered: Searches for superpartners and mechanisms of supersymmetry breaking. Results from recent neutrino experiments and their implications are presented. The central issues of modern cosmology such as inflation, formation of topological defects and dark matter are discussed. The smooth running of the school is largely due to the remarkable support and help provided by the ICTP staff. E. Gava A. Masiero K. S. Narain S. Randjbar-Daemi G. Senjanovic Q. Shaft A. Smimov

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vii

Contents Preface

v

Part I Dualities in Theories with 32 Supersymmetries: A Beginners' Guide S. Mukhi BPS Branes in Supergravity K. Stelle

1

29

Lectures on Orientifolds and Duality A. Dabholkar

128

Lectures on D-Branes, Gauge Theory and M(atrices) W. Taylor IV

192

Black Holes and D-Branes J. M aldacena

272

Aspects of N = 1 String Dynamics S. Kachru

299

Part II Recognizing Superpartners at LEP G. Kane

326

Supersymmetry Breaking G. Dvali

342

Neutrinos S. Pakvasa

371

WIMP and Axion Dark Matter M. Kamionkowski

394

Inflation and the Theory of Cosmological Perturbations R. H. Brandenberger

412

Formation of Topological Defects T. Vachaspati

446

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DUALITIES IN THEORIES WITH 32 SUPERSYMMETRIES: A BEGINNER'S GUIDE Sunil Mukhi Tata Institute of Fundamental Research, Homi Bhabha Rd, Mumbai 400 005, INDIA This is an introductory review of dualities in theories with 32 supersymmetries. I describe the maximally supersymmetric theories in 11 and 10 dimensional spacetime, their spectrum, symmetries and inter-relationships, and their toroidal compactifications. The emphasis is on presenting a few simple ideas explicitly and with clarity.

1

Introduction

The nature of classical supersymmetry has been under investigation for over two decades. The basic supersymmetry algebras in various dimensions were substantially classified, and the field contents of possible supersymmetric Lagrangians written down, quite some time ago a . Many of the new developments which will be reviewed in these notes are essentially quantum mechanical in nature. They are not, therefore, (as is sometimes suggested) re-statements of well-known old results from the days of classical supersymmetry. However, it is true that the understanding of classical supersymmetry, and more specifically the supersymmetry algebra - a precise classical statement that carries over directly into the quantum theory - are the essential scaffolding on which the structure of modern duality symmetries is erected. As we will see, in various situations, conjectured duality symmetries can be thought of as evidence that a quantum theory really exists, when no other firm evidence is available. Some of these dualities involve statements about strongly coupled field and string theories, hence they cannot be actually demonstrated by any known technique. Some people would even say that these symmetries provide definitions of quantum theories, in which case they should not be thought of as conjectures at all. What is certain is that the nonperturbative duality symmetries are extremely natural, and are supported by impressive evidence. The modern approach goes roughly like this: on the basis of some initial evidence, assume a duality symmetry. Find at least one nontrivial consequence of this assumption that is not already known to be true or false. Investigate it. If (as has usually QFor a comprehensive review, see Ref. I.

2

been the case) it turns out to be true, we have one more reason to believe the duality. In practice, the community working on the subject reaches a consensus on the validity of a duality symmetry, when the amount of evidence crosses a critical value. It has long been believed that classical supersymmetry, (together with other conventionally assumed symmetries), can only exist in spacetime dimensions less than or equal to 11. This requires an assumption that there is no tensor or spinor particle with spin greater than 2 (it has generally been believed that field theories with higher spins than this are inconsistent). In these notes I will start with this highest possible dimension and work downwards from there. In this article, I have cited only a few books, review articles and seminal papers that might help the beginning student. Those, in turn, will contain references to the original literature that the student will need to access to gain a deeper understanding and to start on research. 2

Supersymmetry

If we require spins ~ 2, then the maximal number of supersymmetries (in components) is 32. A rough argument for this goes as follows. On-shell, only half the components of a spinor are physical, so in this case we have 16 onshell supersymmetry generators. Taking complex combinations, we can make 8 raising and 8 lowering operators out of these. Each of these raises or lowers the spin projection of a state by ~. Starting with a state of spin projection -s for some s, and assuming it is annihilated by all the lowering operators, we can raise it until we get a maximum spin projection of -s + 4. A supermultiplet must of course contain all spin projections from -s to +s, so with this number of supersymmetries, s 2. With more supersymmetries, s will necessarily be larger. In 4 dimensions, 32 supersymmetries are organized as 8 Majorana supercharges with 4 (off-shell) components each, hence the corresponding theories are said to have N 8 supersymmetry in that case. In 10 dimensions one has 2 Majorana-Weyl supercharges of 16 components each, hence N 2 supersymmetry. The maximal spacetime dimension which admits 32 supersymmetries is d 11. A Majorana spinor has 32 off-shell components in this case, and the supersymmetry is called N 1. Above 11 dimensions, a supercharge will have more than 32 components and hence, as explained above, we are forced to have undesirably large spins. Indeed, in 11 dimensions there is a unique supersymmetric Lagrangian of conventional type upto two-derivative order. It

=

=

=

=

=

3

=

is generally known as D 11 supergravity. A quantum theory ("M-theory") is now believed to exist, for which this supergravity is the effective low-energy Lagrangian for the massless particles. Below, we will encounter considerable evidence for the existence of M-theory. In 10 dimensions there are two distinct supersymmetry algebras with 32 supersymmetries: the type IIA and IIB theories. Each is associated to a definite content of massless fields and a Lagrangian that is unique upto twoderivative order (there is a subtlety for the type IIB case, where a manifestly Lorentz-invariant Lagrangian cannot be written down, as we will see later). The type IIA field content is vectorlike (parity conserving), while the IIB field content is chiral (parity violating) 2. One of the key features of all these theories is that their excitation spectrum contains a variety of different stable objects that occur as soliton solutions, some of which are pointlike ("particles") and others are extended along p space directions ("p-branes"). Conceptually this is no different from the situation with some physically relevant gauge theories in 3 + 1 dimensions, in which the spectrum contains cosmic strings ("l-branes) and domain walls ("2-branes" or "membranes") in addition to the usual fundamental particles. But because the theories of interest here are higher dimensional, and supersymmetric, they admit p-branes for higher values of p, and the p-branes are endowed with many interesting and calculable properties. (For more details of solitonic branes, see the lectures of K. Stelle at this school, and references therein.) We will say all that we can about these three theories: M, IIA, IIB, before going to lower dimensions. Here is a partial summary, in advance, of the interesting facts that we will encounter along the way: (i) M-theory is a quantum theory with massless gravitons and other massless particles among its excitations. It also contains stable 2-branes and 5-branes in its spectrum, but no other stable branes. It is not a string theory, but (as far as we understand it) it is a consistent theory, incorporating quantum gravity, in 11 dimensions. (ii) IIA and IIB theories can both be obtained as limits of M-theory. These are also consistent theories of quantum gravity, in 10 dimensions. In their spectrum they contain, besides massless particles, various kinds of stable p-branes. These include 1-branes or strings. Indeed, the IIA and IIB theories can be phrased as string theories, with a consistent perturbation series, in which the particles are just low-lying excitations of the strings. Thus these theories are better understood than M-theory, as their consistency is established at least in the perturbative regime and loop corrections can be calculated. Nonperturbatively they are best understood as limits of M-theory and this understanding is consequently as limited as that of M-theory.

4

3 3.1

Theories In Their Maximal Dimensions M-theory

The unique supersymmetric classical field theory with fields of spin dimensions, contains the following massless fields: gMN: CMNP: .1.

o/Ma:

~

2 in 11

metric 3 - form potential • 3 MaJorana . ~. spm-'2 lermlOn

(1)

Instead of trying to prove that this is the right field content, we will argue that this is plausibly a supersymmetry multiplet. Let us count on-shell degrees of freedom. In light-cone gauge, where only on-shell degrees of freedom survive, each tensor index can be assigned a set of 9 (transverse) values, while each Majorana spinor index takes 16 values on-shell. Taking into account various symmetries for the bosonic fields, and remembering to project out a spin-~ field from the gravitino, we find:

9 x 10 -1 = 44 2 9x8x7 =84 6 9 x 16 - 1 x 16 = 128

gMN: CMNP: 1/JMa :

(2)

Thus, the total on-shell degrees of freedom match between bosons and fermions. To write the Lagrangian, let e = y'-detg (this and all other conventions are as in Ref. 2 ). Then the classical Lagrangian of 11-dimensional supergravity can be written

(3) where the bosonic part is 1

- 2~2eR -

CB

V2~

- 3456 f

1

MNPQ

48eGMNPQG

MNPQRSTUVWXC G G MNP QRST uvwx

(4)

and the fermionic part is given by

CF

=

1 -

-'2e1/JM

r MNP DN1/JP

_ V2~ (.7. rMNPQRS.I.o/S + 12·7.NrPQ.I.R) G 192 o/M 0/ 0/ NPQR

+4

- fermi terms

(5)

5

=

Here, r M1 ... Mk ;h(rM1 ..• rMk ±(k!-l) terms). The 4th rank antisymmetric tensor GMNPQ is defined as ~ [OMCNPQ + 3 terms]. The above action is invariant under the supersymmetry transformations K. A "2ijr tPM V2

-Tijr[MNtPP]

=

1 -(DM1J)a K.

+ ~ [(r~9RS - 8oLr QRS )1JL GPQRS + 3 - fermi terms

(6)

A useful tip about dealing with complicated supergravity Lagrangians (this is one of the simplest!) is that one usually needs only: (i) the bosonic part CB of the Lagrangian, (ii) the fermionic variation OtPMa in the supersymmetry transformation. So far we have only written down a classical Lagrangian. It is not clear that there is a corresponding quantum theory, since by the usual criteria we are dealing with a highly non-renormalizable Lagrangian. Evidence for an underlying quantum theory will slowly energe. Continuing with the classical theory, we look for stable solitonic solutions of the equations of motion. These are expected to be important if the theory can be quantized, as they would then correspond to nonperturbative, stable quantum states. Generically, solitons in quantum theory are stable only if they carry a quantized charge. The lightest soliton carrying a single unit of this charge cannot decay, by charge conservation. Point particles naturally carry electric charge with respect to an electromagnetic field AI" One manifestation of this is an interaction e f AI' dxl' on the world-line of the particle. A magnetic monopole in 4 dimensions will instead couple to the "dual photon" AI' via f Al'dxl' where A is defined by the duality transform A. AA _ 1 !3[A Apj (7) VLI'

vj -

2{I'VAP V

=

Note that the above transform interchanges the electric field Ei FOi with the magnetic field Bi ~{ijkFjlr in 4 spacetime dimensions. This concept generalizes to extended solitons and higher-rank antisymmetric tensor fields in arbitrary dimensions. The only antisymmetric tensor field in M-theory is the 3-form CMNP. Hence we wish to look for a stable electrically charged object with respect to this 3-form, having an interaction

=

6

J CMNP dx M /\ dx N /\ dx P on its world-volume.

Such an object must have a 3 .spacetime-dimensional world-volume, so it is a 2-brane or membrane. A stable magnetically charged object, on the other hand,will have a worldvolume interaction CMNPQRS dx M /\ ... /\ dx s , where

J

-

qTCMNPQRS]

=

1

7!f™NPQRSABCD {)

[A

C

BCD]

(8)

It follows that this object has a 6 spacetime-dimensional world-volume, so it is

a 5-brane. Thus the potentially stable objects in M-theory are singly charged electric 2-branes and magnetic 5-branes. It is pleasing that the 3-form field of llD supergravity acquires a useful role in this way. Supergravity theories must of course contain gravitons and gravitinos, by definition, but now it emerges that other fields are there to provide charges to stabilize branes. The postulated branes actually do exist as classical solutions of the lowenergy supergravity. We have a family of solutions, one for every integer k which labels the elctric/magnetic charge carried. From our discussion above, the truly stable ones have k = 1. The solutions are given by: 2-brane:

(1 + :6) -2/3 + (1 + ~)

ds 2

dxfAdxvTJfAV

+1/3 dymdynomn

ffAv>.

=

=

k ( 1 + r6

)-1

(9)

=

where J.L, v 0,1,2; m, n 3, ... ,10; r Jymym, and the components of C not appearing above are zero along with the fermion fields. 5-brane:

(1 + ~ ) + (1 + :a) 2/3

-1/3 dxfAdxvTJfA v

ds2

G mnpq

=

dymdynomn

(10)

=

=

=

where this time J.L, v 0, 1, ... ,5; m, n 6, ... ,10; r Jymym and again the components not appearing above, along with the fermions, are set to zero.

7

One is tempted to assume that there is a quantum theory with massless point particles corresponding to the fields in the classical Lagrangian, and also quantum states corresponding to stable 2-branes and 5-branes. This is the conjecture that M-theory exists. We will gradually uncover evidence for this conjecture. One can show that the 2-brane and 5-brane solutions preserve half the supersymmetry of the underlying theory. The remaining half of the supersymme tries is broken in the presence of branes. Explicitly, of the 32 independent supersymmetry variations 8'IjJJ-ta (for 32 independent spinors 7]), 16 are zero when the brane solution for the metric and GMNPQ given by Eq. 9 or Eq. 10 is inserted into the right hand side of Eq. 6. This is closely related to the fact that the most general supersymmetry algebra in lId is: {Qa, Q,6}

=

(rJ-t)a,6PM

+ (rMN)a,6ZMt

+(rMNPQR)a,6Zt:(PQR

(11)

where Z(2) and Z(5) are "central charges" which are non-vanishing precisely on 2 and 5-branes respectively. The branes satisfy an important relation between their mass density (measured by Po) and their charge density (measured by Z(2)' Z(5))' Schematically, half the supercharges satisfy (12) while the other half satisfy (13)

=

=

So, if the branes satisfy P - Z 0 or P + Z 0 then they can preserve half the supersymmetries, otherwise they break all. In the former case we have IFI = IZI on the branes while in the latter we evidently have the strict bound IFI > IZI· This bound is called the Bogomolny-Prasad-Sommerfeld or BPS bound. The stable 2-brane and 5-brane solutions written down in Eqs. 9,10 satisfy IPI = IZI, hence they saturate the bound and are said to be "BPS-saturated". Note that the total number of independent charges (including momenta) occurring on the RHS is 11 (momenta) + 55 (ZMt) + 462 (Ztt(PQR) = 528. This concludes our preliminary analysis of M-theory, but we will return to it soon. A nice review of M-theory can be found in Refs. 3, and there are surely others.

8

3.2

Type IIA Theory

The type IIA and lIB theories can be discovered in a rather analogous way. Just by studying the realizations of 32 supersymmetries in 10 dimensions, one finds that there are two distinct multiplets of massless fields with spins ~ 2. One of these, leading to type IIA theory, is as follows: (14) .1.(1) ../,(2)

\ (1) \ (2)

(F

.)

(15) '''01' erml (the tilde on the I-form and 3-form fields will be explained in the next subsection). The Fermi fields come in pairs of opposite chirality: a is the spinor index for 30(9,1) and a' is the conjugate spinor. These fields are related to those of M-theory in a very interesting way. Suppose M-theory is compactified on a circle (somewhat inconsistently with our earlier notation, we call this dimension 11). The resulting spectrum from the 10d point of view consists of some massless fields (coming from 11d fields independent of the 11 direction) and massive fields from Fourier modes excited in the 11 direction. Consider first just the massless modes: 'P P.OI , 'P P.OI" "01

d= 11 gMN CMNP

d= 10

-+ -+

gp.v; Cp.vP;

= Ap.; g11,11 = ljJ Cp.v,11 = Bp.v

9 p.,11

(16)

Thus the Bose fields oftype IIA theory, listed in Eq. 14, are exactly reproduced. It is easily checked that the same is true for the Fermi fields of IIA theory: they arise by dimensional reduction of the the gravitino of M-theory. What about the Lagrangian? It is clear that taking the M-theory Lagrangian to be independent of xl! will lead to a Lagrangian in 10 spacetime dimensions which necessarily has the right supersymmetries. The single Majorana supercharge in 11 dimensions splits into a pair of Majorana-Weyl spinors in 10 dimensions, one of each chirality. This happens because under dimensional reduction, local massless fields split in a non-chiral way and so we always get vectorlike theories. So dimensional reduction gives a valid way of actually deriving the type IIA Lagrangian. It is illuminating to carry through this derivation. From now on, we will ignore numerical coefficients in front of individual terms in supergravity Lagrangians. Thus we write the bosonic 11d Lagrangian as:

a~1) = ~ (eR+ elGI2 + C 1\ G 1\ G) K,

(17)

9

(we have performed a redefinition C -+ ~). To compactify on a circle, we replace gMN by a matrix like

(18) where Q is an arbitrary constant to be chosen later. This form exhibits the Kaluza-Klein scalar tjJ and gauge field ..41£. However, it is convenient to modify it since otherwise ~ will depend on ..41£' leading to ugly formulae. A better ansatz is

gMN""

[

g/lV

+ ea 4>-AI£Av

4>-]

ea AI£

ea 4> Av

(19)

ea 4>

so that ~ = eH~. If we treat x 11 as an angle-valued coordinate (taking values from 0 to 211'), the radius of the compactification circle is R11 = e~4>. Note that tjJ(x) is a scalar field, so we really mean R11 e~("'("')) e~'" where tjJ is the constant part, or VEV, of tjJ(x). Thus we have

=

9

=

MNR(ll) MN

g/lV

[Rh~O) + Ol£tjJOVtjJ + e- a "'le a 4> FI2 + ...

gl£V

[Rh~O) + 0l£tjJOVtjJ + ea 4> IFI2 + ...]

j (20)

So

(21) Similarly, e¥

J g(10) 10(10) 12 + e ~4> J g(10)e- a 4>IH(10) 12

.Jif1Of [e H IG(10)1 2 + e- H IH(10)1 2]

(22)

Collecting the above results, dropping the (10) superscript on the fields, and temporarily ignoring the C A. G A. G term, the 10d bosonic Lagrangian is:

10

(23) where F = dA, H = dB and G = de are 2-form, 3-form and 4-form field strengths respectively. It is useful to make a Weyl rescaling: g (lO) J.W

-+ ef3 g(lO)

(24)

J.lV

where the constant (3 will be determined by our convenience. The various terms transform under this rescaling as follows, modulo higher-derivative terms which we are ignoring:

.gar

-+

Id¢12 -+ IHI2 -+

e5f3 .gar, e- f3 Id¢12,

RJ.lv

-+ RJ.lv,

R

-+ e- f3 R

IFI2 -+ e- 2f3 IFI 2 IGI2 -+ e- 4f3 IGI2

e- 3f3 IHI 2,

(25)

Thus we arrive at the Weyl-rescaled Lagrangian

.c~O)

=

.gar [e(4 f3 +ij) R +e(4f3+ij)ld¢12 + e(3 f3 +!a)IFI 2 +e(2 f3 -ij)IHI 2 + e(f3+ij)IGI 2]

(26)

Now notice that by taking {3 + ~ = 0, we can make the e terms disappear in front of both IFI2 and IGI2. Also, the factors in front of the other three terms: R, Id¢12, IHI2 become equal. Thus with this choice, and with the conventional choice a = ~, we finally get

;2'

The constant part of e- 2 is like where oX is the coupling constant for the metric, ¢ and B fields. The other two fields appear in a "nonstandard" way, with no coupling factors in front (this partly explains why we placed tildes over them). We have oX e, while R(11) e H so:

=

=

R (11) --

\2/3

A

•

(28)

11

All this has a physical interpretation. In M-theory there is no scalar field, hence no parameter like (¢) which can play the role of an adjustable, dimensionless coupling. M-theory only has a dimensionful constant K, the ll-dimensional Planck scale (which we have set equal to 1 in the above discussion). Thus there is nothing analogous to a weak-coupling limit in M-theory. In type IIA theory, obtained by compactification on a circle, there is such a scalar - the dilaton - and its VEV is related to the radius ofthe 11th dimension. Thus type IIA should admit a weak coupling expansion. Before investigating this expansion, let us examine the spectrum of stable branes in the type IIA theory. The gauge fields and the associated charged branes are: AI'

0 - brane (electric) 1 - brane (electric) 2 - brane (electric)

(29)

Let the 7-form AI'IIAPUTa be the dual of the I-form AI' (in 10 spacetime dimensions) via dA =* dA. A 6-brane couples electrically to this, which is equivalent to saying that is carries magnetic charge under AI" Similarly for the 2-form BI'll and 3-form OI'IIA above, there are dual 6-form and 5-form potentials B and C respectively. Thus we can also have the following magnetically charge branes: AI'

6 - brane (magnetic) 5 - brane (magnetic) 4 - brane (magnetic)

(30)

To summarize, these simple arguments suggest that type IIA theory should have stable p-branes for p = 0,1,2,4,5,6. All the corresponding soliton solutions have been determined, so these branes really do exist. Since IIA theory arose by compactifying M-theory, all its branes should have explanations in M-theory. Indeed we can find them as follows. When Mtheory is compactified on a circle, the resulting theory will have distinct quantum states in its Hilbert space corresponding to 2-branes that are independent ofthe circle and 2-branes that wrap the circle. Similarly, it has quantum states corresponding to 5-branes that are independent of, or wrapped on, the circle. In the limit of a small circle, when the correct description is as type IIA theory, these four kinds of states behave respectively as 2-branes, I-branes, 5-branes and 4-branes.

12

We have used up all the basic branes in M-theory, but have not yet found an M-theoretic explanation for the O-brane and 6-brane of type IIA theory. To understand these branes, note that the gauge field AJS , whose electric charge is carried by the O-brane, is a Kaluza-Klein gauge field arising from the lId metric. So the missing O-brane should also arise in some way from the KaluzaKlein mechanism. Indeed, this is the case. On the circle xll, we can make a mode expansion ofthe lId massless fields:

g2-~(xl ... x lO ; xlI) =

L

ei~ng2-~(xl, ... , x lO )

(31)

nEZ

The fields on the RHS are of course reducible when viewed as 10d tensors. Now observe that a translation xll -+ xll + fR of the eleventh direction sends (lO)(N) ----' ijfER (lO)(n) gMN ---, e gMN

(32)

Hence the local version of this transformation, xll -+ xll + f(xl, .. " x lO ) is just a local U(l) gauge transformation for which AJS is the gauge field. Thus the fields g2-~(n) have charge n under Aw Moreover, except for n = 0, these fields are all massive: since the elevendimensional metric components satisfy a massless wave equation

(33) it follows that the 10-dimensional fields satisfy 0(10) (lO)(n) gMN -

(~)2 R

(lO)(n) gMN -

0

(34)

Thus the Kaluza-Klein fields have equal mass and charge in some units. (Actually in string units, (mass) = t(charge).) In modern language, the KaluzaKlein mechanism automatically gives rise to BPS saturated states! Of course, it is the supersymmetry algebra which guarantees that configurations which are classically BPS saturated actually go over into BPS saturated states in the full quantum theory. The field of lowest nonzero charge will correspond to a stable particle for reasons that we have already cited. So we can identify the multiplet of particles corresponding to g2-~(l), C,U~~) with the multiplet of states of a stable unitcharge O-brane. Similarly, the "Kaluza-Klein monopole", magnetically charged under All' is the 6-brane of IIA theory. So far, we have not needed any detailed information about either M-theory or IIA theory, especially about how to quantize them. Formulae arising from

13

supersymmetry, such as the BPS formula, give us a lot of information once we merely assume that some quantum theory exists. At this stage, however, we can argue that a quantum type-IIA theory really does exist. Among the branes of the IIA theory is a l-brane, or string. One can work out the world-sheet action for this string, and then quantize the resulting string theory. It turns out that the string theory so obtained is perturbatively welldefined, consistent (even finite) and unitary. The perturbation series is given, in powers of ,X2 = e2 ¢, by an expansion in Riemann surfaces. The low-energy spectrum of the string reproduces the IIA supergravity fields, and string interactions reproduce the type of Lagrangian we have written down, with calculable corrections. This shows that "IIA theory" (the theory of massless particles and various branes) does correspond to a well-defined quantum theory, at weak coupling. It is perfectly reasonable to assume that the theory exists also at strong coupling, in which case it follows that the strongly coupled theory is 11-dimensional and can be thought of as a definition of M-theory.

3.3

Neveu-Schwarz and Ramond sectors

At this point we digress a little from the main theme of these notes. Our point of view has been that classical supersymmetry completely dictates the structure of Lagrangians for massless fields in 11 and 10 dimensions, with 32 supersymmetries. Since these Lagrangians admit stable brane solutions, one must expect that upon quantization they lead not just to quantum field theories of point particles, but rather to quantum theories of particles and branes together. In the la-dimensional case, we argued that one such theory (type IIA) can be formulated as a quantum string theory (and we will soon see that this is true also of the other lad theory, type lIB). This fact gives us a whole new perspective on these theories. To do full justice to this, we would need an entire course on string theory, but we will instead make a few brief remarks in this direction and establish some relevant facts for later use. Details can be found in Ref. 2. Quantization of the l-brane, or string, of type IIA theory is somewhat complicated (this is true for any relativistic string theory) by the presence of local symmetries and constraints on the world sheet. At the end of this story, which involves gauge-fixing and related issues, it emerges that the type IIA string has massless states in 10 dimensions which arise as follows. If 0' labels points on the string and T is the time parameter on the world-sheet, then the modes of this string, which we take to be closed, factorize into "left-movers"

14

depending only on (0' - r) and "right-movers" depending only on (0' + r). The total Hilbert space is obtained by tensoring these two factorized spaces subject to some constraints. Massless particles in spacetime arise by keeping the lowest modes in the two sectors. For the type IIA string, the Lorentz quantum numbers of these modes (on-shell) are: left movers: right movers:

=

li}L,la}L li}R,I,B'}R

(35)

where i, i 1, .... ,8 represent the transverse components of an 80(9, 1) vector, a 1, ... ,8 represents the on-shell components of an 80(9,1) spinor, and (3' 1, ... ,8 represents a conjugate spinor of 80(9, 1). The spinor and conjugatespinor states are, as one would expect, spacetime-fermionic. For historical reasons, the vector states are said to be in the "Neveu-Schwarz" (NS) sector, while the spinor states are in the "Ramond" (R) sector. Some 80(9,1) group theory then enables us to build up the massless particle states and hence work out the Lorentz covariant massless fields that must appear in the low-energy Lagrangian. Bosonic fields arise by combining li}L ® li}R which decomposes into a symmetric traceless, an antisymmetric and a trace part. Th corresponding fields are a metric gllv, a 2-form B llv and a scalar (dilaton) ¢. Thus we have already recovered some of the massless fields of type IIA supergravity that were listed in Eq. 14. More bosonic fields arise from la}L ® 1(3'}R, and it is quite easy to work out that these correspond to the remaining fields All and GIlV )..' So we have re-discovered what we already knew, but with an extra insight. The fields g, B, ¢ arise by tensoring world-sheet modes that were bosonic separately in the left-movinj; ~nd right-moving sectors, hence they are NS-NS fields. The other fields A, C arise by combining spinor modes from the leftand right-moving sectors, so they are called R-R fields (this is what the tildes were supposed to denote). This distinction is apparent only from the point of view of string theory, and it turns out to be a very important one. The crucial difference between the two types of fields, which is a consequence of their distinct origins, is that perturbative states in the theory carry charge under the former and not under the latter. Indeed, it can be shown that couplings of perturbative states to the R-R gauge fields are through the field strength tensors rather than the gauge fields themselves, somewhat like magnetic-moment couplings in QED. As one example, the string itself is charged under B. Therefore, if we compactify on a circle with coordinate xi, the winding modes of the string on that circle will be charged under the I-form gauge field A~). == Bili. Also, momentum modes of the string along that circle are charged under another

=

=

15

I-form gauge field A/~) == g/-l;. But no perturbative modes of the string are charged under the R-R gauge fields A/-I or C/-IVA or their components after compactification. We will see, however, that soli tonic branes exist that do carry R-R charge. This will prove important in what follows. Before concluding this section, we note that the type lIB string differs in an apparently very small detail from the IIA string. Instead of the modes given in Eq. 35, it turns out that type lIB string gives rise to the modes left movers: right movers :

li}L,la}L Ij}R,IJ3)R

(36)

where the only difference is a missing prime on the spin or state I,B}R. Thus, there are only spinors andno conjugate spinors of 50(9,1). The NS-NS sector is identical to that of the type I1A theory, but the R-R sector is now quite different. It arises by decomposing la}L ® I,B)R, which gives rise to the bosonic fields ;jJ, B/-Iv, jj~VAP' namely a new scalar, a new 2-form and a 4-form potential whose 5-form field strength is self-dual in 10 dimensions. Thus the massless bosonic spectrum of type lIB supergravity, analogous to that listed in Eq. 14, IS

(37) Evidently the lIB theory has a pair of scalars and a pair of 2-forms, each pair having one NS-NS and one R-R member.

3.4

Type JIB Theory

As we have seen above, type lIB theory has two 2-form gauge potentials B/-Iv and B/-Iv, and a self-dual 4-form potential DtvAp. Following our previous arguments, we expect to find the following branes: B/-Iv

1 - brane (electric)

B/-Iv

5 - brane (magnetic)

B/-Iv

1 - brane (electric)

B/-Iv

5 - brane (magnetic)

-+

D/-IVAP

3 - brane (self - dual)

(38)

Thus we should have two distinct (electric) I-branes, or strings, and two distinct (magnetic) 5-branes. Also, because the 4-form is self-dual, the 3-brane

16

can equally well be described as electric or magnetic, so we just call it "selfdual" b As in the type IIA theory, here too the expected branes can be found as soliton solutions of the classical equations of motion. However, in this case there are some remarkable surprises. Let us write down the bosonic part of the type lIB supergravity action, upto two-derivative order:

.c~:s!) =

V9[e- 2if>(R+ld4>1 2+IHI 2)

+ld¢12 + IH - ¢H12 + 1112] +.0+ I\H I\H where H = dB, H-

(39)

= dB,- 1- = dD- +.

Actually, self-duality implies that 1[1 2 = [ 1\* [ = [ 1\ [ = 0 since [ is a 5-form. So it is not really correct to write the action as above when [ i= O. However, we adopt the procedure of relaxing the self-duality condition in the action, and then imposing it after obtaining equations of motion. This will be adequate for us, although a number of important subtleties are associated to this problem, because of which a covariant action for the fields of type lIB supergravity does not exist. The above action has a global 8L(2, IR) symmetry under which the two 2-form fields transform as a doublet. To exhibit this, it is convenient to make a Weyl transformation. Let 9/Jv -+ e~if>9/Jv. This implies that

-+ eHVu, -+ e-Hld4>1 2, -+ e-~if>1[12

R -+ e- H

IHI2 -+ e- t if>IHI 2 (40)

In this frame (the "Einstein frame") the Einstein-Hilbert action has no dilaton dependence. The action becomes

.c~:s!),E.F. =

v9 [R + (ld4>1 2+ e2if>ld¢12) +e-if>IHI 2+ 1[1 2+ eif>IH - ¢HI2] + D+ 1\ H 1\ H (41)

bWe will be hiding some "high branes": in addition to the above, type lIB theory has a 7-brane and a 9-brane, while type lIA has an 8-brane. Their roles are a bit more subtle, and we will not need them here.

17

Now defining the complex scalar field

T

= ~ + ie-4>, we can write

Id¢>12 + e24>ld~12

It is easy to check that under

T

--+

~;t~

=

IdTI2 (ImT)2

with

(~

(42)

!) E SL(2, ffi), this term

is invariant. Also, if H, iI transform as

(H, -9) --+ (H, -iI)

(~

!)

(43)

under SL(2, ffi), then e4>liI - ~H12 + e-4>IHI 2 is invariant. Thus, the low-energy action of type IIB theory is invariant under a global SL(2, ffi) symmetry, which in particular rotates the two 2-form gauge potentials into each other. One may ask whether this symmetry extends to the full type IIB theory, which includes branes in addition to the massless fields. We have already predicted the existence of I-branes, or strings, carrying unit charge under the potentials B/-w and B/.IV' Under the SL(2, ffi) transformation described above, we would generate a continuous infinity of I-branes carrying various (arbitrary real) charges under B, B. This conflicts with Dirac quantization for branes, unless the SL(2, ffi) transformation matrix has all integer entries. This defines the infinite discrete subgroup SL(2, Z). It follows that the largest subgroup of SL(2, ffi) that can be a symmetry of the full type IIB theory is SL(2, Z). Impressive evidence exists that this subgroup really does correspond to an exact symmetry. This symmetry is often called "S-duality" . One of the simplest pieces of evidence comes from re-examining soliton solutions. Let us label the string carrying unit charge under B/.IV as a (1,0) string, and the one carrying unit charge under the R-R field BI-'v as a (0,1) string. The latter can be obtained from the former by the particular SL(2, Z) transformation:

(44) Moreover, a general SL(2, Z) transformation by the matrix

(~

!) maps

the (1,0) string to a new object, an (a, c) string carrying charge a under B and simultaneously charge e under B. Note that since the matrix has integer entries with ad - be = 1, it follows that a, e are coprime. Thus if this is to be a symmetry then type IIB theory must support strings with arbitrary integer

18

charges (p, q) that are relatively prime. Indeed, it turns out that such soliton solutions do exist. The action of the S-duality group SL(2, Z) on the 5-brane coupling magnetically to BJ.l/.l also generates a family of 5-branes in the obvious way. For the 3-brane, however, the situation is different. Since S-duality leaves the self-dual 4-form potential invariant, it does not generate new 3-branes, and there is only a single species of 3-brane in type lIB theory. The (1,0) string, which is electrically charged under BJ.l/.l is very similar to the string in type IIA theory. In particular, both are weakly coupled when the dilaton expectation value satisfies e(if» « 1. Quantizing this string leads to "type lIB string theory", of which type lIB supergravity is the effective low-energy theory for the massless fields. Observe that 5L(2, Z) contains the transformation T -t -~ which converts a weakly coupled theory into a strongly coupled one. It is therefore a remarkably non-trivial duality symmetry, and has very powerful consequences. Clearly, the evidence that we have uncovered to support its existence does not amount to a proof, which would require an unimaginable level of control over the full nonperturbative dynamics of the theory. In order to relate the type lIB theory to M-theory, we will need an important property of string theory, called "T-duality". The type IIA and type lIB theories are distinct in 10 dimensions, but after compactifying on a circle to 9 dimensions, they are actually equivalent. This is briefly described below. For more details see 4. In the quantization of strings propagating on a circle, we find momentum modes quantized in units of just as for particles. Indeed, these modes correspond to particle-like motion of the string centre-of-mass, and the quantization arises for the usual reasons associated with a compact spactial direction. In addition, there are modes representing a string wound one or more times around the compact direction. These are quantized in units of R. The transformation R -t interchanges these two types of modes. These two types of modes appear symmetrically in various formulae, including those giving the spectrum and interactioml of the theory. Therefore, this transformation (called T-duality) would appear to be a symmetry of string theory - but for one subtlety. The operation R -t changes the spacetime chirality of half the fermions, so it interchanges the type IIA gravitinos ('¢~12, '¢~22,) with type lIB gravitinos ('¢~2, '¢~2J), and similarly for the other fermions. The difference is in the spinor representations: type IIA fermions come in pairs of opposite chirality, namely a spinor and a conjugate spinor of 50(9,1), while type lIB fermions come in pairs with a common chirality, as we have seen above.

ii,

ii

ii

19

Indeed, it can be shown that under the circle T-duality R --t /C, the type IIA and lIB strings are exchanged. As a result, type IIA theory on a circle of radius R is equivalent to type lIB theory on a circle of radius 1/ R, where the string winding modes of one map to the momentum modes of the other. It follows that if we take type IIA theory in 10 dimensions, compactify on a circle of radius R and take the limit R --t 0, we recover type lIB theory in 10 dimensions! It is important to realize that from the type IIA point of view, as R --t 0 the winding modes along the x 10 direction, together with the usual momentum modes in the remaining (noncompact) directions, assemble to give rise to lO-dimensional Lorentz invariance - a string miracle which would be hard to understand without knowing T-duality. This appears to solve our problem about the origin of type lIB theory from M-theory. Recalling that type IIA theory is M-theory compactified on a circle of some radius Rll, it would appear that lIB theory is obtained by compactifying M-theory on a 2-torus (Ru, RiO) and taking R10 --t O. This is not quite correct. Radii of circles as measured in M-theory and string theory are different because of the Weyl rescaling that we made. So we need to examine the issue more closely. Compactify M-theory on a (rectangular) 2-torus and let R l l , R10 be the lengths of the basic cycles ofthe torus in the M-theory metric. We have already shown above that

Ru ,..,. )..2/ 3

(45)

Now let R~~IA) be the radius of x 10 in the type IIA metric. Since

g~~(M) = e-Hg~~(IIA)

(46)

it follows that RIO

e

=

-1/34>R(IIA) 10

R (IIA) 10

(47)

).1/3

Now T-duality is performed in 9 dimensions, and it keeps the 9-dimensional coupling invariant. "For the type IIA theory in 9 dimensions, the coupling is 1

R(IIA)

R(IIB)

___ 10_ _ _ 10_

).~ -).~

).1

Together with R~~I B) = ~, this implies that RiO

(48)

20 R(IIB)

1

10

(IIA) R 10

1 = R 10 A1/ 3

1 -

R

10

Rl/2

(49)

11

We are interested in the limit R~~IB) -+ 00 with AB fixed, so we must take RIO, R11 to zero together, with the ratio ful. R 10 fixed. This is the limit in which Mtheory compactified on a 2-torus gives rise to uncompactified, 10-dimensional type lIB theory. (It is important to keep in mind, however, that on a 2-torus with generic R ll , RIO, M-theory is equivalent to type lIB theory compactified on a circle.) To the extent that M-theory is supposed to be the parent theory underlying the lO-dimensional theories, this should enable us to deduce the nonperturbative duality group SL(2, Z) of type lIB theory that we discussed earlier, and indeed this is the case. The interchange RlO ++ Rll is a symmetry of M-theory (it is part of Lorentz invariance), hence the type lIB theory must have the symmetry AB -+ >.~. One can repeat the above calculation for the case of a slanted 2-torus in the 10-11 directions. In this case one finds that iAB is replaced by a complex quantity r, which in M-theory is the modular parameter labelling possible complex structures of the torus, while in type lIB theory it is the complex scalar field r = J + ie-. Thus the symmetry AB -+ ~ is replaced by r -+ _1. As is well known, T this is just one element of the full global diffeomorphism group of a 2-torus, ~B

which is given by integer matrices

(~ ~)

in SL(2, Z), acting on the modular

parameter r as ar+ b r-+ - - cr+d

(50)

Thus we conclude that, if M-theory has 11d Lorentz invariance, then type lIB theory necessarily has SL(2, Z) symmetry! The mysterious, conjectural S-duality of type lIB theory thus gets geometrized into a completely natural symmetry of M-theory. The assumption that M-theory exists, is really all that is needed. For derivations and more details pertaining to SL(2, Z) duality in 10d, see Ref. 5.

4

Moduli Space

To understand vacuum configurations, or backgrounds of the theories that we have been studying, it is crucial to introduce the concept of moduli space. This is basically the parameter space of the theory, modulo global identifications.

21

Moduli space can be assigned a topology and a metric. The idea is that an infinitesimal vector tangent to moduli space shifts a theory in one background to a theory in a neighbouring background. In string theory, a background is described perturbatively by a conformal field theory (eFT), and a deformation is described by a marginal operator in the eFT. If the collection of all marginal operators is denoted ~i (T, f) then the eFT can be perturbed by shifting the 2-dimensional action: S -t S + ~9i

f

d2 T

~i(T, f) = S + fJS

(51)

I

Now the correlation function on the 2-sphere, in the presence of the perturbation: . -)A> ( -) -68) gij(gi) ( A> ( (52) ~i T, T ~j W, W e = 1z-w 14 defines a metric gij (g) on the parameter space. For type IIA theory in 10 dimensions, the moduli space is the space of vacuum expectation values of the dilaton, or more naturally the space of values of etP , which is the half-line IR+. The moduli space of the lIB theory in 10 dimensions is more complicated. It is the space of vacuum expectation values of T = ¢ + ie-tf> modulo the identification T-t

aT+ b --d' CT+

(53)

Since Im(T) > 0, T lies in the upper half plane (UHP), the moduli space is the quotient (U H P)/SL(2, 7l.). From the M-theory point of view, the picture is as follows. In 11 noncompact dimensions, M-theory has no moduli space since there is no scalar field to take an expectation value. After compactifying on a circle or 2-torus, one expects to find that the moduli space is the parameter space of a circle or 2-torus respectively. In the former case this is labelled by the compactification radius with no further identifications, hence it is IR+ , while in the latter case it is the moduli space of complex structures on the 2-torus, which is well-known to be (U H P)/SL(2, 7l.). Thus the moduli spaces of the two 1O-dimensional N = 2 theories emerge naturally from M-theory. 5

Toroidal Compactification

In this section we discuss toroidal compactifications of M-theory. If we compactify M-theory on a (d + 1)-dimensional torus for d > 2, we expect to obtain

22

larger moduli spaces, but also larger analogues of the duality group SL(2, Z) from geometrical symmetries of the torus. This indeed happens, but there are several interesting surprises. First, let us look at things from the point of view of the type IIA/IIB theories. From our previous discussions, M-theory on a (d + I)-dimensional torus is equivalent to type IIA/IIB theory on a d-torus. Since the latter are string theories, they have a T-duality symmetry group which, on a torus Td of d > 1, is quite nontrivial. This group is perturbatively visible from the point of view of string theory, so whatever happens, the full duality group must contain T-duality. Additionally, the full duality group must contain the SL(2, Z) S-duality of type lIB theory. To start with, T-duality in 9 dimensions is just R -t as we have seen. Thus the T-duality group is Z2. S-duality commutes with this, so the full duality group in 9 dimensions is SL(2, Z) x Z2. For compactification on a d-torus Td with d > 1, we have to first understand T-duality in some detail and then try to discover the maximal duality group, called "U-duality", which combines T and S dualities along with others. We start by recalling a few standard formulae from the conformal field theory of massless compact scalars. These scalars are coordinates of the dtorus, which can be thought of as the quotient of d-dimensional Euclidean space by a lattice r. Let e;, i = 1, ... , d be the generators of r. The generators of the dual lattice r*, denoted by ej, are defined by ei . ej = c5ij. It follows that the inner product of a vector in r with a vector in the dual lattice r* is necessarily an integer. This fact will be useful shortly. The Hamiltonian of such a eFT is given by La + La where

tz

Lo

~ (i1£) 2 + oscillators

La

~(PR)2 + oscillators.

(54)

where d

PL

d

m ·e·+ 2:n-~ 2: ;=1 t

t

,

,

i=1

d

PR

d

2: m ·e. - 2:n-~ t

;=1

s

(55)

, ,

;=1

This all-important formula follows from the fact that on a torus, the string has momentum modes which are integer multiples of and winding modes which are integer multiples of ei.

er,

23

Now the vectors (ei' 0) and (0, en together generate a 2d-dimensionallattice rEB r*. A general vector in rEB r* is (17, tV) with 17 E r, tV E r*. We define a norm on this lattice as follows:

I I(~ I V,W~)112_(~ v W~)(O 1

1) (17) tV --2~. ~ 0

v

W

(56)

where 1 denotes the d x d identity matrix. Since 17 . tV is integer for 17 E rand tV E r*, the norm of any vector according to the above definition is an even integer. Hence the lattice rEB r* is said to be an even lattice. This lattice is also self-dual, since (r EB r*)* = r* EB r which is isomorphic. Note that the norm defined above, using the matrix

(~ ~)

is a Minkowski

norm, since this matrix has d eigenvalues equal to + 1 and d eigenvalues equal to -l. We will be interested in continuous deformations of this even, self-dual lattice which preserve the norm. These are given by all 2d x 2d matrices M acting as

(57) and satisfying

(58) By definition, such matrices M parametrize the orthogonal group O(d, d; ffi.) which is like the Lorentz group for a spacetime with d space and d time directions. Of course, this strange signature is in no way related to the signature of the physical spacetime on which our string theory lives. Rather, it arises from the mathematics of the lattice r EB r* on which this Minkowski norm is natural and leads to the property of being even and self-dual. This action of O( d, d; ffi.) generates a whole family of lattices starting from a given r EB r*. (Not all the lattices so generated have a direct-sum form, however.) All the lattices in this family are equally valid backgrounds for compactification, and in the 2d eFT sense, one interpolates continuously among them by marginal deformations. Thus the matrices M generate a moduli space: a family of backgrounds for string theory. By an obvious change of basis, we can define the group O(d, d; ffi.) as the collection of all real matrices satisfying

(59)

24

However, some of these matrices M do not generate physically new string backgrounds, but only spatial rotations of the old ones. These correspond to matrices M which in the above basis are block diagonal and of the form: M=

(~l ~2)

(60)

Here M 1 , M2 are independent d x d matrices, which form an O(d, IR) x O( d, IR) subgroup of O(d, d, IR). This subgroup consists of independent rotations of the world-sheet left-movers and right-movers, a symmetry of the 2d CFT. Indeed, one can check that Lo and to are separately invariant under this subgroup. As a result, the moduli space is not the full parameter space of O(d, d; IR) but must be quotiented by these symmetries. We write the quotient

O(d, d; IR)/ (O(d, IR) x O(d, IR))

(61)

The dimension of this quotient space is the difference between the dimensions of the groups in the numerator and the denominator, and comes out to be d 2 • Indeed, it has been shown in general that deformations of the lattice r E!1 r* are parametrized by the scalar fields gij, Bij corresponding to components of the metric and the NS-NS 2-form along the internal directions. As g is symmetric and B is antisymmetric, these scalars are d 2 in number as expected. This is still not the end of the story, however. There are discrete global identifications on the moduli space, arising from the group of T-dualities. These are precisely the lattice autmorphisms: the discrete transformations that leave the lattice r E!1 r* invariant. Clearly, all linear transformations of the lattice with integer entries will map the lattice into a sublattice of itself. The ones which preserve the inner product will map it onto itself. Thus all 2d x 2d matrices M with integer entries and satisfying Eq. 59 correspond to automorphisms of the lattice, and they define the infinite discrete group O(d, d; Z). Thus the moduli space in Eq. 61 needs to be further quotiented by this discrete group. Before doing that, let us consider a simple example. For compactification on a circle of radius R, the lattice r E!1 r* is generated by (R,O) and (0, The only 2 x 2 integer matrix, besides the identity, satisfying

it).

MT

is ±

(~ ~)

(~ ~) M = (~ ~)

(62)

itself. This sends (63)

25

it,

so it acts as R -+ a fact which was used earlier. From the above considerations, one may expect that the duality group of type II theories compactified on a d-torus should be O(d, d; Z) and the moduli space should be O(d, d; Z)\O(d, d; ffi)j (O(d, ffi) x O(d, ffi))

(64)

(we follow the now-standard convention of quotienting by continuous groups from the right and discrete groups from the left). This structure of the moduli space is all that can be deduced purely from T-duality. However, the actual duality group and moduli space are larger, because we have not yet taken into account S-duality. This is most evident from the type lIB point of view. Even before compactification, type lIB theory had an S-duality group SL(2, Z) and a nontrivial moduli space SL(2, Z)\(U H P). So the full duality group must contain both S-duality and T-duality, possibly along with other dualities. The upper half-plane can be thought of as the quotient of the group manifold of SL(2, ffi) by U(I). As SL(2, ffi) is the same as SO(2, 1; ffi) and U(I) is the same as SO(2), we can equivalently write the duality group of type lIB in 10 dimensions as SO(2, 1; Z) and its duality group as SO(2, 1; Z)\SO(2, 1; ffi)jSO(2, ffi)

(65)

which is structurally quite similar to the space in Eq. 64. The issue is now tq find the right duality group that combines both T-duality and S-duality (as mentioned before, this is called the V-duality group), and the right moduli space that contains the structure of both Eq. 64 and Eq. 65. These considerations are not enough to fix the maximal V-duality group. A further input is the fact that compactification of type II theories on Td is equivalent to an M-theory compactification on T d +1, hence must possess the geometric symmetries ofthe latter torus. Finally, the classical supergravity Lagrangian with 32 supersymmetries, in various dimensions, has associated continuous duality groups analogous to SL(2, ffi) in 10 dimensions. One expects these to be broken to discrete subgroups, also by analogy with 10 dimensions. Thus, the answer for the V-duality group and associated moduli space has to be found on a case-by-case basis. Let us start with compactification to 8 dimensions on a 2-torus. Based on T-duality alone, we would expect the moduli space to be as in Eq. 64, namely 0(2,2; Z)\0(2, 2; ffi)j (0(2, ffi) x 0(2, ffi))

(66)

of dimension 4. The massless scalar fields in the spacetime Lagrangian that parametrize this moduli space are 999,910,10,99,10, B 9 ,10.

26

However, there are three more massless scalars in this 8-dimensional theory, which from the type IIB point of view are == H(y) = 1 +

k_

k

rd

> 0,

(2.21)

where the constant of integration ¢I has been set equal to zero here for r-+oo simplicity: ¢oo = o. The integration constant k in (2.21) sets the mass scale of the solution; it has been taken to be positive in order to ensure the absence of naked singularities at finite r. This positivity restriction is similar to the usual restriction to a positive mass parameter M in the standard Schwarzschild solution. In the case of the elementary/electric ansatz, with -ldA(C'eC )2. Combining this with (2.16), one finds the relation (2.22) (where it should be remembered that a < 0). Finally, it is straightforward to verify that the relation (2.22) is consistent with the equation of motion for F[n]:

\l2C + C'(C' +

iJ' -

dA' + a¢')

=0 .

(2.23)

In order to simplify the explicit form of the solution, we now pick values of the integration constants to make Aoo = Boo = 0, so that the solution tends to flat empty space at transverse infinity. Assembling the result, starting from the Laplace-equation solution H(y) (2.21), one finds 7,13

= e'" =

ds 2

H(y)

=

-4d

4d

H d(D-2) dxlJ.dxvTJlJ.v + H d(D-2) dymdym Hfl;.

.,

= {+1, -1,

elementary/electric solitonic/magnetic

k 1+--=,

(2.24a) (2.24b) (2.24c)

rd

and in the elementary/electric case, C(r) is given by e

C

2 H- l = VK .

(2.25)

42 In the solitonic/magnetic case, the constant of integration is related to the magnetic charge parameter A in the ansatz (2.6) by

k=

~A. 2d

(2.26)

In the elementary/electric case, this relation may be taken to define the parameter A. The harmonic function H(y) (2.21) determines all of the features of a pbrane solution (except for the choice of gauge for the A(n-l] gauge potential). It is useful to express the electric and magnetic field strengths directly in terms of H: Fml-'l' .. l-'n_l

F m, ... mn

El-'l ... l-'n_l

om(H- 1 )

-Em, ... mnrorH

m = d, ... , D - 1 electric (2.27a)

m = d, ... , D - 1

magnetic, (2.27b)

with all other independent components vanishing in either case. 3

D = 11 examples

Let us now return to the bosonic sector of D = 11 supergravity, which has the action (1.1). In searching for p-brane solutions to this action, there are two particular points to note. The first is that no scalar field is present in (1.1). This follows from the supermultiplet structure of the D = 11 theory, in which all fields are gauge fields. In lower dimensions, of course, scalars do appear; e.g. the dilaton in D = 10 type IIA supergravity emerges out of the D = 11 metric upon dimensional reduction from D = 11 to D = 10. The absence of the scalar that we had in our general discussion may be handled here simply by identifying the scalar coupling parameter a with zero, so that the scalar may be consistently truncated from our general action (2.1). Since a 2 = A - 2dd/(D - 2), we identify A = 2·3·6/9 = 4 for the D = 11 cases. Now let us consider the consistency of dropping contributions arising from the F F A Chern-Simons' term in (1.1). Note that for n = 4, the F(4] antisymmetric tensor field strength supports either an elementary/electric solution with d = n - 1 = 3 (i.e. a p = 2 membrane) or a solitonic/magnetic solution with d = 11 - 3 - 2 = 6 (i.e. a p = 5 brane). In both these elementary and solitonic cases, the F F A term in the action (1.1) vanishes and hence this term does not make any non-vanishing contribution to the metric field equations for our ansatze. For the antisymmetric tensor field equation, a further check is necessary, since there one requires the variation of the F F A term to vanish in

43 order to consistently ignore it. The field equation for A[3] is (1.2), which when written out explicitly becomes

By direct inspection, one sees that the second term in this equation vanishes for both ansatze. Next, we shall consider the elementary/electric and the solitonic/magnetic D = 11 cases in detail. Subsequently, we shall explore how these particular solutions fit into wider, "black," families of p-branes. 3.1

D = 11 Elementary/electric 2-brane

From our general discussion in Sec. 2, we have the elementary-ansatz solution 18

+ ~)-2hdJjdxvTJJjv + (1 + rk6)'hdymdym EJjvA(l + ~)-1 , other components zero. (1

(3.2)

electric 2-brane: isotropic coordinates At first glance, this solution looks like it might be singular at r = O. However, if one calculates the invariant components of the curvature tensor RM N PQ and of the field strength F mJj, 1'21'3' subsequently referred to an orthonormal frame by introducing vielbeins as in (2.8), one finds these invariants to be nonsingular. Moreover, although the proper distance to the surface r = 0 along a t = xO = const. geodesic diverges, the surface r = 0 can be reached along null geodesics in finite affine parameter.19 Thus, one may suspect that the metric as given in (3.2) does not in fact cover the entire spacetime, and so one should look for an analytic extension of it. Accordingly, one may consider a change to "Schwarzschild-type" coordinates by setting r = (f6 - k) '/6. The solution then becomes: 19

(1 -

;6 )2h( -dt2 + da 2 + dp2) + (1 - ;6 )-2df2 + f2dn~

EJjVA (1

- ;6) ,

other components zero,

(3.3)

electric 2-brane: Schwarzschild-type coordinates where we have supplied explicit worldvolume coordinates xl' = (t, a, p) and where dn~ is the line element on the unit 7-sphere, corresponding to the boundary 8M 8T of the 11 - 3 = 8 dimensional transverse space.

44

The Schwarzschild-like coordinates make the surface f = el6 (corresponding to r = 0) look like a horizon. One may indeed verify that the normal to this surface is a null vector, confirming that f = k 1/6 is in fact a horizon. This horizon is degenerate, however. Owing to the 2/3 exponent in the goo component, curves along the t axis for f < k 1/6 remain timelike, so that light cones do not "flip over" inside the horizon, unlike the situation for the classic Schwarzschild solution. In order to see the structure of the membrane spacetime more clearly, let us change coordinates once again, setting f = 16 (1 - R3)_1/S • Overall, the transformation from the original isotropic coordinates to these new ones is effected by setting r = e/6 R 1/2 / (1 - R 3 ) 1/6. In these new coordinates, the solution becomes 19

e

ds 2

=

{R2 (-dt 2 + du 2 + dp2) + te/3 R- 2dR2} +eh [(1- R3)-% - 1mR- 2dR 2 + dn~l

+ e/3dn~

(a) (b)

other components zero. electric 2-brane: interpolating coordinates (3.4) This form of the solution makes it clearer that the light-cones do not "flip over" in the region inside the horizon (which is now at R = 0, with R < 0 being the interior). The main usefulness of the third form (3.4) of the membrane solution, however, is that it reveals how the solution interpolates between other "vacuum" solutions of D = 11 supergravity.19 As R -+ 1, the solution becomes flat, in the asymptotic exterior transverse region. As one approaches the horizon at R = 0, line (b) of the metric in (3.4) vanishes at least linearly in R. The residual metric, given in line (a), may then be recognized as a standard form of the metric on (AdS)4 x 8 7 , generalizing the Robinson-Bertotti solution on (AdSh x 8 2 in D = 4. Thus, the membrane solution interpolates between flat space as R -+ 1 and (AdS)4 x 8 7 as R -+ 0 at the horizon. Continuing on inside the horizon, one eventually encounters a true singularityat f = 0 (R -+ -00). Unlike the singularity in the classic Schwarzschild solution, which is spacelike and hence unavoidable, the singularity in the membrane spacetime is timelike. Generically, geodesics do not intersect the singularity at a finite value of an affine parameter value. Radial null geodesics do intersect the singularity at finite affine parameter, however, so the spacetime is in fact genuinely singular. The timelike nature of this singularity, however, invites one to consider coupling a 8-function source to the solution at f = o. Indeed, the D = 11 supermembrane action,20 which generalizes the NambuGoto action for the string, is the unique "matter" system that can consistently

45

couple to D = 11 supergravity.2o,22 Analysis of this coupling yields a relation between the parameter k in the solution (3.2) and the tension T of the supermembrane action: 18 (3.5) where 1/(2/';,2) is the coefficient of FaR in the Einstein-Hilbert Lagrangian and fh is the volume of the unit 7-sphere S7, i. e. the solid angle subtended by the boundary at transverse infinity. The global structure of the membrane spacetime 19 is similar to the extreme Reissner-Nordstrom solution of General Relativity~4 This global structure is summarized by a Carter-Penrose diagram as shown in Figure 1, in which the angular coordinates on S7 and also two ignorable worldsheet coordinates have been suppressed. As one can see, the region mapped by the isotropic coordinates does not cover the whole spacetime. This region, shaded in the diagram, is geodesically incomplete, since one may reach its boundaries 1{+, 1{- along radial null geodesics at a finite affine-parameter value. These boundary surfaces are not singular, but, instead, constitute future and past horizons (one can see from the form (3.3) of the solution that the normals to these surfaces are null). The "throat" P in the diagram should be thought of as an exceptional point at infinity, and not as a part of the central singularity. The region exterior to the horizon interpolates between flat regions J± at future and past null infinities and a geometry that asymptotically tends to (AdS)4 x S7 on the horizon. This interpolating portion of the spacetime, corresponding to the shaded region of Figure 1 which is covered by the isotropic coordinates, may be sketched as shown in Figure 2.

46

timelike singularity at 1' = 0 (R-,;-oo ) /

const. hypersurface

"throat" P

io spatial infinity

R

const. hypersurface

Only the shaded region is covered by the isotropic coordinates

Figure 1: Carter-Penrose diagram for the D

== 11 elementary/ electric 2-brane solution.

47

., ,

,

,,

,

infinite "throat:" (AdS) x 8 7 ~ 4

Figure 2: The D

= 11 elementary/electric 2-branesolution interpolates between flat at

:r±

space

and (AdS)4 x S7 at the horizon.

3.2 D = 11 Solitonic/magnetic 5-brane Now consider the 5-brane solution to the D = 11 theory given by the solitonic ansatz for F[4J. In isotropic coordinates, this solution is a magnetic 5-brane: 25 v = 0, ... ,5 3kEml ... m4p~ other components zero. magnetic 5-brane: isotropic coordinates (3.6) As in the case of the elementary/electric membrane, this solution interpolates between two "vacua" of D = 11 supergravity. Now, however, these asymptotic geometries consist of the flat region encountered as r -7 00 and of (AdSh x 8 4 as one approaches r = 0, which once again is a degenerate horizon. Combining two coordinate changes analogous to those of the elementary case,

ds 2 F m1 "' m4

(1 + ~ )_1/3dx/ldxv"I/lV + (1 + :3) 2/3dy mdym p

j.l,

48

r

= (1;3 -

k)'/s and f

= k'/s(l- R 6 )-'/3, one has an overall transformation k ' /3 R2

r

= (1 _ R6)'/3

(3.7)

.

After these coordinate changes, the metric becomes d

S

2

-

4W2

dO;]

R 2d JLd v + k'/s [ (1_R6)B/3 dR2 + (1_R6)2/3 x x 'T}JLV magnetic 5-brane: interpolating coordinates

(3.8)

Once again, the surface r = 0 B R = 0 may be seen from (3.8) to be a nonsingular degenerate horizon. In this case, however, not only do the light cones maintain their timelike orientation when crossing the horizon, as already happened in the electric case (3.4), but now the magnetic solution (3.8) is in fact fully symmetric 26 under a discrete isometry R -7 -R. Given this isometry R -t - R, one can identify the spacetime region R ::; 0 with the region R 2 o. This identification is analogoust to the identification one naturally makes for flat space when written in polar coordinates, with the metric dS flat = -dt 2 + dr 2 + r 2d2. Ostensibly, in these coordinates there appear to be separate regions of flat space with r ~ 0, but, owing to the existence of the isometry r -t -r, these regions may be identified. Accordingly, in the solitonic/magnetic 5-brane spacetime, we identify the region -1 < R ::; 0 with the region 0 ::; R < 1. In the asymptotic limit where R -t -1, one finds an asymptotically flat geometry that is indistinguishable from the region where R -t +1, i.e. where r -t 00. Thus, there is no singularity at all in the solitonic / magnetic 5-brane geometry. There is still an infinite "throat ," however, at the horizon, and the region covered by the isotropic coordinates might again be sketched as in Figure 2, except now with the asymptotic geometry down the "throat" being (AdSh x S4 instead of (AdS)4 x S7 as for the elementary /electric solution. The Carter-Penrose diagram for the solitonic/magnetic 5-brane solution is given in Figure 3, where the full diagram extends indefinitely by "tiling" the section shown. Upon using the R -7 -R isometry to make discrete identifications, however, the whole of the spacetime may be considered to consist of just region I, which is the region covered by the isotropic coordinates (3.6). dIn considering this analogy, one should also take into account the possibility of conical singularities. In the case of flat space, a conical singularity with deficit angle (} = 271" arises at the origin R = 0 if one chooses not to make a discrete identification of the two regions R ~ o. This is most easily seen by considering a combined pair of forward and backward cones with deficit angle (} < 271", then taking the limit (} ---+ 271". In this case, as in the case of the magnetic 5-brane geometry, the elimination of conical singularities actually requires making the discrete identification.

49

'1 identified with I R 100 = o. The generalized solution (5.27) corresponds to parallel and similarlyoriented p-branes, with all charge parameters Ao: = 2dk a /V/S.. required to be positive in order to avoid naked singularities. The "centers" of the individual "leaves" of this solution are at the points y = Ya, where a ranges over any number of centers. The metric and the electric-case antisymmetric tensor gauge potential corresponding to (5.27) are given again in terms of H(y) by (2.24a,2.25). In the soli tonic case, the ansatz (2.6) needs to be modified so as

69

to accommodate the multi-center form of the solution: (5.28) which ensures the validity of the Bianchi identity just as well as (2.6) does. The mass/(unit p-volume) density is now [ = 20 D -

d - 1 '""' ~

,fE

A '>l

(5.29)

a

while the total electric or magnetic charge is given by OD-d-l I: Aa , so the Bogomol'ny bounds (4.16) are saturated just as they are for the single-center solutions (2.24). Since the multi-center solutions given by (5.27) satisfy the same supersymmetry-preservation conditions on the metric and antisymmetric tensor as (2.24), the multi-center solutions leave the same amount of supersymmetry unbroken as the single-center solution. From a mathematical point of view, the multi-center solutions (5.27) exist owing to the properties of the Laplace equation (2.20). From a physical point of view, however, these static solutions exist as a result of cancellation between attractive gravitational and scalar-field forces against repulsive antisymmetrictensor forces for the similarly-oriented p-brane "leaves." The multi-center solutions given by (5.27) can now be used to prepare solutions adapted to dimensional reduction in the transverse directions. This combination of a modification of the solution followed by dimensional reduction on a transverse coordinate is called vertical dimensional reduction 23 because it relates solutions vertically on a D versus d plot.h In order to do this, we need first to develop translation invariance in the transverse reduction coordinate. This can be done by "stacking" up identical p branes using (5.27) in a periodic array, i.e. by letting the integration constants ka all be equal, and aligning the "centers" Ya along some axis, e.g. the z axis. Singling out one "stacking axis" in this way clearly destroys the overall isotropic symmetry of the solution, but, provided the centers are all in a line, the solution will nonetheless remain isotropic in the D - d - 1 dimensions orthogonal to the stacking axis. Taking the limit of a densely-packed infinite stack of this sort, one has '""' ka ~·I~ ~ Id a Y -Ya

1

+00

-00

kdz = k (f2 + z2)d/2 fd-l

(5.30a)

hSimiiar procedures have been considered in a number of articles in the literature; see, e.g. Refs~3

70 D-2

".2

k

Lymym m==d

y'1fkr(d 2r(d)

(5.30b)

!)

(5.30c)

where", in (5.30b) is the radial coordinate for the D - d - 1 residual isotropic transverse coordinates. After a conformal rescaling in order to maintain the Einstein frame for the solution, one can finally reduce on the coordinate z along the stacking axis. After stacking and reduction in this way, one obtains a p-brane solution with the same world volume dimension as the original higher-dimensional solution that was stacked up. Since the same antisymmetric tensors are used here to support both the stacked and the unstacked solutions, and since A is preserved under dimensional reduction, it follows that vertical dimensional reduction from D to D - 1 spacetime dimensions preserves the value of A just like the diagonal reduction discussed in the previous subsection. Note that under vertical reduction, the worldvolume dimension d is preserved, but d = D - d - 2 is reduced by one with each reduction step. Combining the diagonal and vertical dimensional reduction trajectories of "descendant" solutions, one finds the general picture given in the plot of Figure 6. In this plot of spacetime dimension D versus world volume dimension d, reduction families emerge from certain basic solutions that cannot be "oxidized" back up to higher-dimensional isotropic p-brane solutions, and hence can be called "stainless" p-branes!3 In Figure 6, these solutions are indicated by the large circles, with the corresponding A values shown adjacently. The indication of elementary or solitonic type relates to solutions of supergravity theories in versions with the lowest possible choice of rank (n ::; Dj2) for the supporting field strength, obtainable by appropriate dualization. Of course, every solution to a theory obtained by dimensional reduction from D = 11 supergravity (1.1) may be oxidised back up to some solution in D = 11. We shall see in Section 6 that what one obtains upon oxidation of the "stainless" solutions in Figure 6 falls into the interesting class of "intersecting branes" built from four basic "elemental" solutions of D = 11 supergravity.

71

D

11 /

. I

/

/

I

9

.

/

I

/

/

I

/

8 / /

/

/

/ /

.1./ .".

1/

/' I

I

,,4

/

/ 'I

/

/

/

/

/ /

I I

•

7

/ /

ta4 J"f'

/

/

/

6

o

elementary

•

5

••

3

self-dual

"stainless"

Kaluza-Klein descendants ~

4

/

)

soJitonic

values

/ /

vertical reduction trajectories

2 /

diagonal reduction trajectories

/

o instanton

2 particle

string

3

4

membrane 3-brane

5

6

7

4-brane

5-brane

6-brane

Figure 6: Brane-scan of supergravity p-brane solutions (p :::: (D - 3))

d

72

5.4

The geometry of (D - 3)-branes

The process of vertical dimensional reduction described in the previous subsection proceeds uneventfully until one makes the reduction from a (D, d = D - 3) solution to a (D - 1, d = D - 3) solution! In this step, the integral (5.30) contains an additive divergence and needs to be renormalized. This is easily handled by putting finite limits ±L on the integral, which becomes J!:Ldi(r2 + i2)-1/2, and then by subtracting a divergent term 2lnL before taking the limit L -t 00. Then the integral gives the expected In f harmonic function appropriate to two transverse dimensions. Before proceeding any further with vertical dimensional reduction, let us consider some of the specific properties of (D - 3}-branes that make the next vertical step down problematic. Firstly, the asymptotic metric of a (D - 3)brane is not a globally flat space, but only a locally flat space. This distinction means that there is in general a deficit solid angle at transverse infinity, which is related to the total mass density of the (D - 3}-brane.34 This means that any attempt to stack up (D - 3}-branes within a standard supergravity theory will soon consume the entire solid angle at transverse infinity, thus destroying the asymptotic spacetime in the construction. In order to understand the global structure of the (D - 3)-branes in some more detail, consider the supersymmetric string in D = 4 dimensions.12 In D = 4, one may dualize the 2-form AI''' field to a pseudoscalar, or axion, field X, so such strings are also solutions to dilaton-axion gravity. The p-brane ansatz gives a spacetime of the form M4 = M2 X ~2, where M2 is D = 2 Minkowski space. Supporting this string solution, one has the 2-form gauge field AI''' and the dilaton ¢. These fields give rise to a field stress tensor of the form

T/L,,(A,¢) Tmn(A, ¢)

= =

-116 (a 2 + 4) Om KOmK1J/L" ~(a2 - 4) (omKonK - !6mn (OpKOpK)} ,

(5.31)

where a is as usual the dilaton coupling parameter and e- K = H = 1 8GTln(r}, with r = "';ymym, m = 2, 3. If one now puts in an elementary string source action, with the string aligned along the j.t, v = 0, 1 subspace, so that T mn (source) = 0, then one has the source stress tensor (5.32) iSolutions with worldvolume dimension two less than the spacetime dimension will be referred to generally as (D - 3)-branes, irrespective of whether the spacetime dimension is D or not.

73 By inspection of the field solution, one has Tmn(A, 1» = 0, while the contributions to T{tv from the A{tv and 1> fields and also from the source (5.32) are both of the form diag(p, -p). Thus, the overall stress tensor is of the form TMN = diag(p, -p, 0, 0). Consequently, the Einstein equation in the transverse m, n indices becomes Rmn - ~gmnR = 0, since the transverse stress tensor components vanish. This equation is naturally satisfied for a metric of the form of the p-brane ansatz, because this form causes the transverse components of the Ricci tensor to be proportional to the Ricci tensor of a D = 2 spacetime, for which Rmn - ~gmnR == 0 is an identity, corresponding to the fact that the usual Einstein action, ,;=gR, is a topological invariant in D = 2. Accordingly, in the transverse directions, the equations are satisfied simply by by 0 = O. In the world-sheet directions, the equations become (5.33)

or just R

= 167rGp,

(5.34)

and as we have already noted, R = Rmm. Owing to the fact that the D = 2 Weyl tensor vanishes, the transverse space 1;2 is conformally flat; Eq. (5.34) gives its conformal factor. Thus, although there is no sensible Einstein action in the transverse D = 2, space, a usual form of the Einstein equation nonetheless applies to that space as a result of the symmetries of the p-brane ansatz. The above supersymmetric string solution may be compared to the cosmic strings arising in gauge theories with spontaneous symmetry breaking. There, the Higgs fields contributing to the energy density of the string are displaced from their usual vacuum values to unbroken-symmetry configurations at a stationary point of the Higgs potential, within a very small transverse-space region that may be considered to be the string "core." Approximating this by a delta function in the transverse space, the Ricci tensor and hence the full curvature vanish outside the string core, so that one obtains a conical spacetime, which is flat except at the location of the string core. The total energy is given by the deficit angle 87rGT of the conical spacetime. In contrast, the supersymmetric string has a field stress tensor T{tv(A, 1» which is not just concentrated at the string core but instead is smeared out over spacetime. The difference arises from the absence of a potential for the fields A{tv, 1> supporting the solution in the supersymmetric case. Nonetheless, as one can see from the behavior of the stress tensor Tmn in Eq. (5.31), the transverse space 1;2 is asymptotically locally flat (ALF), with a total energy density given by the overall deficit angle measured at infinity. For multiple-centered string

74

solutions, one has (5.35) Consequently, when considered within the original supergravity theory, the indefinite stacking of supersymmetric strings leads to a destruction of the transverse asymptotic space. A second problem with any attempt to produce (D - 2)-branes in ordinary supergravity theories is simply stated: starting from the p-brane ansatz (2.3,2.6) and searching for (D - 2) branes in ordinary massless supergravity theories, one simply doesn't find any such solutions. 5.5

Beyond the (D - 3)-brane barrier: Scherk-Schwarz reduction and domain walls

Faced with the above puzzles about what sort of (D - 2)-brane could result by vertical reduction from a (D -"3)-brane, one can simply decide to be brave, and to just proceed anyway with the established mathematical procedure of vertical dimensional reduction and see what one gets. In the next step of vertical dimensional reduction, one again encounters an additive divergence: the integral J~L dz In(y2 + Z2) needs to be renormalized by subtracting a divergent term 4L(1nL - 1). Upon subsequently performing the integral, the harmonic function H (y) becomes linear in the one remaining transverse coordinate. While the mathematical procedure of vertical dimensional reduction so as to produce some sort of (D - 2)-brane proceeds apparently without serious complication, an analysis of the physics of the situation needs some care~5 Consider the reduction from a (D, d = D - 2) solution (a p = (D - 3) brane) to a(D - I,d = D - 3) solution (a p = (D - 2) brane). Note that both the (D - 3) brane and its descendant (D - 2)-brane have harmonic functions H(y) that blow up at infinity. For the (D - 3)-brane, however this is not in itself particularly remarkable, because, as one can see by inspection of (2.24) for this case, the metric asymptotically tends to a locally flat space as r ~ 00, and also in this limit the antisymmetric-tensor one-form field strength (5.36) tends asymptotically to zero, while the dilatonic scalar cP tends to its modulus value cPoo (set to zero for simplicity in (2.24)). The expression (5.36) for the field strength, however, shows that the next reduction step down to the (D -1, d = D - 2) solution has a significant new feature: upon stacking up (D - 3) branes prior to the vertical reduction, thus producing a linear harmonic function in

75

the transverse coordinate y, H(y)

= const. + my

,

(5.37)

the field strength (5.36) acquires a constant component along the stacking axis f-t reduction direction z, (5.38) which implies an unavoidable dependence j of the corresponding zero-form gauge potential on the reduction coordinate: A[o](x,y,z)

= mz + X(x,y)

(5.39)

From a Kaluza-Klein point of view, the unavoidable linear dependence of a gauge potential on the reduction coordinate given in (5.39) appears to be problematic. Throughout this review, we have dealt only with consistent Kaluza-Klein reductions, for which solutions of the reduced theory are also solutions of the unreduced theory. Generally, retaining any dependence on a reduction coordinate will lead to an inconsistent truncation of the theory: attempting to impose a z dependence of the form given in (5.39) prior to varying the Lagrangian will give a result different from that obtained by imposing this dependence in the field equations after variation. The resolution of this difficulty is that in performing a Kaluza-Klein reduction with an ansatz like (5.39), one ends up outside the standard set of massless supergravity theories. In order to understand this, let us again focus on the problem of consistency of the Kaluza-Klein reduction. As we have seen, consistency of any restriction means that the restriction may either be imposed on the field variables in the original action prior to variation so as to derive the equations of motion, or instead may be imposed on the field variables in the equations of motion after variation, with an equal effect. In this case, solutions obeying the restriction will also be solutions of the general unrestricted equations of motion. The most usual guarantee of consistency in Kaluza-Klein dimensional reduction is obtained by restricting the field variables to carry zero charge with respect to some conserved current, e.g. momentum in the reduction dimension. But this is not the only way in which consistency may be achieved. In jNote that this vertical reduction from a (D - 3)-brane to a (D - 2)-brane is the first case in which one is forced to accept a dependence on the reduction coordinate z; in all higherdimensional vertical reductions, such z dependence can be removed by a gauge transformation. The zero-form gauge potential in (5.39) does not have the needed gauge symmetry, however.

76

the present case, retaining a linear dependence on the reduction coordinate as in (5.39) would clearly produce an inconsistent truncation if the reduction coordinate were to appear explicitly in any of the field equations. But this does not imply that a truncation is necessarily inconsistent just because a gauge potential contains a term linear in the reduction coordinate. Inconsistency of a Kaluza-Klein truncation occurs when the original, unrestricted, field equations imply a condition that is inconsistent with the reduction ansatz. If a particular gauge potential appears in the action only through its derivative, i. e. through its field strength, then a consistent truncation may be achieved provided that the restriction on the gauge potential implies that the field strength is independent of the reduction coordinate. A zero-form gauge potential on which such a reduction may be carried out, occurring in the action only through its derivative, will be referred to as an axion. Requiring axionic field strengths to be independent of the reduction coordinate amounts to extending the Kaluza-Klein reduction framework so as to allow for linear dependence of an axionic zero-form potential on the reduction coordinate, precisely of the form occurring in (5.39). So, provided A[o] is an axion, the reduction (5.39) turns out to be consistent after all. This extension of the Kaluza-Klein ansatz is in fact an instance of Scherk-Schwarz reduction.36 ,37 The basic idea of Scherk-Schwarz reduction is to use an Abelian rigid symmetry of a system of equations in order to generalize the reduction ansatz by allowing a linear dependence on the reduction coordinate in the parameter of this Abelian symmetry. Consistency is guaranteed by cancellations orchestrated by the Abelian symmetry in field-equation terms where the parameter does not get differentiated. When it does get differentiated, it contributes only a term that is itself independent of the reduction coordinate. In the present case, the Abelian symmetry guaranteeing consistency of (5.39) is a simple shift symmetry A[o] -7 A[o] + const. Unlike the original implementation of the Scherk-Schwarz reduction idea~6 which used an Abelian U(l) phase symmetry acting on spinors, the Abelian shift symmetry used here commutes with supersymmetry, and hence the reduction does not spontaneously break supersymmetry. Instead, gauge symmetries for some of the antisymmetric tensors will be broken, with a corresponding appearance of mass terms. As with all examples of vertical dimensional reduction, the 6. value corresponding to a given field strength is also preserved. Thus, p-brane solutions related by vertical dimensional reduction, even in the enlarged Scherk-Schwarz sense, preserve the same amount of unbroken supersymmetry and have the same value of 6.. It may be necessary to make several redefinitions and integrations by parts in order to reveal the axionic property of a given zero-form, and thus to pre-

77

pare the theory for a reduction like (5.39). This is most easily explained by an example, so let us consider the first possible Scherk-Schwarz reduction k in the sequence of theories descending from (1.1), starting in D = 9 where the first axion field appears~5 The Lagrangian for massless D = 9 maximal supergravity is obtained by specializing the general dimensionally-reduced action (5.13) given in Section 2 to this case:

Fa [ R- ~(8cPd2 -

3

../'t

-

~(8cP2)2 - ~e-2(h+2d 2 -

~(8¢>2)2 - ~(8¢>3)2

(5.44) where the dilaton vectors are now those appropriate for D = 8; the term LFFA contains only m-independent terms. It is apparent from (5.44) that the fields Ai~l, and Ai~i have become

AiN

massive. Moreover, there are field redefinitions under which the fields X, Ai~t) and Ai~t) may be absorbed. One way to see how this absorption happens is to notice that the action obtained from (5.44) has a set of three Stueckelberg-type II am grateful to Marcus Bremer for help in correcting some errors in the original expression of Eq, (5.44) given in Ref.35

79

AiN, Ai;l

Af;l

gauge transformations under which and transform according to their standard gauge transformation laws. These three transformations are accompanied, however, by various compensating transformations necessitated by the Chern-Simons corrections present in (5.44) as well as by m-dependent shift transformations of x, Ai~t) and respectively. Owing to the presence

Ai;)3),

of these local shift terms in the three Stuecelberg symmetries, the fields and

Af;)3) may be gauged to zero.

X, Af~)3)

After gauging these three fields to zero, one

. )

(3)

(2)

(1)

has a clean set of mass terms m (5.44 for the fields A[l) ,A[l) and A(2) . As one descends through the available spacetime dimensions for supergravity theories, the number of ax ionic scalars available for a Scherk-Schwarz reduction step increases. The numbers ofaxions are given in the following Table: Table 1: Supergravity axions versus spacetime dimension.

Each of these axions gives rise to a distinct massive supergravity theory upon Scherk-Schwarz reduction,35 and each of these reduced theories has its own pattern of mass generation. In addition, once a Scherk-Schwarz reduction step has been performed, the resulting theory can be further reduced using ordinary Kaluza-Klein reduction. Moreover, the Scherk-Schwarz and ordinary Kaluza-Klein processes do not commute, so the number of theories obtained by the various combinations of Scherk-Schwarz and ordinary dimensional reduction is cumulative. In addition, there are numerous possibilities of performing Scherk-Schwarz reduction simultaneously on a number of axions. This can be done either by arranging to cover a number of axions simultaneously with derivatives, or by further Scherk-Schwarz generalisations of the Kaluza-Klein reduction process.38 For further details on the panoply of Scherk-Schwarz reduction possibilities, we refer the reader to Refs. 35 ,38 The single-step procedure of Scherk-Schwarz dimensional reduction described above may be generalised to a procedure exploiting the various cohomology classes of a multi-dimensional compactification manifold.39 The key to this link between the Scherk-Schwarz generalised dimensional reduction and the topology of the internal Kaluza-Klein manifold K is to recognise that the single-step reduction ansatz (5.39) may be generalised to A[n-l](X,

y, z) =

W[n-l)

+ A[n-l](X, y)

,

(5.45)

80

is an (n - 1) form defined locally on K, whose exterior derivative is an element of the cohomology class Hn(K, R). For example, in the case of a single-step generalised reduction on a circle S1, one has n[l] = mdz E H1(Sl,R), reproducing our earlier single-step reduction (5.39). As another example, consider a generalised reduction on a 4-torus T4 starting in D = 11, setting A[3](X,y,z) = W[3] +A[3](X,y) with n[4] = dw[3] = where

n[n]

=

W[n-l]

dw[n-1]

E H 4 (T4, R). In this example, one may choose to write W[3] locally as W[3] = mZl dZ2 1\ dZ 3 1\ dz4 . All of the other fields are reduced using the standard Kaluza-Klein ansatz, with no dependence on any of the Zi coordinates: The theory resulting from this T4 reduction is a D = 7 massive supergravity with a cosmological potential, analogous to the D = 8 theory (5.44). The same theory (up to field redefinitions) can also be obtained 35 by first making an ordinary Kaluza-Klein reduction from D = 11 down to D = 8 on a 3-torus T 3 , then making an S1 single-step generalised Scherk-Schwarz reduction (5.39) from D = 8 to D = 7. Although the T4 reduction example simply reproduces a massive D = 7 theory that can also be obtained via the single-step ansatz (5.39), the recognition that one can use any of the Hn(K, R) cohomology classes of the compactification manifold K significantly extends the scope of the generalised reduction procedure. For example, it allows one to make generalised reductions on manifolds such as K3 or on Calabi-Yau manifolds~9 For our present purposes, the important feature of theories obtained by Scherk-Schwarz reduction is the appearance of cosmological potential terms such as the penultimate term in Eq. (5.44). Such terms may be considered within the context of our simplified action (2.1) by letting the rank n of the field strength take the value zero. Accordingly, by consistent truncation of (5.44) or of one of the many theories obtained by Scherk-Schwarz reduction in lower dimensions, one may arrive at the simple Lagrangian mdz 1 1\ dZ 2 1\ dZ3 1\ dZ 4

(5.46) Since the rank of the form here is n = 0, the elementary / electric type of solution would have worldvolume dimension d = -1, which is not very sensible, but the solitonic/magnetic solution has d = D - 1, corresponding to a p = D - 2 brane, or domain wall, as expected. Relating the parameter a in (5.46) to the reduction-invariant parameter .:l by the standard formula (2.18) gives .:l = a 2 - 2(D -l)/(D - 2); taking the corresponding p = D - 2 brane solution from (2.24), one finds (5.47a)

81

(5.47b) where the harmonic function H(y) is now a linear function of the single transverse coordinate, in accordance with (5.37)~ The curvature of the metric (5.47a) tends to zero at large values of iyi, but it diverges if H tends to zero. This latter singularity can be avoided by taking H to be

H = const. + Miyi

(5.48)

where M = ~mJK. With the choice (5.48), there is just· a delta-function singularity at the location of the domain wall at y = 0, corresponding to the discontinuity in the gradient of H. The domain-wall solution (5.47,5.48) has the peculiarity of tending asymptotically to fiat space as iyi -+ 00, within a theory that does not naturally admit fiat space as a solution (by "naturally," we are excluding the case a¢ -+ -(0). Moreover, the theory (5.46) does not even admit a non-fiat maximally-symmetric solution, owing to the complication of the cosmological potential. The domain-wall solution (5.47,5.48), however, manages to "cancel" this potential at transverse infinity, allowing at least asymptotic fiatness for this solution. This brings us back to the other facets of the consistency problem for vertical dimensional reduction down to (D - 2)-branes as discussed in subsection 5.4. There is no inconsistency between the existence of domain-wall solutions like (5.47,5.48) and the inability to find such solutions in standard supergravity theories, or with the conical-spacetime character of (D - 3)-branes, because these domain walls exist only in massive supergravity theories like (5.44), with a vacuum structure different from that of standard massless supergravities. Because the Scherk-Schwarz generalised dimensional reduction used to obtain them was a consistent truncation, such domain walls can be oxidised back to solutions of higher-dimensional massless supergravities, but in that case, they have the form of stacked solutions prepared for vertical reduction, with non-zero field strengths in the reduction directions, as in our example (5.38). 6 6.1

Intersecting branes, scattering branes Multiple component solutions

Given the existence of solutions (5.24) with several active field strengths F{;~,l' but with coincident charge centers, it is natural to try to find solutions where mDomain walls solutions such as (5.47) in supergravity theories were found for the D = 4 case in Ref.4o and a review of them has been given in Ref. 41

82 the charge centers for the different Fr'~] are separated~2 This will lead us to a better understanding of the ~ =I- 4 solutions shown in Figure 6. Consider it number of field strengths that individually have ~ = 4 couplings, but now look for a solution where C of these field strengths are active, with centers Yon (t = 1, ... , C. Let the charge parameter for FO: be .xo:. Thus, for example, in the magnetic case, one sets (6.1)

In both the electric and the magnetic cases, the.xo: are related to the integration constants kO: appearing in the metric by kO: = .x 0: / d. Letting ..i(t) = Dxilo=o) as 1=_12

J

·d·

dtd8(i'V-I tJ-Dxt-xJ dt

1 .. k + -A3! 'J-kDxtDxJ Dx ) .

(6.18)

One may additionally 50 have a set of spinorial N = 1 superfields 'l/Ja, with Lagrangian - ~ hab 'l/Ja V t 'l/Jb, where hab is a fibre metric and V t is constructed using an appropriate connection for the fibre corresponding to the 'l/Ja. However, in the present case we shall not include this extra superfield. In order to have extended supersymmetry in (6.18), one starts by positing a second set of supersymmetry transformations of the form JXi = 'T}Ii j Dx j , and then requires these transformations to close to form the N = 2 algebra (6.17); then one also requires that the action (6.18) be invariant. In this way, one obtains the equations

12 i = N jk - Ii[j,k) 1'kl Ik JI j

-1

(6.19a)

0

(6.19b)

1'ij

(6.19c)

92 r7(+)I k . v (i J)

ali (Im jAlmlkl]) - 2Jm[ia[mAjklll)

°

0,

(6.19d) (6.1ge)

where (6.19a,b) follow from requiring the closure of the algebra (6.17) and (6.19c-e) follow from requiring invariance of the action (6.18). Conditions (6.19a,b) imply that M is a complex manifold, with Iij as its complex structure. The structure of the conditions (6.19) is more complicated than might have been expected. Experience with d = 1 + 1 extended supersymmetry 50 might have lead one to expect, by simple dimensional reduction, just the condition V'~+) Ij k = O. Certainly, solutions of this condition also satisfy (6.19c-e), but the converse is not true, i.e. the d = 1 extended supersymmetry conditions are "weaker" than those obtained by dimensional reduction from d = 1 + 1, even though the d = 1 + 1 minimal spinors are, as in d = 1, just real singlecomponent objects. Conversely, the d = 1 + 1 theory implies a "stronger" condition; the difference is explained by d = 1 + 1 Lorentz invariance: not all d = 1 theories can be "oxidized" up to Lorentz-invariant d = 1 + 1 theories. In the present case with two D = 9 black holes, this is reflected in the circumstance that after even one dimensional oxidation from D = 9 up to D = 10, the solution already contains a pp wave element (so that we have a D = 10 "wave-on-a-string" solution), with a light plane metric that is not Poincare invariant. Note also that the d = 1 "torsion" A[3] is not required to be closed in (6.19). d = 1 supersymmetric theories satisfying (6.19) are analogous to (2,0) chiral supersymmetric theories in d = 1 + 1, but the weaker conditions (6.19) warrant a different notation for this wider class of models; one may.call them 2b supersymmetric 0'-models~9 Such models are characterized by a Kahler geometry with torsion. Continuing on to N = 8, d = 1 supersymmetry, one finds an 8b generalization 49 of the conditions (6.19), with 7 independent complex structures built using the octonionic structure constantS' 'Pab C: Jxi = 'fJa Ia ijDxj, a = 1, ... 7, with (Ia)8 b = Jab, (Ia)b 8 = -iSba, (Ia)b c = 'Pabc, where the octonion multiplication rule is eaeb = -iSab + 'Pab cec . Models satisfying such conditions have an "octonionic Kahler geometry with torsion," and are called OKT models~9 Now, are there any non-trivial solutions to these conditions? Evidently, from the brane-probe and Shiraishi analyses, there must be. For our two D = 9 black holes with a D = 8 transverse space, one may start from the ansatz ds 2 = H(y)ds 2(JE8), Aijk = nijkf8fH, where n is a 4-form on JE8. n We let the conventional octonionic "0" index be replaced by "8" here in order to avoid confusion with a timelike index; the lE 8 transverse space is of Euclidean signature.

93

Then, from the 8b generalization of condition (6.19d) one learns 11Sabc = CPabc and 11abcd = -*CPabcd; from the 8b generalization of condition (6.1ge) one learns 8ij 8 i 8 j H = O. Thus we recover the familiar dependence of p-brane solutions on transverse-space harmonic functions, and we reobtain the brane-probe or Shiraishi structure of the black-hole modulus scattering metric with Hrelative = 1

where

7

kred

+ IY1

k red - Y2

16 '

(6.20)

determines the reduced mass of the two black holes.

Duality symmetries and charge quantisation

As one can see from our discussion of Kaluza-Klein dimensional reduction in Section 5, progression down to lower dimensions D causes the number of dilatonic scalars ¢ and also the number of zero-form potentials of I-form field strengths to proliferate. When one reaches D = 4, for example, a total of 70 such spin-zero fields has accumulated. In D = 4, the maximal (N = 8) supergravity equations of motion have a linearly-realized H = SU(8) symmetry; this is also the automorphism symmetry of the D = 4, N = 8 supersymmetry algebra relevant to the (self-conjugate) supergravity multiplet. In formulating this symmetry, it is necessary to consider complex self-dual and anti-self-dual combinations of the 2-form field strengths, which are the highest-rank field strengths occurring in D = 4, higher ranks having been eliminated in the reduction or by dualization. Using two-component notation for the D = 4 spinors, these .. ' e. as a comp1ex · t'IOns t rans£orm as F[ij] comb ma Ct.{3 and F-aMij]' z, J = 1, ... ,8,z. 28-dimensional dimensional representation of SU(8). Since this complex representation can be carried only by the complex field-strength combinations and not by the I-form gauge potentials, it cannot be locally formulated at the level of the gauge potentials or of the action, where only an SO(8) symmetry is apparent. Taking all the spin-zero fields together, one finds that they form a rather impressive nonlinear a-model with a 70-dimensional manifold. Anticipating that this manifold must be a coset space with H = SU(8) as the linearlyrealized denominator group, Cremmer and Julia 51 deduced that it had to be the manifold E 7 (+7)/SU(8); since the dimension of E7 is 133 and that of SU(8) is 63, this gives a 70-dimensional manifold. Correspondingly, a nonlinearlyrealized E7( +7) symmetry also appears as an invariance of the D = 4, N = 8 maximal supergravity equations of motion. Such nonlinearly-realized symmetries of supergravity theories have always had a somewhat mysterious character. They arise in part out of general covariance in the higher dimensions, from

94

which supergravities arise by dimensional reduction, but this is not enough: such symmetries act transitively on the a-model manifolds, mixing both fields arising from the metric and also from the reduction of the D = 11 3-form potential A[3] in (1.1). In dimensions 4 ::; D ::; 9, maximal supergravity has the sets of a-model nonlinear G and linear H symmetries shown in Table 2. In all cases, the spinzero fields take their values in "target" manifolds G/H. Just as the asymptotic value at infinity of the metric defines the reference, or "vacuum" spacetime with respect to which integrated charges and energy/momentum are defined, so do the asymptotic values of the spin-zero fields define the "scalar vacuum." These asymptotic values are referred to as the moduli of the solution. In string theory, these moduli acquire interpretations as the coupling constants and vacuum B-angles of the theory. Once these are determined for a given "vacuum," the classification symmetry that organizes the distinct solutions of the theory into multiplets with the same energy must be a subgroup of the little group, or isotropy group, of the vacuum. In ordinary General Relativity with asymptotically flat spacetimes, the analogous group is the spacetime Poincare group times the appropriate "internal" classifying symmetry, e.g. the group of rigid (i.e. constant-parameter) Yang-Mills gauge transformations. Table 2: Supergravity IT-model symmetries.

9 8 7 6 5 4 3

G GL(2,lR) 8L(3, lR) X SL(2, lR) SL(5, lR) 80(5,5) E 6 (+6) E7(+7) E S (+8)

H

SO(2) SO(3) X 80(2) SO(5) 80(5) X 80(5) USP(8) 8U(8) SO(16)

The isotropy group of any point on a coset manifold G/H is just H, so this is the classical "internal" classifying symmetry for multiplets of supergravity solutions. 7.1

An example of duality symmetry: D = 8 supergravity

In maximal D = 8 supergravity, one sees from Table 2 that G = SL(3, lR) X SL(2, lR) and the isotropy group is H = SO(3) X SO(2). We have an (11 3 = 8) vector of dilatonic scalars as well as a singlet and a triplet :rtf]

Ft!t

95

(i, j, k = 1,2,3) of I-form field strengths for zero-form potentials. Taken all together, we have a manifold of dimension 7, which fits in precisely with the dimension of the (SL(3, R) x SL(2, R))/(SO(3) x SO(2)) coset-space manifold: 8 + 3 - (3 + 1) = 7. Owing to the direct-product structure, we may for the time being drop the 5-dimensional SL(3, R)/S0(3) sector and consider for simplicity just the 2-dimensional SL(2, R)/SO(2) sector. Here is the relevant part of the action: 52

I~L(2)

f d xFo[R8

!V'MaV'Ma - !e- 20"V'MxV'M x

- 2 \,eO" (F[4])2 - 2 \,XF[4]*F[4]] (7.1)

where *FMNPQ = I/(4'Fu)eMNPQX1X2X3x4 FX1X2X3X4 (the e[8] is a density, hence purely numerical). On the scalar fields (a, X), the SL(2, R) symmetry acts as follows: let >. = X + ieO"j then (7.2)

with ab- cd = 1 is an element of SL(2, R) and acts on >. by the fractional-linear transformation \ a>. + b (7.3) /\ ----t c>. + d . The action of the SL(2, R) symmetry on the 4-form field strength gives us an example of a symmetry of the equations of motion that is not a symmetry of the action. The field strength F[4J forms an 8L(2, lR) doublet together with (7.4)

i.e.,

(~i:~)

----t

(AT)-l

(~i:~)

(7.5)

One may check that these transform the F[4J field equation V' M(eO" FMNPQ

+ X*FMNPQ)

= 0

(7.6)

into the corresponding Bianchi identity, V'M*FMNPQ

= O.

(7.7)

Since the field equations may be expressed purely in terms of F[4], we have a genuine symmetry of the field equations in the transformation (7.5), but since

96 this transformation cannot be expressed locally in terms of the gauge potential A[3], this is not a local symmetry of the action. The transformation (7.3,7.5) is aD = 8 analogue of ordinary Maxwell duality transformation in the presence of scalar fields. Accordingly, we shall refer generally to the supergravity O"-model symmetries as duality symmetries. The F[4] field strength of the D = 8 theory supports elementary/electric p-brane solutions with p = 4 - 2 = 2, i.e. membranes, which have a d = 3 dimensional worldvolume. The corresponding solitonic/magnetic solutions in D = 8 have world volume dimension d = 8 - 3 - 2 = 3 also. So in this case, F[4] supports both electric and magnetic membranes. It is also possible in this case to have solutions generalizing the purely electric or magnetic solutions considered so far to solutions that carry both types of charge, i.e. dyons?2 This possibility is also reflected in the combined Bogomol'ny bound' for this situation, which generalizes the single-charge bounds (4.16): (7.8)

where U and V are the electric and magnetic charges and 0"00 and XOO are the moduli, i.e. the constant asymptotic values of the scalar fields O"(x) and X(x). The bound (7.8) is itself SL(2, IR) invariant, provided that one transforms both the moduli (O"oo,Xoo) (according to (7.3)) and also the charges (U, V). For the simple case with 0"00 = XOO = 0 that we have mainly chosen in order to simplify the writing of explicit solutions, the bound (7.8) reduces to £2 ~ U 2 +V 2 , which is invariant under an obvious isotropy group H = SO(2).

7.2 p-form charge quantisation conditions So far, we have discussed the structure of p-brane solutions at a purely classical level. At the classical level, a given supergravity theory can have a continuous spectrum of electrically and magnetically charged solutions with respect to any one of the n-form field strengths that can support the solution. At the quantum level, however, an important restriction on this spectrum of solutions enters into force: the Dirac-Schwinger-Zwanziger (DSZ) quantisation conditions for particles with electric or magnetic or dyonic charges.53 ,54 As we have seen, however, the electric and magnetic charges carried by branes and appearing in the supersymmetry algebra (1.5) are forms, and the study of their chargequantisation properties involves some special features not seen in the D = 4 Maxwell case?5 DIn comparing (7.8) to the single-charge bounds (4.16), one should take note that for F[4J in (7.1) we have ~ = 4, so 2/~ = 1.

97

We shall first review a Wu-Yang style of argument~3 (for a Dirac-string argument, see Ref. 54 ,56) considering a closed sequence W of deformations of one p-brane, sayan electric one, in the background fields set up by a dual, magnetic, p = D - p - 4 brane. After such a sequence of deformations, one sees from the supermembrane action (4.1) that the electric p-brane wavefunction picks up a phase factor

1

exp ( (p iQe + I)! Jw A Ml ... Mp+t dXlMA

...

Ad Mp+l) X

,

(7.9)

where A[p+lJ is the gauge potential set up (locally) by the magnetic fJ-brane background. A number of differences arise in this problem with respect to the ordinary Dirac quantisation condition for D = 4 particles. One of these is that, as we have seen in subsection 4.1, objects carrying p-form charges appearing in the supersymmetry algebra (1.5) are necessarily either infinite or are wrapped around compact spacetime dimensions. For infinite p-branes, some deformation sequences W will lead to a divergent integral in the exponent in (7.9); such deformations would also require an infinite amount of energy, and so should be excluded from consideration. In particular, this excludes deformations that involve rigid rotations of an entire infinite brane. Thus, at least the asymptotic orientation of the electric brane must be preserved throughout the sequence of deformations. Another way of viewing this restriction on the deformations is to note that the asymptotic orientation of a brane is encoded into the electric p-form charge, and so one should not consider changing this p-form in the course of the deformation any more than one should consider changing the magnitude of the electric charge in the ordinary D = 4 Maxwell case. We shall see shortly that another difference with respect to the ordinary D = 4 Dirac quantisation of particles in Maxwell theory will be the existence of "Dirac-insensitive" configurations, for which the phase in (7.9) vanishes. Restricting attention to deformations that give non-divergent phases, one may use Stoke's theorem to rewrite the integral in (7.9):

where Mw is any surface "capping" the closed surface W, i.e. a surface such that BMw = W; :\ INS) are the scalar 2 superpartners in the worldvolume. The Ramond sector gives the fermionic superpartners. If there are n identical parallel D-branes, then the open string can begin on a D-brane labeled by i and end on one labeled by j (Figure 2). The label of

I

f"

Figure 2: An open string beginning on the i-th D-brane and ending on the j-th D-brane.

the D-brane is what in early string theory was called the Chan-Paton index at each end. Let us denote a general state in the open string sector by 11P, ij).,ij.

144

Here i, j are Chan-Paton indices, Aij is the Chan-Paton wave-function, 'IjJ is the state of the worldsheet fields, and by reality of the string wave function, At = A. The massless excitations of the open string now give rise to a supersymmetric U(n) gauge theory on the worldvolume. The spectrum of these nonperturbative states provides many non-trivial checks of duality. For example, 5L(2, Z) predicts a whole tower of 'dyonic' (p, q) strings that that have charge p with respect to B and charge q with respect to B' [50, 69]. Many of these predictions have now been confirmed providing substantial evidence for the correctness of duality. T-duality has a simple action on D-branes [46]. T-duality along a longitudinal direction of a p-brane turns it into a (p-1)-brane, and T-duality along a transverse direction turns it into a (p + 1) brane. This follows from the observation that T-duality is a one-sided parity transform so it turns Dirichlet and Neumann boundary conditions into each other. 3

3.1

Orbifolds General Remarks

Given a manifold M with a discrete symmetry G, one can construct an orbifold M' = MIG. If the symmetry acts freely on M, i.e., without any fixed points, then M' is also a smooth manifold. If there are fixed points then M' is singular near the fixed points. If we now consider strings moving on a target space M, then we are naturally led to the concept of orbifolds in conformal field theory. Consider a theory A with a discrete symmetry group G. One can construct a new theory A' = orbifold of A by G, A' = A/G. For simplicity we shall take G to be a Z2 == {I, a} generated by an involution a because in fact all examples used in these lectures are Z2 orbifolds. In point particle theory, we simply take the Hilbert space of A and keep only those states that are invariant under G to obtain the Hilbert space of A'. However, the particle propagation would be singular near the fixed points of G. In closed string theory, we must also add the "twisted sectors" that are localized near the fixed points. In twisted sectors, the string is closed only up to an action by an element of the group. What is surprising is that after the inclusion of twisted sectors, string propagation on the orbifold is nonsingular even near the fixed points. In string theory, there is a well-defined procedure for adding twisted sectors. Twisted sectors are necessary for modular invariance which is the requirement that the string path integral be invariant under the modular group.

145

For a torus, the modular group is SL(2, Z). The modular group is the group of global diffeomorphisms of the surface. Invariance with respect to this group is essential to avoid global gravitational anomalies which would render the theory inconsistent. This requirement necessitates the inclusion of twisted sectors. We refer the reader to [19, 30] for details of modular invariance. Physically, unitarity is what requires twisted sectors. Even if you excluded twisted states at tree level, once you include interactions, they will appear in loops because an untwisted string can split into a string twisted by an element 9 and another string twisted by 9- 1 . For Z2 orbifolds there are only two sectors: one untwisted and the other twisted by a. In each sector we must perform the projection onto Z2 invariant states with the projector ~(1 + a). Here a is the operator that represents the action of a on the Hilbert space. In the sector twisted by a, all worldsheet fields, which we generically refer to as , satisfy the boundary condition

(0' + 21r, r) = a(O', r).

(38)

For Z2 orbifolds, level matching is necessary and sufficient to ensure modular invariance at one loop. Level matching requires that

(39) where EL and ER are the energies of any two states of the left-moving and right-moving Hilbert spaces respectively. In the next subsection I illustrate this procedure by constructing Type IIA theory as a Z2 orbifold of Type-lIB. 3.2

Type-IIA theory as an orbifold

We orbifold Type-lIB theory by the symmetry group

Z2 == {I, (_I)FL}.

(40)

Untwisted Sector: After the projection ~(1 + (_I)FL), all R-R and R-NS states are removed but the NS-NS and NS-R states Iii) 0 (Ii) EB Ib)) survive. We are left with gij, Bij, ¢J and a single gravitino 1f;ib. Twisted Sector: The twisting of boundary conditions affects only the left-moving fermion sa because other fields are invariant under (-1 )FL .

sa (0' + 21r) Sa (0' + 21r)

=

(_I) FL sa(O') = _sa (0')

sa (0'),

Xi (0' + 21r) = Xi(O').

(41)

146

sa

Therefore, the mode expansion of the coordinates Xi and in the twisted sector is the same as in the untwisted sector. The oscillators are integer moded as before and, in particular, the right-moving ground states are given by the representation of the zero mode algebra {sg, SS} oab. We thus obtain, as in the untwisted sector,

=

(42) as the right-moving ground states. The oscillators of the left-moving fields are moded with half-integer modings so as to satisfy the boundary condition 41

(43) The ground state energy is a sum of zero point energies of the oscillators. For a single complex boson twisted by a phase e21ri f/, the ground state energy is given by the formal sum 2::=0 !(n + 1]). It can be evaluated as ((0,1]), where (( k , 7]) = 2::=0 ~ (n + 7]) - k is the Riemann zeta-function which regularizes the sum. The ground state energy of a single complex boson is [19] 1 1 --1]) . 12 + -7](1 2

(44)

The ground state energy of a fermion with the same twisting is negative of the above. Now, in the left-moving twisted sector, there are 4 (complex) bosons which are untwisted (7] 0) and and integer moded, and 4 complex fermions that are twisted with (1] = 0) and are integer moded. Adding the zero point energies of these fields we get that the ground state energy in the twisted sector is - ~. The ground state is therefore tachyonic because The mass-shell condition is M2 = 4/o/(N - ~) and the level matching condition for physical states is 1 N--=N (45)

=

2

The ground state does not satisfy the physical state condition. Moreover, it is odd under the action of (-1 )FL and is any way projected out by the Z2 projection. The first excited state S~~IO) 2

satisfies the constraints and the Z2 invariance. It gives rise to massless states

la) ® (Ii) EBlh»).

(46)

147

We see that an additional gravitino ?/Jjcx has appeared in the twisted sector with chirality opposite to the one that was projected out. The product la) ® Ib) can be reduced as in Eq.ll (47) Now, because .AI and .A2 have opposite chirality, only products such as ri and ijk appear. We thus obtain a vector A; and a 3-form Cijk. Altogether what we have obtained is precisely the spectrum of Type-IIA theory, which has two spinors and §b to begin with: Bosons: NS-NS: metric gij, 2-form Bij, dilaton I/J, R-R: vector Ai, 3-form Cijk, Fermions: NS-R: gravitino ?/Jib' R-NS: gravitino ?/Jja.

r

so.

4

Type-I String as an Orientifold

An important and simple example which illustrates most of the features of the orientifold construction is Type-I theory in ten dimensions. In this section we shall work through this example in detail, after some general remarks about orientifolds.

4.1

General Remarks About Orientifolds

In general, a symmetry operation of a string theory A can be a combination of target spacetime symmetry and orientation-reversal on the world sheet. The group of symmetry can then be written as a union

Given such a symmetry of A, one can construct a new theory A' = A/G. In section §3 we had implicitly assumed that G 2 is empty and that the orbifold symmetry consists of only target space symmetries. If G 2 is non-empty, the resulting theory A' is called an "orientifold" of A [17,47,4,32, 33,31,48]. In most examples discussed recently, one starts typically with a ZN orbifold of toroid ally compactified Type lIB theory and then orientifolds it further by a symmetry Z2 = {I, Q,B}, where ,B is a Z2 involution of the orbifold. If

148

the orbifold group ZN is generated by the element a, then the total orientifold symmetry is G = {I, a, ... , aN-I, 0,8, 0,80', ... , 0,8aN - 1 } or symboliZN U o(,8ZN ). We describe below some general features of the cally, G orientifold construction. (1) Unoriented Surfaces: An orientifold is obtained, like an orbifold, by gauging the symmetry G. A non-empty OG2 means that orientation reversal, accompanied by an element of G2, is a local gauge symmetry; a string and its orientation reversed image are gauge equivalent and must be identified. Therefore, the string perturbation theory of the orientifold includes unoriented surfaces like the Klein bottle. (2) Closed String Sector: The closed string sector of the theory A' consists of states in the Hilbert space of A that are invariant under G and which survive the orientifold projection. It is completely analogous to the untwisted sector of an orbifold after the projection. Typically, starting with oriented closed strings, one gets unoriented closed strings after the projection. (3) Tadpole Cancellation and Orientifold Planes: Orientifolds often but not always have open strings in addition to the closed strings. The open string sector in orientifolds is analogous to, but not exactly the same as, the twisted sectors in orbifolds. In the case of orbifolds, twisted sectors are necessitated by the requirement of modular invariance. In the case of orientifolds, the one-loop diagrams in string perturbation theory include unoriented and open surfaces for which there is no analog of the modular group. There is, however, a consistency requirement for these surfaces that is analogous to the requirement of modular invariance for the torus. This is the requirement of 'tadpole cancellation'. These loop diagrams can have a divergence in the tree channel corresponding to a tadpole of a massless particle. Cancellation of all tadpoles is necessary for obtaining a stable string vacuum. This requirement is very restrictive and it more or less completely determines when and how the open string should be added. Physically, nonzero tadpoles imply that the equations of motion of some massless fields are not satisfied. They occur for the following reason. The planes that are left invariant by the elements of G2 are called the 'orientifold planes'. Like a D-brane, an orientifold plane is a p-dimensional hyperplane which couples to an R-R (p+1)-form which we generically refer to as Ap +1 • The charge of the orientifold plane can be calculated by looking the R-R tadpole, i.e., emission of an R-R closed string state in the zero momentum limit. If the orientifold plane has a nonzero charge then it acts as a source term in the equations of motion for the (p+ 1)-form field Ap +1 :

=

(48)

149

where Hp+2 is the (p+2)-form field strength of Ap+1 , Jp+1 and h-p are the 'electric' and 'magnetic' sources. Consistency of the field equations requires that fEk *hO-k = 0, for all surfaces Ek without a boundary. In particular, there can be no net charge on a compact space. This is the analog of Gauss law in electrodynamics. The field lines emanating from a charge must either escape to infinity or end on an opposite charge. In a compact space, the field lines have nowhere to go to and hence must end on an equal and opposite charge. The only way the negative charge of a p-dimensional orientifold plane in a compact transverse space can be neutralized is by adding the right-number of Dirichlet p-branes so that Gauss law is satisfied and all tadpoles cancel. (4) Open String Sector and Surfaces with Boundaries: D-branes are hyperplanes where open strings can end. Inclusion of D-branes introduces the open string sector in the theory. The action of the group G is represented in the D-brane sector by some matrices, which we denote by 'Y. The 'Y matrices act on the Chan-Paton indices:

Tadpole cancellation together with the requirement that the 'Y matrices furnish a representation of the symmetry G in the D-brane sector determine not only the number of D-branes but also the form of the 'Y matrices. When n Dbranes coincide, the world volume gauge group is U (n). After the projection onto G-invariant states, we are left with a subgroup of U(n). The group as well as the representations are usually uniquely determined by the consistency requirements discussed above. 4.2

Orientifold Group and Spectrum of Type-I

Let me illustrate the statements in the previous subsection in the context of Type-I theory. Let me first give the orientifold group and the closed and open string spectrum before discussing tadpole cancellation and consistency conditions. Type-I theory is an orientifold of Type-lIB theory with orientifold symmetry group

Z2 = {I, S1}. Closed String Sector:

(51)

150

The closed string sector of Type-I theory contains unoriented strings that are invariant under orientation-reversal. The massless states are simply the states of Type-lIB that are invariant under n. From §2.2 we see that only gij, c/J, j , and a symmetric combination of the two gravitini survive the projection. Open String Sector: Open string sector arises from the addition of D-branes that are required to cancel the charge of the orientifold plane. Orientation reversal is a purely worldsheet symmetry, so it leaves the entire nine-dimensional space invariant. Thus, the orientifold plane is a 9-plane. It turns out to have -32 units of charge with respect to the lO-form non-propagating field from the R-R sector. This charge can be canceled by adding 32 Dirichlet 9-branes which each have unit charge. The world-volume theory of the D9-branes gives rise to gauge group U(32) but only an 80(32) subgroup is invariant under the action of n. Type-I supergravity super Yang-Mills theory is anomaly free only if the gauge group is 80(32) or Eg x Eg. It is satisfying that the spectrum determined by the requiring worldsheet consistency is automatically anomaly free [11, 9,

B:

10].

4.3

Loop Channel and Tree Channel

A massless tadpole leads to a divergence in tree channel. For calculating tadpoles it is useful to keep a field theory example in mind. Let us consider a very massive charged particle in field theory with charge Q. At low momentum, the charge acts as a stationary source for a massless photon. One can calculate the charge Q of the particle by calculating the amplitude for vacuum going into a single photon in the background of this charge. (Figure 3) Alternatively, one can calculate the interaction between two particles each of

®--------------Q

®-------------~ 1 Q

qT

Q

Figure 3: A massless tadpole leads to a divergence in tree channel

charge Q at zero momentum exchange. The Feynman diagram has 1/q2 where q is momentum exchange and the residue is proportional to Q2. If we write 1/q2 as dlexp(-q 2 1), then the zero momentum divergence corresponds to the divergence of this integral coming from very long propagation times I. D-branes and orientifold planes can be treated similarly. A D-brane is like a very massive charged particle. The interaction between the i-th D-brane and the j-th D-brane due to closed string exchanges between the two branes

It

151

can be computed by evaluating a cylinder diagram with one boundary on the i-th brane and the other boundary on the j-th brane. In string theory, unlike in particle theory, because of conformal invariance the tree channel and loop channel diagrams are related. For example, as shown in Figure 6, the tree channel cylinder diagram can also be viewed ru3 a loop-channel diagram that evaluates the loop of an open string with one end stuck at the i-th brane and the other end at the j-th brane. Similarly, the interaction between an orientifold plane and the i-th D-brane is given by the Mobius strip diagram which has one boundary stuck at the i-th brane and one crosscap stuck at the orientifold plane. Recall that a crosscap is a circular boundary with opposite points on the boundary identified. Because some of the elements of the orientifold group leave the orientifold plane invariant, the closed string that emanates from the plane has further identifications under the symmetry and it looks like a crosscap. In summary, we can imagine that a crosscap is stuck at the orientifold plane and the boundary is stuck at a D-brane. With an orientifold with charge Q and with N D-branes of unit charge, the total charge is (Q + N)2, which can be written as Q2 + N 2 + 2Q N. The term N 2 is proportional to the interaction between the D-branes and is computed by the cylinder diagram, the interaction 2QN between the D-branes and orientifold planes is computed by the Mobius strip diagram and the interaction between orientifold planes Q2 is computed by the Klein bottle diagram. An efficient way to evaluate these diagrams is to compute them in loop channel and then factorize them in tree channel. The loop-counting parameter in string theory is the Euler character. A k-th order term in string perturbation theory which goes as the k-th power of the string coupling constant A corresponds to Riemann surfaces with Euler

Figure 4: A Surface with two boundaries, one crosscap and one handle

character k - 1. The Euler character of a Riemann surface with b boundaries, c crosscaps, and h handles is given by

x= 2-

2h - b - c.

(52)

152

A surface with no crosscaps is orient able , otherwise it is nonorientable. We are interested in the first quantum correction, i. e., Riemann surfaces with X O. There are four surfaces that contribute: a torus (one handle), a Klein Bottle (two crosscaps), a Mobius strip (one boundary, one crosscap), and a cylinder (two boundaries) ( Figure 5). Let 0"1 and 0"2 be the coordinate of the surface.

=

IIOBIUS STIIP

TORUS

CYlINDER

Figure 5: Surfaces with X

KLEIN BOTTLE

= 0 in the orientifold perturbation theory

Then we may time slice along constant 0"1 or along constant 0"2. For a tours, both two time-slicings give a loop diagram, but for the other three surfaces one time slicing give a loop diagram and the other time slicing gives a tree diagram. For these three surfaces, we would like to determine, for later use, what a closed string of length 21l' propagating for time 21l'1 in the tree channel corresponds to in the loop channel. The simplest surface is the cylinder. Consider, as shown on the left in Figure 6, a closed string of length 21l' in tree channel, propagating between two D-branes for a Euclidean time 21l'1. Time runs sideways in the diagram. Now, use the conformal invariance of string theory to conform ally rescale the coordinates by and take time to run upwards to get the diagram on the right in Figure 6. This diagram represents an open string of length 1l' beginning on one D-brane and ending on the other D-brane, propagating in a loop for a time 2m == 1l'/I. We conclude that t = 1/21 for the cylinder. For the Klein bottle, consider the double cover of the bottle, viz., a torus

ft-

153

I------l 21t

t

21t

21t 1

Tree channel

1t

Loop channel

Figure 6: Two ways to view a cylinder.

with coordinates 0 ~ 0"1 ~ 411"1 and 0 ~ 0"2 ~ 211", with the identifications '" 0"1 + 411"1 and 0"2 '" 0"2 + 211". The Klein bottle is obtained by a Z2 identification of the torus: 0"1

(53) We can choose two different fundamental regions. If we choose the fundamental region as on the top of Figure 7, then we get the tree channel diagram. It represents a closed string propagating between two orientifold planes for time 211"1. If we choose the fundamental domain taking time to run upwards now, as on the bottom of Figure 7, then we have a closed string propagating in a loop and undergoing a twist o. We have to rescale by 1/21 to obtain a closed string of length 211" in the loop, which gives t = 1/41. Similarly, for the Mobius strip we consider the double cover, viz., a cylinder with coordinates 0 ~ 0"1 ~ 411"1 and 0 ~ 0"2 ~ 211", with the identification 0"1 '" 0"1 + 411"1. The Mobius strip is obtained by the same Z2 identification as in Eq. 53. Again, we can choose two different fundamental regions. The fundamental region as on the top of Figure 8 gives the tree channel diagram which represents a closed string propagating between an orient ifold plane and

154

00

2ltl

Tree channel

,,

--------------------------~

21t!

E

o

)

4ltl

21t

Loop channel

Figure 7: Two ways to view the Klein Bottle.

a D-brane. The fundamental domain as on the bottom of Figure 8, taking time to run upwards now, represents an open string, with both ends on the D-brane, propagating in a loop and undergoing a twist Q. We have to rescale by 1/41 to obtain an open string of length 11" in the loop, which gives t = 1/81. To summarize, a closed string of length 211" propagating for time 211"1 in the tree channel corresponds to an open string of length 11", or a closed string of length 211" propagating for time 2m in the loop channel. For fixed I in the tree channel, the loop channel time t for different surfaces is given by 1

Cylinder:

t

=-

KleinBottle :

t

=-

2l 1

41

155

(J 2xl

Tree channel

21t1

,

1t~1§)~(~ o

n

4nl

Loop channel

Figure 8: Two ways to view a Mobius strip.

(54)

MobiusStrip :

We also need to know how the boundary conditions in tree channel map onto boundary conditions in loop channel. In the tree channel, ( ~ (71 ~ 21rl, ~ (72 ~ 21r) the periodicity and boundary conditions on a generic world-sheet field ¢ in the g-twisted sector (see Figure 9) are as follows:

°

°

=

¢(21rl, 1r + (72)

= Oh 2 ¢(21rl, (72)

KB:

¢(O, 1r + (72) Oh 1 ¢(O, (72), ¢( (71, 21r + (72) = li¢( (71, (72)

MS:

¢(O, (72) E Mi, ¢(21rl, 1r + (72) = Oh¢(21rl, (72) ¢( (71, 21r + (72) = li¢( (71, (72)

156

,. (

D.h l

,

a)

b)

g

.

,

.,.'",.

I

J J I

,,

J

D.h2

"\ ,'

\

\

.

J " J X J .,

I

•

," " ,

t-g

I

\

'-"

c)

I

I

J I

I I

,

,

\

\

j

Figure 9: a) Klein bottle. b) Mobius strip. c) Cylinder.

c: (55) Here Mi is the submanifold where the i-th D-brane is located. The tilde on the group elements allows for additional ± signs that depend on the GSO projection to accompany the action of the group element for world-sheet fermions. The definitions in Eq. 55 are consistent only if KB:

COhd 2 = COh 2)2

MS:

(Oh)2

c:

=

9

= g, gMi = M; (56)

otherwise the corresponding path integral vanishes. The loop channel for the Klein bottle and the Mobius strip (0 S; (71 S; 411"1, o S; (72 S; 11") is obtained geometrically by taking the uppe~ strip 11" S; (72 S; 211", inverting it from right to left, multiplying the fields by (Oh 2)-1, and gluing it to the right side of the lower strip. This construction ensures that the fields are smooth at (71 = 211"1. The periodicity conditions are KB:

MS:

¢( (71,11" + (72) = oh 2 ¢( 411"l - (71, (72), ¢( 411"l, (72) = 9' ¢(O, (72) ¢(71, 11" + (72) = Oh¢(411"l - (71, (72), ¢(O, (72) E Mi, ¢(411"1, (72) E Mi (57)

157

where !i' = Oh 2 (Ohd- 1 . Rescaling the coordinates to standard length for string in loops the respective amplitudes are KB:

cJo~.gl (nh2( _1)1+1 e1r (Lo+'i o)/21)

MS:

op'!!... op'!!.;j

C:

(nh( -1)1 e1rLo/41)

(9(-1)1 e1rLo / 1) ,

(58)

where the closed string trace is labeled by the spacelike twist g' and the open string traces are labeled by the Chan-Paton labels.

4.4 Tadpole Calculation We have followed in these lectures the formalism and notations of Gimon and Polchinski [20]. A similar formalism was used for bosonic orientifolds by Pradisi and Sagnotti in earlier work .[47]. The tadpole constraints that we are about to describe were applied to orientifolds also in refs. [32,37,5,4,49]. An equivalent but technically different method for calculating tadpoles is to construct the boundary state and the crosscap state. We do not use the boundary state method in these lectures but the details can be found in [9, 10, 11, 38]. One-loop amplitude calculates the one loop cosmological constant in spacetime as the sum of zero point energies of all the fields in the spectrum of the string. Let us look, for example, the sum of zero point energies of the fields in the open string sector:

L n;p - L n;p = - L bosons

fermions

i

2;°10 (

f d p~ log(p2 + m;) (-1 10

)F;,

(59)

)

where m; is the mass and F; the spacetime fermion number of a state i. Now we use the identity log A

f

= -lim ~A-< = -lim ~(£ ~e-21rAt) = Spin(4)J x Spin(4)E SU(2)1I x SU(2)2I X SU(2hE

X

SU(2hE,

(100)

where the subscript I is for internal, E is for external. With this embedding, the representations decompose as 8v 8s 8e

(4v, 1) Ef) (1,4v) (2s, 2s) Ef) (2e, 2e) (2s,2e) Ef) (2e, 2s)

(2,2,1,1) Ef) (1,1,2,2), (2,1,2,1) Ef) (1,2,1,2), (2,1,1,2) Ef) (1,2,2,1).

(101)

169

The orbifold group is a Z2 subgroup of SU(2)LJ which acts as -1 on the doublet representation 2. Untwisted sector: The states in the untwisted sector are obtained by keeping the Z2 invariant states of the original10-dimensional states. (8v

Ef)

8e) ® (8v

Ef)

For example, the bosons (labeled by SU(2hE are [4(1,1) ® 4(1,1)] [2(1,2) ® 2(1,2)]

Ef) Ef)

(102)

8e). X

SU(2hE quantum numbers

[(2,2) ® (2,2)] [2(2,1) ® 2(2, 1)]

(103)

This gives rise to a graviton, 25 scalars, 5 self-dual and 5 anti-self-dual 2-forms. The fermions can be obtained similarly which give the superpartners required by supersymmetry. Together, we get the gravity multiplet and five tensor multiplets. Twisted Sector: There are 16 twisted sectors coming from the 16 fixed points. The bosonic fields and fermionic fields are twisted according to their transformation property under the Z2. We see from that four fermions that transform as 2(2,1) and four bosons that transform as (2,2) are Z2 invariant and are not twisted where as the four other are twisted. The ground state energy is zero because there are equal number of bosons and fermions that are twisted. The untwisted fermions have zero modes. By steps analogous to those that led to Eq. 8 in §2.1, the zero mode algebra gives rise to a four dimensional representation (2,1) Ef) 2(1, 1). Therefore the massless representation is [(2,1)

Ef)

2(1, 1)] ® [(2,1)

Ef)

2(1, 1)]

(104)

which gives precisely the particle content of a tensor multiplet. Therefore, the twisted sector contributes 16 tensor multiplets. The massless spectrum of Type-IIB on a K3 orbifold thus consists of a gravity multiplet and 21 tensor multiplet together from the untwisted and the untwisted sector. There are 105 scalars that parametrizes the moduli space O(21,5;Z)\O(21,5;R)jO(21;R) x O(5;R). The spectrum of Type-lIB is chiral. A chiral theory can have gravitational anomalies In 4k + 2 dimensions Up to overall normalization the gravitational anomalies are

-- -..12..(trR2)2 +3 245trR4 288 60'

170

(105) Here 13 / 2 , 11 / 2 , and lA refer to the anomalies for the gravitino, a right-handed fermion, and a self-dual two-form (1,3) respectively d. To get a consistent theory, the total gravitational anomalies must cancel. This requirement is very restrictive and in fact completely determines the spectrum for the theory with (0, 2) supersymmetry. Using the formulae from Eq. 105 it is easy to check that gravitational anomalies cancel only when there are precisely 21 tensor multiplets along with the gravity multiplet [61]. This, as we have seen, is the spectrum of II-B compactified on K3.

5.4

Type IIA string on K3

Type-IIA compactified on a K3 gives a non-chiral theory in six-dimensional Minkowski spacetime with (1,1) supersymmetry. There are only two supermultiplets that are possible. 1. The gravity multiplet: a graviton (3,3), a scalar (1,1), four vectors 4(2,2), a 2-form (3,1) \17 (1,3), gravitini 2(2,3) \17 2(3, 2) two fermions 2(2,1),2(1,2) 2. The vector multiplet: a vector (2,2), four scalars 4(1,1), gauginoes 2(2,1),2(1,2). The spectrum can be found as in the previous section. Untwisted sector: Now the ten-dimensional states are (8v \17 8s) 0 (8v \17 8e).

(106)

Keeping Z2 invariant states we obtain the gravity multiplet and 4 vector multiplet. Twisted Sector: Now, the fermions that not twisted have different quantum number on the dIn 4k + 2 dimensions, the CPT conjugate of a left-handed fermion is also left-handed. Therefore, gravitational couplings can be chiral and consequently gravitational anomalies are possible. Contrast this with the 4k dimensions as in the familiar case of four dimensions where the CPT conjugate of a left-handed fermion is right-handed and a CPT-invariant theory is automatically nonchiral unless there are gauge charges in addition to gravity that distinguish between left and right.

171

left and on the right. Therefore, the representation of the fermion zero mode algebra is different on the left and the right. The massless representation are given by the product

[(1,2) EEl 2(1, 1)] Q9 [(2,1) EEl 2(1, 1)]

(107)

which gives precisely the particle content of a vector multiplet. Therefore, the twisted sector contributes 16 tensor multiplets one from each of the fixed points. The massless spectrum of Type-IIA on a K3 orbifold thus consists of the gravity multiplet and 20 vector multiplets. There are 80 scalars that parametrize the moduli space 0(20,4; Z)\0(20, 4; R)/0(20; R) x 0(4; R). 5.5

F-theory on K3

Until recently string compactifications basically solved vacuum Einstein equations in the low-energy limit for some compact manifold K, ~j

=0,

(108)

In fact, unbroken supersymmetry in the remaining noncom pact dimensions requires that the compact manifold be a Calabi-Yau manifold with SU(n) holonomy, in particular with vanishing first Chern class. Instead of solving the vacuum Einstein equations, one can imagine solving the equations with some nonzero background fields. In particular, one can ask if there are consistent solutions of the Type-lIB string where the complex field .A = X + ie- varies. When all other massless fields of Type-lIB theory are set to zero, the equations of motion for the graviton gM N and the scalar .A can be derived from the action

(109) A particularly interesting nontrivial solution of this action is Type-lIB compactified on a 2-sphere (S2 == Cpl). The spacetime of this compactification is of the form

(110) where M8 is flat Minkowski spacetime with coordinates XO, Xl, ... , X 7 and S2 is the compactification sphere with coordinates X 8 , X9. Now, the sphere which has nonzero curvature Rij =j:. O. In fact, the first Chern class of S2 is

172

the Euler character of the sphere which is nonzero, so S2 is obviously not a Calabi-Yau manifold. The way equations of motion are still satisfied is that the spacetime contains 24 7-branes. A 7-brane, as we shall see, can be thought of a special topological defect which couples to..\. The worldvolume of the 7-brane fills the noncompact M8, so in the transvers S2 it looks like a point. The energy momentum tensor 7';1 of ..\ and the metric are precisely such that they solve the Einstein equation 1 A Rij - 29ijR = Iij·

(111)

To describe the solution, let us first discuss a single 7-brane. Let z = X 8 + iX 9 be the complex coordinate on the plane transverse to the 7-brane. The coordinates XO, Xl, ... , X7 are along the worldvolume of the 7-brane. The equation of motion for ..\ that follows from Eq. 109 is

(..\ - '\)88..\ - 28..\8..\ =

o.

(112)

This equation is solved by any ..\ that is a holomorphic function of z

8..\(z, z) =

o.

Not any holomorphic function will do.

(113) Recall that ..\ parametrizes, after

SL(2, Z) identification, the fundamental domain of the moduli space of a torus (Figure 1). To get a well-defined solution we want a one-to-one map from the fundamental domain to the complex plane. We have already seen that the j-function defined in Eq. 95 gives precisely such a map. Therefore, instead of looking for ..\ as a function of z it is convenient to look for j(l) as function of z. Furthermore, the resulting configuration should have finite energy to be an acceptable solution. The simplest solution that satisfies all the requirement is

j(..\(z)) = .!:. z

(114)

which has the right properties. If we suppress 6 of the coordinates along the 7-brane (say X2, ... , X7) then the 7-brane looks like a cosmic string in four dimensions XO , Xl, X 8 , X9. This in fact is nothing but the "stringy" cosmic string solution discussed by Greene et. al.[29]. Near z = 0 j has a pole. The only pole of j is at q = exp 211"i"\ = 0 at ..\2 -+ 00. For large ..\2 we have,

j(..\) '" exp -211"i..\.

(115)

173

The solution looks like 1

(116)

,\ = -2.logz 7rZ

near z = O. If we go around the origin on a circle at infinity in the z plane with z -+ ze 2 ?ri then ,\ -+ ,\ + 1. This is very much like a global cosmic string or a vortex line in superfluid helium. A cosmic string is a topological defect in which the phase angle () of the order-parameter field has a winding number.

(117) The RR-field a in Type-II string is very similar to the phase of the order parameter () /27r. An important difference is that the total energy of global string or a vortex line has an infrared divergence because very far from the core the superfluid in a vortex lfndergoes huge rotation. By contrast, the energy density of the stringy cosmic string is finite. This is possible because the ,\ field can undergo 5L(2, Z) jumps away from the core. Near the core z = 0, the ,\ field has a nontrivial monodromy or jump under the element T of 5L(2, Z), T : ,\ -+ ,\ + 1, but far away it can undergo jumps under other elements of 5L(2, Z). The nontrivial monodromy of ,\ around the point z = 0 means in string theory that there is a 7-brane at this point that is magnetically charged with respect to the scalar ,\. Indeed, near the origin, this is exactly like a D7-brane in Type-lIB which is magnetically charged with respect to the RR scalar. In other words, it couples to the 8-form RR potential As that is dual to a,

(118)

dAs = *da.

Let us now look at the effect of the 7-brane on the metric. Because of the energy density contained in the field '\, the metric in z plane has a conical deficit near z = 0 with conical deficit angle o. The metric near such a point Zi can be found explicitly [29], and has the form ds 2 =

dzdz Iz - z;jl/6

'"

/

r- 1 6(dr 2 + r 2d'IjJ2)

'

(119)

A metric of the form (120)

can be written in the form (121)

174

with t = r1-A/(l- A). We can now redefine the angle ¢> = (1- A)~

(122)

to bring it to the standard flat metric of the plane in polar coordinates

de +

t 2 d¢>2. But then ¢> goes from 0 to 211" - 211" A. Therefore, the deficit angle is

a=

211"A which for the metric in Eq. 119 is 11"/6. The deficit angle measures the total curvature or = f R where R is the Ricci scalar. So far, Z is a coordinate on the noncompact complex plane. If we put precisely 24 7-branes on the plane then the plane curls up into a sphere. This is because the total contribution to the deficit angle from all the cosmic strings now adds up to 411" to make up the solid angle of a sphere. For the sphere, the Euler character 2 and f R 211"X 411". A sphere with a collection of 24 F-theory 7-branes is a compact manifold. In fact, one can associate with it an elliptically fibered K3, with the sphere as the base, if the T parameter of the fiber torus is identified with A. The elliptic K3 is described by Eq. 99 in §5.3

a

=

y2

=

= x 3 + J(z)x + g(z).

(123)

where J and 9 are polynomials of degree 8 and 12 respectively. The locations of 7-branes are determined by the zeroes of the discriminant .6. == 4P + 27g2 which is the denominator of the j function. Since J is a polynomial of order 8 and 9 is a polynomial of order 12, there will, in general, be 24 zeroes of the discriminant which correspond to the locations of the 24 7-branes. Near every zero Zi, we have

j(A) '" (z 1

~ Zi)

1 q

A(Z) :::::: -2. log(z - Zi), 1I"Z

(124)

(125)

corresponding to a single 7-brane. The compactification of Type-lIB of S2 that we have described has been called an 'F-theory' compactification on K3. 'F-theory' refers to a possible 12dimensional theory which when compactified on T2 would give Type-lIB. In general, consider an elliptically fibered Calabi-Yau manifold K which is a fiber bundle over a base manifold B with a torus as a fiber whose complex structure parameter is T. Even-though K is a smooth manifold, there will be points in the base manifolds where the fiber becomes singular, and the parameter T can have a nontrivial SL(2, Z) monodromyaround these points. An F-theory compactification on K refers to a compactification of Type-lIB theory on B, where the coupling A is identified with T.

175

6 6.1

Applications of Orientifolds to Duality General Remarks on Duality

Apart from T-duality, two other dualities will be relevant to us in the following discussion. Both are 'strong-weak' dualities which relate the strong coupling limit of a ten-dimensional string theory to the weak coupling limit of the dual theory. (1) Duality between the Type-I string and the SO(32) Heterotic string. The massless spectrum as well as the form of the low energy effective action agrees [66] under the duality transformation which takes the coupling constant of Type-I string to the inverse coupling constant of the Heterotic string. The spectrum of some of the solitons in the two theories has also been checked to be in agreement with duality [12, 35, 43]. (2) Self-duality of the Type-lIB string. We have already discussed this. The element S of the SL(2, Z) duality group takes the coupling constant to the inverse coupling constant in the dual theory. For a detailed discussion of the evidence for these dualities, see [53]. In this section we would like to use these dualities, T-duality, and our knowledge of orbifolds and orientifolds, to deduce two more dualities in lower dimensions. 6.2

Duality of Type-IIA on K3 and Heterotic on T4

The main principle that we use in this subsection is 'fiberwise application of duality', which we explain below. Consider a theory A compactified on KA that is dual to another theory B compactified on KB. This duality can be used to deduce some further dualities. Consider A and B compactified respectively on EA and EB, which are obtained by fibering KA and KB over~. By this, we mean that locally EA looks like KA x ~. The moduli mo: of the fiber KA can vary as a function of the coordinates of the base manifold~. As long as the moduli of KA vary slowly we expect to be able to use the original duality to derive a new duality between A on EA and B on E B . There are two possibilities that aries. (i) The first possibility is that the fiber is smooth at all points on the base manifold~. In this case, the duality between A on EA and B on EB follows from the 'adiabatic argument' of Vafa and Witten [64]. The idea is that we can choose the size of ~ to be very large. Then locally A compactified on EA has KA X M n as the target spacetime and and B on EB has KB X M n as the target spacetime. Knowing the duality between the two, we can assert the new duality. The fibered structure will become apparent to a local observer only after circumnavigating the (very large) manifold ~ and so will not be relevant

176

to local physics. We can thus establish the duality between A on EA and B on EB in the limit of large~. Now, if we adiabatically reduce the volume of ~2 we expect that the duality will continue to hold. (ii) The second possibility is that the fiber is smooth everywhere on ~ except at a few discrete points where it degenerates ( Figure 13). The total

Figure 13: A torus degenerates into a sphere by pinching a handle.

manifold EA can still be smooth, it is only the fibration that becomes singular. In this case, the adiabatic argument is strictly not applicable. Near the points where the fiber degenerates, the argument breaks because the moduli vary rapidly. However, in a large number of examples constructed so far, the resulting theories do appear to be dual as long as the number of singular points is a set of "measure zero". Heuristically, even with the singular points, duality is forced by the duality in the bulk. It is instructive to apply these arguments to the special case of fibered manifolds that are orbifolds. Take a smooth manifold M with a Z2 symmetry {I, s}. Let KA and KB have some Z2 symmetry {I, hAl and {I, hB}. Take (126)

Here there are two possibilities. (i) If s has no fixed points on M, then we have the possibility (i) above. The orbifold EA can be viewed at all points as a fiber bundle with ~ =

M/ {I, s} as the base space and KA as the fiber. The fiber is smooth everywhere. There is a twist hA along the fiber as we move along a closed curve from a point p on 1: and its image s(p). (ii) If s leaves some points on M fixed, then we have the possibility (ii). The orbifold EA still has the structure of a fiber bundle with base manifold 1: = M/{l,s} and fiber KA everywhere except the fixed points. At the fixed points the fiber degenerates to KA/{l, hAl giving us a singular fibration. e As an illustration of the second possibility we now 'derive' the duality between Type-IIA on K3 and Heterotic on T4. Let us take A to be the TypelIB theory in ten dimensions and B to be the Type-lIB theory that is S-dual to A (KA and KB are null sets). Compactify both sides on a torus T4 with coordinates X 6, X7, X 8, X 9, i.e., M = T4. Now, take hA = (_1)FL, hB = 0, and s to be the reflection We have used the duality relations

s JIB -+ JIB,

(127)

from §2.3. Therefore, from the earlier arguments, we get the duality

T4

T4

lIB on {I, (-1)FLh789} == lIB on {1,Oh789}

(128)

Now we can use T-dualities to turn these orientifolds into more familiar ones. We use two observations. (1) If we T -dualize one of the coordinates, say X 6 , we get Type-IIA theory on the dual circle. Moreover, the symmetry 16789 ( -1 )FL in lIB maps under this T-duality onto h789 in the dual IIA. To see this, recall that if we T-dualize in a direction Xi

(129) then the left-moving fermions transform as

(130) eThere is a third possibility that 8 leaves the entire manifold M invariant. In this case the resulting orbifolds do not have the fibered structure even locally. In such a situation orbifolding does not commute with duality [56, 55] and cannot be used to deduce new dualities. For example, Type-I is an orientifold of Type-lIB by {l,n} and Type-IIA is an orbifold of Type-lIB by {I, (_l)FL }. Now, n is conjugate to (_1)FL by the element S of the SL(2, Z) group but Type-I is not dual to Type-IIA.

178

where Pi = rri. Now using the properties ofthe r matrices, we see that Pi does not square to identity but instead Pl = (-1 )FL. Furthermore Pi Pj = - Pj Pi. Note that the reflection h789 acts as P6P7P8P9sa on the left-moving fermions, and similarly on the right-moving fermions. Using the properties of Pi matrices we see that (131) Therefore, the orbifold Type-lIB on T 4/{I, (_I)FL h789} is T-dual to Type-IIA on the K3 orbifold T4/{1, h789}. (2) T-duality of all four coordinates maps h789n in lIB into n in lIB by a reasoning similar to the above. (132) Hence, Type-lIB on the orientifold T4/{1,n/6789} is T-dual to Type-lIB on the orientifold T4/ {I, n}, which is nothing but Type-Ion T4. From the duality between Type-I and Heterotic in ten dimensions, it is dual to Heterotic on T4. We conclude from Eq.128 and points (A) and (B) above that IIA on K3 == Heterotic on T4.

(133)

This equivalence has been established at a special point in the moduli space where the K3 becomes the orbifold T4 /Z2.' The duality gives a one-one map between the massless fields on both sides. By giving expectation values to the massless scalar field we can move around in the moduli space and establish the duality at all points in the moduli space. Thus, starting with the 8L(2, Z) duality ofII-B with 32-supercharges (and the duality between Type-land Heterotic), we can get all the structure of the more interesting 'string-string' duality [36] between II-A on K3 Heterotic on T4 with 16-supercharges. This is quite an explicit construction, and all we needed to do was to keep track of a few discrete symmetries and follow the orientifold and orbifold construction. Let us quickly check if the spectrum of Heterotic on T4 matches with the dual Type-IIA spectrum. The moduli space of Heterotic on T4 is the Narain moduli space [41] 0(20,4, Z)\0(20, 4, R)/0(20, R) x 0(4, R) which is identical to the moduli space of Type-IIA on K3. At a generic point in the moduli space, the gauge group SO(32) is broken to U(1)16. In six dimensions we get 4 vector bosons 9I'm, 4 vectors from B I'm, and 16 vectors from the original gauge fields in ten dimensions, A~ (I = 1, ... ,16 is the gauge index, m = 6, ... ,9 is the internal index, and J.l is the Minkowski index J.l = 0.... ,5).

179

Altogether there are 24 vector bosons, exactly as in the case of TypeIIA (§5.4), which transform in the vector representation of the duality group 80(4,20, R). I shall now describe one more duality with 16 supersymmetries before moving onto theories with 8 supercharges. 6.3

Duality oj F-theory on K3 and Heterotic on T2

Another interesting application of orientifolds is in connection with F-theory. In this subsection we concern ourselves with F-theory compactification on K3 to eight-dimensional Minkowski spacetime, but these considerations are applicable to more general compactifications. We have seen in §5.5 that to obtain an F-theory compactification, we start with an elliptically fibered K3 that is described by y2

= x3

+ J(z)x + g(z),

(134)

where J is a polynomial of degree 8 and 9 is a polynomial of degree 12. Such a K3 represents 24 stringy cosmic strings on a 2-sphere located at the 24 points where the torus degenerates. Typically, the coupling constant field A will vary as we move from point to point in the base manifold. Consequently, there will be a non-vanishing RR background. Moreover, the field is allowed to undergo 8L(2, Z) jumps. Some of the elements of the 8L(2, Z) like 8 are nonperturbative. Therefore, for a generic K3, such backgrounds cannot be described perturbatively as conformal field theories. There is a special limit of the K3 for which the modular parameter A of the fiber torus does not vary as we move on the sphere. This is achieved by choosing J3 /9 2 = constant. 9 = 3

J = 0:2

(z) = polynomial of degree 4

by rescaling y and x we can set = given by

I1!=1 (z -

Za). Then the j function is

4(240:)3 ..,...,...:--,-:--::- = constant, 27 + 40:3

j(A)

(135)

therefore, A, which is the image under j-l, is also a constant at all points over the sphere. However there is a nonzero 8L(2, Z) monodromy

R=

-1 ( 0

(136)

180

as we go around a point Z = Za. This is the hyperelliptic involution of the torus which reflects both periods of the torus without changing its modular parameter. The discriminant is 4

~

=

(4a 3

+ 27) II (z -

(137)

Za)6

a=l

which shows that the 24 7-branes are bunched in groups of six at four points {za}. The metric on the base can be read of from 119 by putting 6 7-branes at one point. (138) There is a conical deficit angle of'Tr at four points, otherwise the metric is flat. In other words, the base is T 2/Z 2 and the fiber is T2 at all points except at the fixed points. It is easy to see that this K3 that is nothing but the orbifold T 4 /Z 2 • Conversely, the K3 orbifold T4/Z2 that we constructed in §5.1 can be viewed as an elliptic fibration over S2. The K3 has coordinates (Zl' Z2) and the orbifold symmetry is (Zl' Z2) '" (-Zl' -Z2). It can be viewed as T4 T~l) x T~2) - Z2 - {1, R1R2}

(139)

Let us take Zl to be the coordinate of the first torus T~, and Z2 to be the coordinate ofthe base. Let Ri be the operation Zj --t -Zj. Then the orbifold Eq. 139 can be viewed as a fiber bundle with T~ as the fiber and TV{l, R2} as the base. The base manifold T2/Z 2 is nothing but a sphere. To see this note that the Z2 symmetry acts as Z2 --t -Z2 has four fixed points each with deficit angle 1800 'Tr. So the total deficit angle is 4'Tr giving us f R 4'Tr i.e. the correct Euler character 2 for a sphere S2. So locally, away from the fixed points of R2 the orbifold looks like T~l) x S2 If we go around a fixed point of R2 then the coordinate Zl of the T~l) is twisted by Rl, i.e., it is inverted (Zl --t -zt), but its modular parameter A is unchanged. This is precisely the Z2 monodromy R in Eq. 136. Therefore, in this limit, for this special configuration of 7-branes, the field A is constant everywhere in space and this F-theory compactification can be described as an perturbative orientifold [57]. Such an identification of F-theory with an orientifold is very useful. F-theory on the K3 orbifold above is nothing but the orientifold

=

=

T2

T2

IIB on {1, RIs9} == {1, O(-1)FLIs9} '

(140)

181

if we identify Rl in Eq. 139 with R = O( _l)FL and R2 in Eq. 139 with 189 . There are 4 orientifold fixed planes and 16 7-branes are required to cancel the charge of the orient ifold planes. This fact is obvious if we note further that after T-dualizing in the 89 directions, O(-l)FLI89 goes to 0: (141) Therefore, this orientifold Eq. 140 is T-dual to the orientifold T2 JIB on {I, O}'

(142)

which is nothing but Type-I theory compactified on T2. After two T-dualities the 32 9-branes turns into 7-branes. Because of the identification 189 they have to move in pairs so effectively there are only 16 of them on the orientifold. Thus, at this special point in the moduli space, F-theory is nothing but a T-dual of Type-Ion T2 which in turn is dual to Heterotic on T2. We have thus established the duality F theory on K3

== Heterotic on T2.

(143)

Apart from its use in understanding this duality, the orientifold limit of Ftheory has other very interesting applications. To obtain the orientifold limit, we had to place sixteen 7-branes at the four orientifold planes in four bunches offour. On the other hand, on the F-theory side, we have 24 F-theory 7-branes in four bunches of six. What happens is that as we move the D-7branes away from the orientifold 7-branes, then the orientifold 7-plane splits nonperturbatively into two 7-planes [57]. Thus, in F-theory, the orientifold planes and the D-branes are on an equal footing and are related nonperturbatively. An orientifold plane and four D7-branes turn into six F-theory 7-branes. This splitting of the orientifold plane is very similar to the splitting of the SU (2) point into the monopole point and the dyon point in Seiberg-Witten theory in 3+1 dimensions with gauge group SU(2) and four quark flavors [52). This similarity is not an accident but a precise consequence of using D3-brane probes to probe the geometry near the orientifold plane. The world volume theory of D3-brane probe near an orientifold plane and four D-7branes has exactly the same structure as a Seiberg-Witten theory [3]. 7

Orientifolds in Six Dimensions with (0,1) Supersymmetry

One important application of orientifolds is in the construction of models in six dimensions with (0,1) supersymmetry which has only 8 supercharges. With

182

only 8 supercharges, instead of 16 or 32, supersymmetry is much less restrictive and therefore much more interesting dynamics is possible. At the same time, supersymmetry is still sufficiently restrictive to be a useful guide for checking the properties of these theories as well their possible duals. The massless supermultiplets of (0,1) supersymmetry in terms of representations of the little group SU(2) x SU(2) are as follows: 1. The gravity multiplet: a graviton (3, 3), a self-dual two-form (1,3)' a gravitino 2(2, 3). 2. The vector multiplet: a gauge boson (2, 2), a gaugino 2(1,2). 3. The tensor multiplet: an anti-self-dual two-form (3,1), a fermion 2(2,1), a scalar (1,1). 4. The hypermultiplet: four scalars 4(1,1), a fermion 2(2,1). Cancellation of gravitational anomalies places restrictions on the matter content. Consider V vector multiplets, H hypermultiplets and T tensor multiplets. Then the (trR4) term in the anomaly polynomial Eq. 105 cancels only if H - V = 273 - 29T.

(144)

The (tr R2)2 term is in general nonzero, and needs to be canceled by the GreenSchwarz mechanism. For example, if we compactify the heterotic string or 1 and H V + 244. Type-I string on a smooth K3 we obtain T The dynamics of (0,1) theories in six dimensions offers many surprises like the possibility of exotic phase transitions in which the number of (chiral) tensor multiplets changes, or the appearance of enhanced gauge symmetry when an instanton shrinks to zero scale size. Given the limitations of time it is not possible to discuss the detailed construction in the same depth as we did in earlier sections. In this section I briefly survey two interesting phenomena: (i) enhanced gauge symmetry and appearance of USp(2k) symmetry, and (ii) appearance of multiple tensor multiplets, and describe the utility of orientifolds in this context. To organize the discussion, a useful starting point is the heterotic string compactified on [(3. To begin with, in ten dimensions there are two consistent heterotic strings that have N = 1 supersymmetry.

=

=

183

(1) The heterotic string with gauge group SO(32): The strong coupling limit of this theory is Type-I string. We have already used this duality in §6. (2) The heterotic string with gauge group Es xEs: The strong coupling limit of this theory is M-theory on MID x (S1 jZ2) [34]. The generator of Z2 reflects the coordinate of the circle so the resulting orbifold (S1 j Z 2) is nothing but a line segment I : [0, 7T R]. MID is ten-dimensional Minkowski spacetime. The two Es factors live on the two 'end of the world' boundaries of this manifold. Gravity lives in the bulk and on the boundary. In the bulk we have eleven dimensional supergravity which corresponds to N = 2 Type-IIA supergravity in ten dimensions upon dimensional reduction on a circle. Boundary conditions break it to N = 1 supergravity of heterotic string. The supergravity sector of the heterotic string is identical to the Type-I theory. It contains, along with a graviton, a dilaton, and the fermions, a 2-form field B with field strength H which satisfies the Bianchi identity dH = tr(R/\ R) - tr(F /\ F),

(145)

where R is the curvature 2-form and F is the gauge field strength 2-form. Integrating the Bianchi identity gives the constraint that the integral of the right hand side of Eq. 145 should vanish on a manifold without a boundary. In particular, when we compactify the string on K3, the integral of the right hand side over K3 should vanish. Now JK3 tr(R /\ R) is the Euler character of K3 which equals 24. Therefore, the integral JK3 tr(F /\ F), which is the Pontryagin index or the instanton number of the gauge field on K3, should also equal 24. A consistent heterotic compactification on K3 is possible only if the gauge field is also nontrivial such that there are 24 instantons on K3. An instanton on K3 looks like a soli tonic 5-brarie [60] in ten dimensions that fills flat six-dimensional Minkowski spacetime. The heterotic compactification can become singular in certain limits. One singularity that will concern us here is when the scale size of an instanton shrinks to zero. When a 5-brane instanton shrinks, the geometry in the transverse space becomes singular and it develops a long throat. Deep down the throat the coupling constant grows and the perturbative description of the 5-brane breaks down. We would like understand what happens near such a singularity. From our experience with the moduli space of supersymmetric gauge theories and string theories, typically a singularity in the moduli space indicates that additional states are becoming massless at that point in the moduli space. The singularity occurs because we have incorrectly integrated out these states

184

that are massless. We expect that the singularities in the instanton moduli space also has a similar physical explanation. The small instanton singularity has a completely different physical resolutions in the two heterotic strings, each remarkable in its own way. (i) Small instantons in SO(32) heterotic string: If there are k small instantons that coincide, then there are additional massless gauge bosons which give rise to an additional USp(2k) gauge symmetry. In heterotic theory, this remarkable conclusion was arrived at from considerations of the ADHM construction of instantons[68]. This phenomenon, which cannot be described in a weakly coupled conformal field theory in heterotic compactifications, has a simple perturbative description in terms of a Dirichlet 5-brane in the dual Type-I orientifold theory. In particular, the enhanced USp(2k) symmetry when k small instantons coincide can be understood in terms of coincident 5-branes with a specific symplectic projection in the open string sector that is determined by the consistency of the world-sheet theory. (ii) Small instantons in Es x Es heterotic string: These are even more unusual to understand. The picture is clearer in the dual M-theory. Consider the dual compactification of M-theory on K3 x (S1 /Z2). The gauge fields and the corresponding instantons in the two Es factors are confined to the two boundaries. M-theory contains a solitonic 5-brane which carries unit charge exactly like the heterotic instanton 5-brane. When one of the instanton 5-brane in the boundary Es gauge theory shrinks to zero size, the resulting singularity corresponds to an M-theory solitonic 5-brane stuck to the boundary. One of the possibilities consistent with the Bianchi identity of H and anomaly cancellation is that the M-theory 5-brane can be emitted from the 'end of the world' into into the bulk [51]. The worldvolume theory of an M-theory 5-brane contains a tensor multiplet in its worldvolume theory [8]. The 5-brane fills the noncompact six-dimensional space-time M6. Therefore, in the process of emission of an M-theory 5-brane into the bulk, the number of tensor multiplets in the six-dimensional theory can increase by one. Multiple tensor multiplets are not possible with usual Calabi-Yau compactifications. However, as we shall see in §7.2, one can easily construct orientifolds that have this property. 7.1

Symplectic Gauge Groups

Let us consider Type-lIB theory on a K3 orbifold T 4 /Z2 and orientifold this theory further by {I, O} to obtain Type-I theory on the K3 orbifold. Details of this orientifold can be found in [20]. Here we shall point out some of the

185

salient features. (a) The orientifold group is (146) (b) Fixed planes of n are orientifold 9-planes filling all space, which are identical to those in Type-I theory in ten dimensions discussed in §4. The charge of the orientifold plane is -32 as before requiring 32 D9-branes to cancel it. Fixed planes of h789n are orientifold 5-planes located at the 16 fixed planes of h789 each of charge -2. The net charge is again -32 requiring 32 D5-branes. (c) There are now four open string sector: 55, 59, 95, 99, depending on what type of brane the two ends of an open string end on. (d) A D-5 brane has the same charge as a small instanton and is dual of the soli tonic 5-brane in the heterotic string. (e) Tadpole cancellation determines the matrix 'YO,9, which implements the n projection in the 99 sector, to be symmetric as in §4. Therefore, the gauge group of the 32 9-branes is an orthogonal subgroup of U(32) after the n projection. This group is further broken by the h789 projection. (f) Consistency requirements determine that the matrix 'YO,5, which implements the n projection in the 55 sector, must be antisymmetric if 'YO,9 is symmetric. This follows from a somewhat subtle argument by [20] that involves considerations of factorization and the action of n 2 in the 59 sector. We do not repeat the argument here and refer the reader to [20] for details. When 2k 5-branes coincide, we get a symplectic subgroup U Sp(2k) of U (2k) after the n projection, by the arguments discussed at the end of §4.5. Thus, the small instantons and the enhanced U Sp(2k) gauge symmetry have a very simple perturbative description in terms of D 5-branes in Type I. There are many interesting aspects of this model which have been analyzed in great detail in [6]. A nonperturbative description using F-theory can be found in [58]. 7.2

Multiple tensor Multiplets

We now describe models with (0,1) supersymmetry with multiple tensor multiplets. With conventional Calabi-Yau compactifications, the only way to obtain (0,1) supersymmetry is to compactify either the Heterotic or the Type-I theory on a K3. These compactifications give only a single tensor multiplet. For Type-II strings, compactification on a K3 leads to N = 2 supersymmetry as we saw in §5.3. One cannot obtain lower supersymmetry with Calabi-Yau compactification. One way to reduce supersymmetry further is to

186

take an orientifold so that only one combination of the left-moving and the right-moving supercharges that is preserved by the orientation-reversal survives. If we wish to obtain a large number of tensor multiplets, a natural starting point for orientifolding is the Type-lIB theory compactified on K3 which has 21 tensor multiplets of (0,2) supersymmetry. A tensor multiplet of (0,2) supersymmetry is a sum of a tensor-multiplet (T) and a hyper-multiplet (H) of (0, 1) supersymmetry. If we use the projection (1+0)/2 then we get Type I theory on K3. Under 0, the 4-form Dijkl and the 2-form Bij are odd. Therefore all zero modes of these fields are also projected out. Only the zero modes of the field j survive which gives one self-dual and one anti-self-dual tensor in six dimensions. The self-dual tensor is required in the gravity multiplet. So, we end up with a single tensor multiplet. This counting suggests a generalization. If the K3 orbifold has a Z2 symmetry with generator S, then we can consider taking {I, OS} as the orientifold group. If we want N = 1 supersymmetry, then the requirement is that S should not break supersymmetry further. This is ensured if it leaves the holomorphic two-form of the K3 invariant. But, S can have nontrivial action on other harmonic forms of the K3. Some of the zero modes of Dijkl, can be even with respect to the combined action of OS even if they are odd with respect to 0 alone. These can give rise to additional tensor multiplets that we are interested in. Concretely, let us consider an example of such a symmetry discussed in [14]. Consider a K3 orbifold T 4 /Z 2 that we have been discussing in §5. Such a K3 admits a Z2 involution

B:

S

(Zl' Z2) --+ (-Zl

1

1

+ 2' -Z2 + 2)·

(147)

S leaves the holomorphic 2-form dz 1 1\ dz 2 of the K3 orbifold, and in fact all forms coming from the untwisted sector of the orbifold invariant. Let us look at its action on the twisted sector. It is easy to see from Figure 10 that Stakes the 16 fixed points of h789 into each other so out of the 16 a anti-self-dual 2-forms coming from the twisted sector, eight are odd and eight are even. To obtain an anti-self-dual 2-form in six dimensions as a zero mode of the 4-form in ten dimensions, we use separation of variables to write the 4form as D(4) = B~2) 1\ !';, 0: = 1, ... ,19 where f~ is one of the anti-self-dual harmonic 2-forms on K3 which depends only on the coordinates of K3 and B~2) depends only on the non-compact coordinates. Because f~ is harmonic, B~2) is a massless field in six dimensions. By the self-duality of D(4) in ten dimensions, B~2) is anti-self-dual in the six Minkowski dimensions. Now, if

187

we use the combined projection, (1+~S) instead of (1~n) then eight tensors coming from the eight f~ 's that are odd under S survive and the remaining are projected out. In addition there is one more tensor multiplet that comes from the zero mode of j as in Type-I. Altogether, we get T = 9. The orientifold group is

B:

(148) We shall not discuss the open string sector here but it can be found in [14]. One interesting aspect of models with multiple tensors is worth mentioning. The cancellation of gauge and gravitational anomalies in these models requires an extension of of the Green-Schwarz mechanism found by Sagnotti [49] in which more than one tensors participate in the anomaly cancellation. Details of anomaly cancellation for the model described above can be found in [14]). The model has an M-theory dual [54] that makes use of the observation that Type-lIB on K3 is dual to M-theory on T 5 /Z 2 [18, 67]. There are a number of other ways to obtain multiple tensors in string models. Orientifolds of K3 orbifolds where the orbifold group is other than Z2 typically give multiple tensors [24, 15, 25]. Yet another interesting variation is to accompany the action of n with additional phases in the twisted sectors of the Z2 orbifold symmetry [45, 7, 16,27]. This is the analog of discrete torsion for ZN x ZN orbifolds [63]. Acknowledgments

I would like to thank the organizers of the ICTP Summer school for inviting me deliver these lectures, and Rajan Pawar and M. R. Shinde for help with figures and with TeXing the manuscript. References

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Lectures on D-branes, Gauge Theory and M(atrices)* Washington Taylor IV Department of Physics, Princeton University, Princeton, New Jersey 08544, U.S.A.

vatiOPrinceton.edu These notes give a pedagogical introduction to D-branes and Matrix theory. The development of the material is based on super Yang-Mills theory, which is the lowenergy field theory describing multiple D-branes. The main goal of these notes is to describe physical properties of D-branes in the language of Yang-Mills theory, without recourse to string theory methods. This approach is motivated by the philosophy of Matrix theory, which asserts that all the physics of light-front Mtheory can be described by an appropriate super Yang-Mills theory.

Table of Contents 1. Introduction 2. D-branes and Super Yang-Mills Theory 3. D-branes and Duality (T-duality and S-duality in D-brane SYM theory) 4. Branes and Bundles (Constructing (p ± 2k)-branes from p-branes) 5. D-brane Interactions 6. M(atrix) theory: The Conjecture 7. Matrix theory: Symmetries, Objects and Interactions 8. Matrix theory: Further Developments 9. Conclusions 1

1.1

192 199 203 213 224 239 244 258 263

Introduction

Orientation

In the last several years there has been a revolution in string theory. There are two major developments responsible for this revolution. i. It has been found that all five string theories, as well as ll-dimensional supergravity, are related by duality symmetries and seem to be aspects of one underlying theory whose fundamental principles have not yet been elucidated. ·Based on lectures given at the Trieste summer school on particle physics and cosmology, June 1997

193

ii. String theories contain Dirichlet p-branes, also known as "D-branes". These objects have been shown to playa fundamental role in nonperturbative string theory. Dirichlet p-branes are dynamical objects which are extended in p spatial dimensions. Their low-energy physics can be described by supersymmetric gauge theory. The goal of these lectures is to describe the physical properties of D-branes which can be understood from this Yang-Mills theory description. There is a two-fold motivation for taking this point of view. At the superficial level, super Yang-Mills theory describes much of the interesting physics of Dbranes, so it is a nice way of learning something about these objects without having to know any sophisticated string theory technology. At a deeper level, there is a growing body of evidence that super Yang-Mills theory contains far more information about string theory than one might reasonably expect. In fact, the recent Matrix theory conjecture 1 essentially states that the simplest possible super Yang-Mills theory with 16 supersymmetries, namely N = 16 super Yang-Mills theory in 0 + 1 dimensions, completely reproduces the physics of eleven-dimensional supergravity in light-front gauge. The point of view taken in these lectures is that many interesting aspects of string theory can be derived from Yang-Mills theory. This is a theme which has been developed in a number of contexts in recent research. Conversely, one of the other major themes of recent developments in formal high-energy theory has been the idea that string theory can tell us remarkable things about low-energy field theories such as super Yang-Mills theory, particularly in a nonperturbative context. In these lectures we will not discuss any results of the latter variety; however, it is useful to keep in mind the two-way nature of the relationship between string theory and Yang-Mills theory. The body of knowledge related to D-branes and Yang-Mills theory is by now quite enormous and is growing steadily. Due to limitations on time, space and the author's knowledge there are many interesting developments which cannot be covered here. As always in an effort of this sort, the choice of topics covered largely reflects the prejudices of the author. An attempt has been made, however, to concentrate on a somewhat systematic development of those concepts which are useful in understanding recent progress in Matrix theory. These lectures begin with a review of how the low-energy Yang-Mills description of D-branes arises in the context of string theory. After this introduction, we take super Yang-Mills theory as our starting point and we proceed to discuss a number of aspects of D-brane and string theory physics from this point of view. In the last lecture we use the technology developed in the first four lectures to discuss the recently developed Matrix theory.

194

Figure 1: D-branes (gray) are extended objects on which strings (black) can end

1.2

D-branes from string theory

We now give a brief review of the manner in which D-branes appear in string theory. In particular, we give a heuristic description of how supersymmetric Yang-Mills theory arises as a low-energy description of parallel D-branes. The discussion here is rather abbreviated; the reader interested in further details is referred to the reviews of Polchinski et a1.2 ,3 or to the original papers mentioned below. In string theory, Dirichlet p-branes are defined as (p + I)-dimensional hypersurfaces in space-time on which strings are allowed to end (see Figure 1) . From the point of view of perturbative string theory, the positions of the Dbranes are fixed, corresponding to a particular string theory background. The massless modes of the open strings connected to the D-branes can be associated with fluctuation modes of the D-branes themselves, however, so that in a full nonperturbative context the D-branes are expected to become dynamical p-dimensional membranes. This picture is analogous to the way in which, in a particular metric background for perturbative string theory, the quantized closed string has massless graviton modes which provide a mechanism for fluctuations in the metric itself. The spectrum of low-energy fields in a given string background can be simply computed from the string world-sheet field theory 4 . Let us briefly review the analyses of the spectra for the string theories in which we will be interested. We consider two types of strings: open strings, with endpoints which are free to move independently, and closed strings, with no endpoints. A superstring theory is defined by a conformal field theory on the (1 + 1)dimensional string world-sheet, with free bosonic fields XI' corresponding to the position of the string in 10 space-time coordinates, and fermionic fields '1/;1' which are partners of the fields XI' under supersymmetry. Just as for the classical string studied in beginning physics courses, the degrees of freedom on the open string correspond to standing wave modes of the fields; there are twice as many modes on the closed string, corresponding to right-moving and

195

left-moving waves. The open string boundary conditions on the bosonic fields XJJ can be Neumann or Dirichlet for each field separately. When all boundary conditions are Neumann the string endpoints move freely in space. When 9 - p of the fields have Dirichlet boundary conditions, the string endpoints are constrained to lie on a p-dimensional hypersurface which corresponds to a D-brane. Different boundary conditions can also be chosen for the fermion fields on the string. On the open string, boundary conditions corresponding to integer and half-integer modes are referred to as Ramond (R) and NeveuSchwarz (NS) respectively. For the closed string, we can separately choose periodic or antiperiodic boundary conditions for the left- and right-moving fermions. These give rise to four distinct sectors for the closed string: NS-NS, R-R, NS-R and R-NS. Straightforward quantization of either the open or closed superstring theory leads to several difficulties: the theory seems to contain a tachyon with M2 < 0, and the theory is not supersymmetric from the point of view of ten-dimensional space-time. It turns out that both of these difficulties can be solved by projecting out half of the states of the theory. For the open string theory, there are two choices of how this GSa projection operation can be realized. These two projections are equivalent, however, so that there is a unique spectrum for the open superstring. For the closed string, on the other hand, one can either choose the same projection in the left and right sectors, or opposite projections. These two choices lead to the physically distinct IIA and lIB closed superstring theories, respectively. From the point of view of 10D space-time, the massless fields arising from quantizing the string theory and incorporating the GSa projection can be characterized by their transformation properties under spin(8) (this is the covering group of the group SO(8) which leaves a lightlike momentum vector invariant). We will now simply quote these results from [4]. For the open string, in the NS sector there is a vector field A JJ , transforming under the 8v representation of spin(8) and in the R sector there is a fermion 1/J in the 8. representation. The massless fields for the IIA and lIB closed strings in the NS-NS and R-R sectors are given in the following table: NS-NS IIA lIB

rP, BJJv 9 JJV , rP, B JJV 9JJv,

R-R A~),A~~p A(O),A~~,A~~pa

The IIA and lIB strings have the same fields in the NS-NS sector, corresponding to the space-time metric 9JJv, dilaton rP and antisymmetric tensor field B JJv • In addition, each closed string theory has a set of R-R fields. For the IIA theory there are I-form and 3-form fields. For the lIB theory there is a second

196

scalar field (the axion), a second 2-form field, and a 4-form field A(4) which is self-dual. The NS-NS and R-R fields all correspond to space-time bosonic fields. In both the IIA and lIB theories there are also fields in the NS-R and R-NS sectors corresponding to space-time fermionic fields. Until recently, the role of the R-R fields in string theory was rather unclear. In one of the most important papers in the recent string revolution 5, however, it was pointed out by Polchinski that D-branes are charge carriers for these fields. Generally, a Dirichlet p-brane couples to the R-R (p + I)-form field through a term of the form

(1) where the integral is taken over the (p + I)-dimensional world-volume of the p-brane. In type IIA theory there are Dirichlet p-branes with p = 0,2,4,6,8 and in type lIB there can be Dirichlet p-branes with p = -1,1,3,5,7,9. The D-branes with p > 3 couple to the duals of the R-R fields, and are thus magnetically charged under the corresponding R-R fields. For example, a Dirichlet 6-brane, with a 7-dimensional world-volume, couples to the 7-form whose 8-form field strength is the dual of the 2-form field strength of A(1). Thus, the Dirichlet 6-brane is magnetically charged under the R-R vector field in IIA theory. The story is slightly more complicated for the Dirichlet 8-brane and 9-brane 3 j however, 8-branes and 9-branes will not appear in these lectures in any significant way. In addition to the Dirichlet p-branes which appear in type IIA and lIB string theory, there are also solitonic NS-NS 5-branes which appear in both theories, which are magnetically charged under the NS-NS two-form field B,.,v. In the remainder of these notes p-branes which are not explicitly stated to be Dirichlet or NS-NS are understood to be Dirichlet p-branesj we will also sometimes use the notation Dp-brane to denote a p-brane of a particular dimension. It is interesting to see how the dynamical degrees of freedom of a D-brane arise from the massless string spectrum in a fixed D-brane background 6. In the presence of a D-brane, the open string vector field A,., decomposes into components parallel to and transverse to the D-brane world-volume. Because the endpoints of the strings are tied to the world-volume of the brane, we can interpret these massless fields in terms of a low-energy field theory on the Dbrane world-volume. The p + 1 parallel components of A,., turn into a U(I) gauge field Acr on the world-volume, while the remaining 9 - p components appear as scalar fields x a . The fields x a describe fluctuations of the D-brane world-volume in transverse directions. In general throughout these notes we

197

will use f-l, 1/, ... to denote lOD indices, a, /3, ... to denote (p + l)-D indices on a D-brane world-volume, and a, b, ... to denote (9 - d)-D transverse indices. One way to learn about the low-energy dynamics of a D-brane is to find the equations of motion for the D-brane which must be satisfied for the open string theory in the D-brane background to be conform ally invariant. Such an analysis was carried out by Leigh 7. He showed that in a purely bosonic theory, the equations of motion for a D-brane are precisely those of the action S = -Tp

J

dP+1e e-4>

J-det(G

a (3

+ Ba/3 + 27ra' Fa/3)

(2)

where G, Band

..(Pf(V) - A4) In the gauge theory, the Higgs branch has 5 complex dimensions. In our problem, there is only one relevant dimension in the moduli space, parametrized by z. This implies the existence of a tree-level superpotential 8W = V1~

+ V1~ + V223 + V 224

320 Integrating out the Vs appearing in c5W, we find that they vanish. Then the Pfaffian constraint obtained by integrating out .x reveals

A4 V12

V34=-

This yields the required complex I-dimensional moduli space. What about the pole? Consider adding a term

to the superpotential. Then altogether we have

W = Wo

+ c5W + c5W

Now integrating out V13, V14 , V23, V24 and

.x, we find

This is the desired pole, with z '" V12! Note that nonperturbative quantum phenomena in the SU(2) gauge theory 2

can explain classical heterotic string singularities because e- ~ is playing the 1 role of e - ~. This is the analogue of the special properties of RR fields used by Strominger in explaining classical singularities of N=2 string vacua with quantum loops of light solitons 3 . 7

K3 Fibrations and Es x Es Strings

We will now briefly discuss a similar analysis of singular Es x Es string compactifications on K3 fibrations. This will reveal a novel phenomenon which we did not encounter in the SO(32) case. We will basically motivate and summarize the results of 25 , where all of the details can be found. One of the longest standing mysteries of string theory has been the vacuum degeneracy problem. It is hard to see how a dynamical selection principle could choose between disconnected components of the space of vacua. For the case of compactifications with N=2 supersymmetry, the situation improved considerably with the work of4. The case of N=I vacua is qualitatively different from the N=2 case. N=I is the first case where chiral gauge representations occur. No transition which occurs in weakly coupled Lagrangian field theory can change the net number of chiral generations. Nonetheless, we will find in this section that in fact

321

some vacua of the Es x Es string with different generation numbers can be connected by phase transitions 25. These transitions involve going through a point in moduli space where there is no free long distance description of the physics.

7.1

Fibration Picture of the Phase Transition

As in §6, we consider compactification on a K3 fibration M with base sphere C. Call the radius of C R. Above the scale 1/ R, the physics can be analyzed in terms of an effective six-dimensional compactification on K3. Suppose an Es small instanton develops in the generic fiber. For simplicity, suppose it develops in an Es factor which is on a locus with unbroken E7. Then as we discussed in §5.2, moving the small instanton fivebrane off into the bulk (i.e., turning on the scalar in its tensor multiplet) yields a vacuum with one less 1/256 localized on the wall it left. At the intersection ofthe two branches, the physics is governed by a nontrivial RG fixed point. Let us now consider the theory below the scale 1/ R, i.e. in four dimensions. For definiteness let us consider an SU(3) bundle V embedded in one Es on the Calabi-Yau threefold M. This leads to an E6 gauge symmetry in spacetime. In a class of examples, an analysis of the 4d spectrum reveals the following 25: Each 6d 1/256 hypermultiplet descends to a set of 27s or to a 27 in the 4d theory. There is also some matter which does not descend from matter in the 6d theory, i.e. which is not associated with the generic fiber. This matter is therefore associated with singular fibers. In the 4d theory, a zero-size instanton in the generic fiber corresponds to a fivebrane wrapped around the base pl. As shown in 42 it remains consistent with supersymmetry for the fivebrane to move away from the end of the world, as long as it is wrapped on a holomorphic curve. In this phase, since of the the 1/2 56s has been removed in six dimensions, the corresponding 27s or 27 have been removed in the four-dimensional theory. One can argue that no additional states from singular fibers are produced in the transition. So we find that the net generation number has shifted. This is possible because at the point of transition, there is no weakly coupled description of the infrared physics: The theory is governed by a nontrivial RG fixed point obtained by ''fibering'' the 6d fixed point over a sphere. It would be very interesting to obtain a better understanding of such exotic 4d N=1 fixed points; examples have been constructed on brane probes in 43.

322

7.2

Instanton Effects

So far, we have discussed the 4d theory below the scale 1/ R set by the KaluzaKlein modes on the sphere. There are also lower dynamical scales in the problem, set by the strength of worldsheet instanton effects. As we mentioned in §5.2, in addition to the fundamental string there are two noncritical strings of interest when the fivebrane moves off into the interval. If ¢ parametrizes its position on S1 / Z2, then the tension of these noncritical strings goes like ¢ and ¢. So euclidean world sheets of these strings wrapping C will generate effects that go like e- R2 4> and e-R2(~-4».

;1 -

The physics of these instanton effects can be classified according to the splitting type of V over C at the small instanton singularity, as in §6. Actually, in general there is another vector bundle V2 embedded in the second Ea. At the point where the fivebrane is "absorbed" into the second E a , V2 is also singular over C and its splitting type is of interest. There are a few different cases one can consider, and we mention two of the simplest below. Note that T-duality relates this theory, compactified on an additional circle, to the SO(32) theories we discussed in the previous section. So in many cases, results about threedimensional SU(2) gauge dynamics are (indirectly) related to the phenomena under study here. Case I: If h°(V1c ® 0(-1)) and h°(V21c ® 0(-1)) are both greater than two, then there is no worldsheet instant on generated superpotential on the ¢ branch. This follows from a standard counting of worldsheet instanton zero modes 37, which are associated to these two cohomology groups for the two kinds of noncritical strings. For these splitting types, there are too many zero modes for a potential to be generated. Furthermore, in these cases fundamental string worldsheet instanton effects also cannot remove the "small instanton" point in the classical moduli space, where the nontrivial fixed point theory occurs. The physics as analyzed at the scale 1/ R persists in the full quantum theory. Such vacua provide examples where the chirality change can occur while remaining in the moduli space of vacua, i.e. at zero cost in energy. Case II: If hO (VIc ® O( -1)) or h°(V21c ® O( -1)) is zero, a nonperturbative worldsheet instanton superpotential destabilizes the ¢ branch where the fivebrane is on the interval. One can see this in the Ea theory by considering tensionful string worldsheet instantons of the first or second Ea. The number of zero modes on the worldsheet of such a string, for these splitting types, will be small enough for a nontrivial instanton-generated superpotential. This is consistent with known results about SU(2) gauge theories in three dimensions 44,45. In this case, the chirality change which seems possible between the scale J.l generated by the instanton superpotential and 1/ R is obstructed by quantum

323

effects below /-t. However, the potential barrier which prevents chirality change in such cases can be hierarchically smaller than the string scale. This indicates that in these cases chirality change can "almost" occur, i.e. it is possible at a small cost in energy. In closing: The insights from string duality have already been used to great effect in "solving" vacua with extended supersymmetry. In 4d N=l compactifications, there has been progress in understanding methods of supersymmetry breaking and in removing an obstruction to unifying vacua. New techniques (particularly from F-theory/heterotic duality) are being developed 46,47, and one should expect further progress in the near future. Acknowledgments

Much of the work reported on in these notes was done in collaboration with E. Silverstein. I would also like to thank O. Aharony, P. Aspinwall, T. Banks, J. Distler, J. Louis, and N. Seiberg for discussions on some of these topics. References 1. J. Polchinski, Rev. Mod. Phys. 68 (1996) 1245, hep-th/9607050;

2. 3. 4.

5. 6. 7. 8. 9. 10.

J. Schwarz, hep-th/9607201; P. Aspinwall, hep-th/9611137; C. Vafa, hep-th/9702201. P. Green and T. Hubsch, Phys. Rev. Lett. 61 (1988) 1163; P. Candelas, P. Green, and T. Hubsch, Phys. Rev. Lett. 62 (1989) 1956. A. Strominger, Nucl. Phys. B451 (1995) 96, hep-th/9504090. B. Greene, D. Morrison, and A. Strominger, Nucl. Phys. B451 (1995) 109, hep-th/9504145; T. Chiang, B. Greene, M. Gross, and Y. Kanter, hep-th/9511204; A. Avram, P. Candelas, D. Jancic, and M. Mandelberg, Nucl. Phys. B465 (1996) 458, hep-th/9511230. S. Kachru and C. Vafa, Nucl. Phys. B450 (1995) 69, hep-th/9505105. S. Ferrara, J. Harvey, A. Strominger, and C. Vafa, Phys. Lett. 361B (1995) 59, hep-th/9505162. P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B258 (1985) 46. K. Uhlenbeck and S.T. Yau, Comm. Pure. App. Math. Vol XXXIX (1986) S257. T. Hubsch, Calabi- Yau Manifolds, World Scientific, 1992. M. Dine, N. Seiberg, X.G. Wen, and E. Witten, Nucl. Phys. B278 (1986) 769, Nucl. Phys. B289 (1987) 319.

324 11. E. Silverstein and E. Witten, Nucl. Phys. B444 (1995) 161, hepth/9503212. 12. J. Polchinski and E. Witten, Nucl. Phys. B460 (1996) 525, hepth/9510169. 13. M. Douglas, hep-th/9512077. 14. See e.g. B.S. Acharya, Nucl. Phys. B475 (1996) 579, hep-th/9603033 and references therein. 15. C. Vafa, Nucl. Phys. B469 (1996) 403, hep-th/9602022. 16. C. Vafa and E. Witten, Nucl. Phys. B447 (1995) 261, hep-th/9505053. 17. S. Sethi, C. Vafa and E. Witten, Nucl. Phys. B480 (1996) 213, hepth/9606122. 18. E. Witten, Nucl. Phys. B403 (1993) 159, hep-th/9301042. 19. J. Distler and S. Kachru, Nucl. Phys. B413 (1994) 213, hep-th/9309110. 20. J. Distler and S. Kachru, Nucl. Phys. B442 (1995) 64, hep-th/9501111. 21. T. Chiang, J. Distler, and B. Greene, hep-th/9702030. 22. P. Candelas, X. De la Ossa, P. Green, and L. Parkes, Nucl. Phys. B359 (1991) 21. 23. N. Seiberg, Phys. Rev. D49 (1994) 6857, hep-th/9402044. 24. N. Seiberg and E. Witten, Nucl. Phys. B426 (1994) 19, hep-th/9407087. 25. S. Kachru and E. Silverstein, hep-th/9704185. 26. E. Witten and J. Bagger, Phys. Lett. 115B (1982) 202. 27. J. Blum and K. Intriligator, hep-th/9705044; P. Aspinwall and D. Morrison, hep-th/9705104, and references therein. 28. E. Witten, Nucl. Phys. B460 (1996) 541, hep-th/9511030. 29. M. Atiyah, N. Hitchin, V. Drinfeld, and Y. Manin, Phys. Lett. 65A (1978) 185. 30. P. Horava and E. Witten, Nucl. Phys. B460 (1996) 506, hep-th/9510209. 31. O. Ganor and A. Hanany, Nucl. Phys. B474 (1996) 122, hepth/9602120. 32. N. Seiberg and E. Witten, Nucl. Phys. B471 (1996) 121, hepth/9603003. 33. M. Duff, R. Minasian, and E. Witten, Nucl. Phys. B465 (1996) 413, hep-th/9601036. 34. N. Seiberg, Phys. Lett. B388 (1996) 753, hep-th/9608111; D. Morrison and N. Seiberg, Nucl. Phys. B483 (1997) 229, hepth/9609070; M. Douglas, S. Katz, and C. Vafa, hep-th/9609071; O. Ganor, D. Morrison and N. Seiberg, Nucl. Phys. B487 (1997) 93, hep-th/9610251.

325

35. S. Kachru, N. Seiberg, and E. Silverstein, Nucl. Phys. B480 (1996) 170, hep-th/9605036. 36. I. Affleck, M. Dine, and N. Seiberg, Nucl. Phys. B24I (1984) 493. 37. J. Distler, Phys. Lett. BI88 (1987) 431. 38. S. Kachru and E. Silverstein, Nucl. Phys. B482 (1996) 92, hepth/9608194. 39. E. Witten, Nucl. Phys. B474 (1996) 343, hep-th/9604030. 40. R. Donagi, A. Grassi, and E. Witten, Mod. Phys. Lett. All (1996) 2199, hep-th/9607091. 41. G. Curio and D. Lust, hep-th/9703007. 42. E. Witten, Nucl. Phys. B471 (1996) 135, hep-th/9602070. 43. O. Aharony, S. Kachru, and E. Silverstein, Nucl. Phys. B488 (1997) 159, hep-th/9610205. 44. O. Aharony, A. Hanany, K. Intriligator, N. Seiberg and M. Strassler, hep-th/9703110. 45. I. Affleck, J. Harvey and E. Witten, Nucl. Phys. B206 (1982) 413. 46. R. Friedman, J. Morgan, and E. Witten, hep-th/9701162. 47. M. Bershadsky, A. Johansen, T. Pantev, and V. Sadov, hep-th/9701165.

326

RECOGNIZING SUPERPARTNERS AT LEP GORDON L. KANE Randall Lab of Physics, University of Michigan, Ann Arbor, MI48109-1120 There are several hints from data that behave as they would if a few events of superpartner production had been observed at LEP and FNAL. In an higgsinoLSP world the properties of the events and their rates are consistent with what is expected from supersymmetry. In such a world additional small effects are expected only in BR(b -+ s')'), R b , as and again the data is consistent. Although it is remarkable that this set of phenomena can have a consistent interpretation in terms of new physics, the data (though as strong as it could be given the limited available statistics) is not compelling and could be just bad luck. Once LEP has collected at least 50 pb- 1 per detector at Vs ~ 190 GeV, and possibly sooner, the situation will be clear. The observed events involve isolated, energetic photons. Finally, I discuss learning the basic parameters of the supersymmetric effective Lagrangian from data, and connecting them to the theory at the unification scale.

1

Introduction

In 1986, Haber, Quiros and 11 argued that events at colliders with energetic, isolated ,,/'s could occur as the signature of superpartner production. This occurs over a significant part of the parameter space, where the LSP is mainly higgsino, and heavier neutralinos that are largely photino-like decay into "/+ higgsino. In 1996 we learned about the CDF ee-n event 2, with two isolated, energetic ,,/'s. In the Standard Model the probability 4 of getting such an event is less than 10- 4 . Analysis 4 showed that in SUSY the probability is '" 1 if a few of the SUSY masses (me ,jJ,M1 ,M2 ) and tan,B are in certain narrow regions. This is not trivial - a number of conditions have to be satisfied. Based on this SUSY higgsino-LSP interpretation of this event, we predicted 4,5 a number of possible confirming tests at LEP and FNAL. Given some uncertainties in masses one could not be sure which channels might appear. One such channel is e+ e---t

N2N2

where Ni are the neutralino mass eigenstates, Nl LSP, N2 next lightest, etc. The analysis of the CDF event implies N2 is mostly photino, and the 1986 analysis plus extensive more recent calculations showed that 6

327

,,!VI.

is the dominant decay. Thus the signature is e+ e- -+ In spring of 1997 it became known 7 that about (depending on cuts) 6 such events were reported from the 161 + 172 GeV LEP runs, combining the four detectors, in the kinematic region of interest. Since the LSP mass is ~ 50 Ge V, the missing invariant mass J.jf must be over about 100 GeV; since much of the background must be in the Z peak that is an important cut. I estimate 8 the background in the relevant region is about 1.5 events, so the 6 reported events are very encouraging. (The 6 events are consistent with cuts J.jf > 4f z , E, > 5 Ge V, I cos 0, I < 0.85, for which cuts I estimate 1.5 events background. They are also consistent with E, > 1 GeV, IcosO,1 < 0.95, giving a background of about 6 events. But then 2/3 of the events should have 0.85 < I cos 0, I < 0.95 and 1 < E, < 5, but they don't.) Further analysis 5 suggested another possible channel e+e- ---t

N2N3

with N3 decaying invisibly (to vv, with v -+ vNt) and N2 -+ ,Nl, so the signature is a single photon plus J.jf > M N3 + M NI . The reported spectrum 7 of, + J.jf events follows the SM expectations, except for an excess at large J.jf just where it would be expected for such a signal. All of this is based on a complete model that is consistent with all other data 5, not a simple condition to satisfy. In particular, the LEP events behave exactly like they should ifthe parameters had the values required by the FNAL event. If these events are indeed signals, that will be confirmed at LEP in 1997 or 1998 (depending on how much luminosity is gathered). If it is just bad luck that they behave as a signal should, we will know that soon too. The relevant parameters imply a light Higgs boson mass too, so if the events are real hO should also be detected soon. It would be very nice if clean, unambiguous experimental signals could appear one day. But a little thought tells us that is unlikely - probably impossible. Consider colliders. At least until the LHC, which is unlikely to produce its first paper relevant to supersymmetry until well over a decade from now, what will happen is that as energy and/or luminosity increases at LEP and FNAL a few events of superpartner production will occur. Perhaps such events have already occurred. Each event has two escaping LSP's, so it is never possible to find a dramatic Z-like two-body peak, or even a W-like peak with one escaping particle. Even worse, often several channels look alike to detectors so simple features can be obscured. And usually there are SM processes that can fake any particular signature, as well as ways to fake signatures because detectors are imperfect.

328

Thus to make progress it is essential to proceed with limited amounts of incomplete information. Without theory input it is entirely possible that signals would not be noticed, hidden under backgrounds since there was no guide to what cuts to use (as we just saw for the "f./! events). Further, a particular signal might be encouraging but not convincing - only when combined with other signals that were related by the theory but not directly experimentally could a strong case be made. What about information from decays? In the best cases, such as b --+ where there is no tree level decay, the SUSY contribution could be comparable to the SM one, say a large effect of order 30%. Then to get a significant effect the combined errors of the theoretical calculation of the SM value and the experiment have to be below 10%. Only b --+ of known decays can approach that; presently the theoretical error 9 is at about that level, and the experimental error about 20%. At best, in a couple of years this could provide compelling evidence of new physics. If so, taken alone several interpretations would be possible, but with theoretical input it could be combined with collider data to determine which were consistent. When there is a tree level SM contribution, such as for Rb = r (Z --+ bb) / r (Z --+ hadrons) the SUSY effect has to be a loop and can be '" so experimental errors have to be several times smaller. Just from statistics that requires'" 106 events, which is unlikely. Sometimes several decays are related in a particular model, in which case the combined predictions can be tested and the results are somewhat more significant (that is the situation for b --+ plus Rb plus a.). Further input could come from proton decay to channels favored by SUSY, from occurrence of decays forbidden in the SM such as J-l --+ ef or K --+ J-le, from neutron or electron electric dipole moments, from non-SM CP violation, or other rare phenomena. From the point of view of giving lectures, whether the events are real or not the techniques and analyses are examples of what must be done when superpartners are eventually discovered, so it is worthwhile to present them. The entire analysis is done in terms of an effective SUSY Lagrangian at the EW scale, without assumptions about parameters, which is the appropriate approach once there is data. The important parameters for this analysis are tan,8 (the ratio of the two vacuum expectation values), 1 ;S tan,8 ;S 70; J-l, which can be thought of as a generalized higgsino mass (a priori J-l can have any value -Mpl ;S J-l ;S MPl); and Ml and M 2 , the U(l) and SU(l) gaugino soft breaking masses, o;S M 1 , M2 ;S Te V. For discussions of b --+ 8, and Rb we will also need the mass and mixing angle of the right-handed stop. We denote the four neutralino mass eigenstate, by Ni and the two chargino mass eigenstates by Ci.

8"

8,

!%,

8,

329 2

Comparisons

Almost all of the studies of supersymmetric models can be classified according to what is the LSP, and fall in three categories, as shown in Table 1. On the left are listed several criteria that are often used to compare and test models. The first world listed has a light gravitino LSP (GLSP), the second an LSP that is mainly bino (BLSP) and the third mainly higgsino (hLSP). The implications for SUSY-breaking and experimental signature are very different for the three cases - it will be easy to recognize which is being observed once one is detected. (GLSP) corresponds to gauge-mediated SUSY-breaking, and the other two gravity-mediated SUSY-breaking.) Some of the individual criteria will be described in more detail below in the (hLSP) section; see also ref. 2. To be complete one would also add a fourth column for an unstable LSP. Then one would expect no large missing energy events, no events with prompt "{'s, etc. In a GLSP world there is a small window for a signal at LEP but no phenomenological reason J.o expect one there. If the CDF event? were interpreted as evidence for a GLSP world, which is difficult given constraints but not completely excluded, many such events would occur at FNAL after the collider starts running again in 1999; otherwise there is only a small window at FNAL. Such a world could probably be detected at LHC or a lepton collider about 2010. In a BLSP world there is no evidence today for sparticles, and all hints must disappear (no more ee"{"{lh events at FNAL; BR(b -t s"{) -t SM, Rb -t SM, a~ther -t 0; no future excess of "{"{$ events at LEP; baryogenesis not at EW scale; etc.). There are small windows at LEP and FNAL but no phenomenological reason for sparticles to be there. Such a world probably be detected 3't LHC or a lepton collider about 2010. In a hLSP world sparticles may have already been observed. Confirmation will occur at LEJ>3 once 50 pb- 1 fdetector at VB ~ 190 GeV has been accumulated or before. In the next section the hints for an hLSP world and some of the tests are described in a little more detail.

a;z -

3

The CDF e "e"

"{"{FIr

Event

The reported 2 CDF event is interesting for four reasons. (1) Most important, 4 the probability of the Standard Model giving such an event is extremely small, less than 10- 4 from naive use of the theory and probably considerably smaller when experimental considerations are included. While one event can never be

330

Table 1. Evidence/Criteria Absence of FCNC

Number of parameters

GLSP

BLSP

hLSP

Yes, if messenger Mechanisms exist, but don't scale low know if they are applicable Presently all about same - GLSP somewhat more than others now, but will be more predictive after squarks and sleptons observed. maybe

no

yes

maybe bs/, no for Rb, a,

no

yes

no

no

yes

not LSP

ok

yes

g, light i at FNAL?

no

no

ok

EW baryogenesis

no

no

ok

no reason

no reason

yes

(a) either 2/,'s in every event; or 2 charged leptons in every event; or long-lived particles that decay in or outside of detector (b) medium ¥

(a) no /,'s; (b )events with no charged leptons; (c) larger ¥ (d )trilepton + JfJ events at FNAL, LHC

(a) 0, 1, or 2 /,'s (b) large ¥ (c) not only LSP but also ii, N3 invisible

CDF e "e" "(y$T

LEP TY $ events Cold Dark Matter

typical signatures that distinguish

331

a convincing signal, it is important to understand that this event is interesting because it should not have occurred in the Standard Model; the background is low. (2) The event has ,...., 50 GeV FIr, as expected for SUSY events. (3) It also has hard isolated ,'s, predicted long ago 1 as one likely way to detect superpartners. (4) A number of conditions, cross sections branching ratios etc. have to come out right or such an event could not qualify as a SUSY candidate. It is not easy to satisfy the conditions. 4,5 Here I want to emphasize that one way4 to get the photons, and to interpret this event as production and decay of superpartners, is for the LSP to be mainly higgsino, and for the next-to-LSP to be mainly photino. That interpretation implies the parameters of figure 1, in a way completely independent of the Rb, b-t as data. It only uses general features of the CDF event, the presence of energetic isolated ,'s and ItT. The logic that leads to the above conclusions depends mainly 0E the neutralino mass matrix. A large branching ratio is required for e -t Nze, so N z must be mostly a combination of .:y and Z, since higgsinos would couple proportional to the fermion mass. A large branching ratio Nz -t N1 , can give the photons, if one of N1 or N z is mainly photino, the other higgsino. Together these require N1 ,...., .:y, N1 ,...., h. To see if this can be realized in practice, consider the symmetric neutralino mass matrix

s"

(Mz - Md swcw M1s?v + Mzc?v

o Mz Il sin 2/1

This is in the basis .:y, Z, hsym , h anti . Here Cw = cos Ow ,sw = sin Ow. We see that Ml ~ M z minimizes .:y - Z mixing, while tan /1 near 1 minimizes h sym - hanti mixing, both leading to high purity for N z ,...., .:y, Nl ,...., h sym . The mass ordering needed can come if Il < 0, IIlI < M 1 c?v + Mzs?v '== M1 '== M2 . All this can be seen analytically, and numerical analysis shows there are no other solutions. It is highly non-trivial that this worked. One can also examine the Nz decay. Nz could have two body decays to 01 W, ee, iliJ, NlZ, Nlh. But all of them are suppressed either by phase space, or vanishing vertex if Nz is mainly .:y. Then one has to look at loops, with WO l or tt in the loop for the kinematically favored decay_,Nl . One finds that over a large region, particularly if M 1 ~ M z, one has N 1, as the dominant decay. Recently the possibility that the positron in the CDF event is a r+ has been raised. That does not affect our interpretation in any detail since the photons and missing energy still arise the same way. It does affect the basis production

332

e'e'rr"'6r

1.2 :5 tan P:5 1.8 11--Mw M\ :5M2-MZ

Figure 1 mechanism which would then be via chargino production (or something exotic like slepton mixing). There are a number of additional channels one can look for at FNAL, such as ev, c;iJ2 , and so on. They have signatures with typicaliy one prompt "'/ and possibly a charged lepton. The total cross section for events with 2",/s is small, less than 60 tb. 4

LEP "'Y"'Y 11 and "'Y $ Events

Combining LEP data from 161 and 172 GeV, and four detectors, about six events 7 have been reported for e+ e- -+ "'/"'/+ nothing, with missing invariant mass If! in the region above Mz + 10 GeV. We can view the occurrence of such events as a prediction of the higgsino - LSP picture suggested by the presence of the photons in the CDF event, or equivalently as a prediction of the Rb + b -+ 8",/ + O. The righthand side of the previous equation is a fermionic state (goldstino) I'1/» = QIO). This fermion is created out of the vacuum by the supercurrent

(6) where

(7) VEvacuum is the goldstino decay constant. Thus, Evacuum =I 0 signals a supersymmetry breakdown. Equivalently, supersymmetry is spontaneously broken if some of the auxiliary components of the chiral or gauge superfields (F or D components) get vacuum expectation values (VEVs). This can be seen by recalling the supersymmetric transformation of the fermions which can play the role of goldstinos. Such can be the Fermi component of chiral or gauge

f =

344

superfield (or a combination of them). Under supersymmetric transformation (. '" 4': A, while squarks and sleptons do so only at two loops, m2 '" (4':) 2A2. Because of the different dimension ali ties between fermionic and scalar mass terms, this implies that the gaugino and squark mass scales are of the same order, m>. '" m. On the other hand, in the case of the Higgs parameters, we find BI-' '" p,A, as parameters with different dimensionalities are generated at the same loop level. Since A must be in the range 10-100 Te V to generate appropriate squark and gaugino masses, then either p, is at the weak scale and BI-' violates the naturalness criterion 40, or BiJ is at the weak scale and p, is unacceptably small. We will refer to this puzzle as to the p,-problem in G MSB theories. Following 38, let us discuss this problem in somewhat more detailes. If the p,-term is present in the superpotential in the limit of exact supersymmetry, then naturally it can only be of the order of the Planck scale MpL or some other fundamental large mass scales. We assume therefore that the p,-term is forbidden in the original superpotential but generated, together with B iJ , by the following effective operators

f ~2 f

~

M

d4 ()H Hxt

(50)

'

d4 ()H Hxxt

.

(51)

Here X is a superfield which parametrizes the breaking of supersymmetry, and M is the messenger mass scale. Replacing X i~ eqs. (50) and (51) with the

356

VEV of its auxiliary component F, we obtain Jl and BiJ terms of the order of A and A2 respectively, with A = F j M. In theories where gravity mediates supersymmetry breaking, M has to be identified with M pL and the operators in eqs. (50) and (51) are present in the theory as non-renormalizable interactions. The existence of the operator in eq. (50) requires however a non-minimal Kahler potential 39 . In GMSB theories we want to obtain the operators in eqs. (50) and (51) after integrating out some heavy fields. Since the operators in eqs. (50) and (51) break a Peccei-Quinn symmetry, they cannot be induced by gauge interactions alone. The simplest way to generate a Jl-term at the one-loop level is then to couple the Higgs superfields to the messengers in the superpotential: (52) Assuming that a single superfield X describes the supersymmetry breaking (53) the diagram of fig. la gives (54)

f(x) = (x Inx2)j(l- x 2 ). However, the couplings in eq. (52) also generate the diagram of fig. Ib, which contributes to BJj at the one-loop level: (55) Finally the diagram of fig. lc generates the soft-breaking masses mlI and m1. Unexpectedly, the leading order contribution here cancels and the scalar Higgs masses are generated only at higher order '" 16~2 F4 j M6. However the cancellation is valid only in the simple case of eq. (53) in which the ratios Ai = Fid Mi (i = 1,2) for the two messengers are the same. If this is not the case then

mlI = 1~:2 (Al -

m1

A2)2 g().1!).2) [1

and similarly for with), replaced by 2(1 - x 2 )]j(l- x 2 )3. From eq. (54) and (55) we obtain

A;

+ O(F2 jM4)] ,

here g(x) = x 2 [(1

(56)

+ x 2)lnx 2 +

(57)

357

This problematic relation is the expression of the J.L-problem in GMSB theories. It is just a consequence of generating both J.L and BJi at the one-loop level through the same interactions. If the dominant contributions to m'JI and miI come from two-loop gauge and three-loop stop contributions 36,

2 _ 2"3 (0'411"2)2 A2 '

mH -

2 = mH2 [1 -

mii

-4h;

311"2

(0'3) 0'2 2In (11"-0'3 ) 1,

(58)

then eq. (57) is actually inconsistent with electroweak symmetry breaking. This is true because the condition for the stability of the Higgs potential, 21BJil < m'JI + miI + 2J.L2, cannot be satisfied when J.L is determined by electroweak symmetry breaking 2 2 t 2f3 _ mz 2 1 12 = m H - m ii an

J.L

tan 2 f3 - 1

2 '

(59)

with tan f3 = (fJ) / (H). Extra contributions to m'JI and miI can allow the electroweak symmetry breaking, but only at the price of introducing a considerable fine tuning among parameters. The alternative to coupling each Higgs superfield separately to the messengers, as done in eq. (52), is to couple the Higgs bilinear H fI to a heavy singlet superfield S. Of course, for the mechanism to work, S cannot develop a VEV in the supersymmetric limit. Now, possibly through a coupling to a different singlet N which has tree-level non-zero VEV, a J.L-term can be induced at the one-loop level through the graph in fig. 2a 46. However with an extra insertion of the spurion superfield X, the same interaction gives rise to a BJi contribution through the graph in fig. 2b and we recover the unwanted relation of eq. (57). Nevertheless, this is not always the case. Suppose that we couple N to the messengers in the same way as we coupled H to the messengers in eq. (52). From the previous example, we know that the diagram of fig. Ic suffers from a cancellation of the leading contribution in the case of the minimal secluded sector given in eq. (53). This means that also the diagram in fig. 2b vanishes at the leading order. Extra X and xt or extra N and Nt insertions in the loop of fig. 2b can generate a non-zero BJi but this will be suppressed by terms O(F2 / M4) or O( (N)2 / M2), if M is larger than the other scales. However we do not consider this mechanism entirely satisfactory, since it relies on an accidental cancellation occurring only for a very specific supersymmetry-breaking structure.

358

Finally we mention that in the literature the possibility of adding extra light Higgs superfields in GMSB theories has also been considered 36. If a light singlet S is present with a superpotential

(60) then j.t and BJj are generated whenever (S) and (Fs) are non-zero. However, in GMSB theories, trilinears, bilinears and the soft-mass of S are suppressed with respect to the Hand H soft masses, and a non-zero VEV for S requires an appreciable fine-tuning 36. This problem can be overcome but its solution may require additional quark superfields coupled to S in order to induce a large soft mass m~ 36. As shown in 38 it is in principle possible to have a mechanism which satisfies the following criteria of naturalness: i) the different supersymmetry-breaking Higgs parameters are generated by a single mechanism; ii) j.t is generated at one-loop, while BJj is generated at two loop; iii) all new coupling constants are of order one; iv) there are no new particles at the weak scale. The crucial point is that j.t, instead of being generated by the operator (50), arises from the operator

(61) Here Da is the supersymmetric covariant derivative. This operator can be generated from the diagram offig. 3. The crucial point is that a BJj-term cannot be induced from such diagram even if we added extra X and xt insertions in the loop of fig. 3. This is because a D2 acting on any function of X and xt always produces an antichiral superfield. The mechanism requires at least two singlets, 5 and N, such that only 5 couples at tree level to H H and to the messengers. We forbid the coupling of N to the messengers and a mass term 52 in the superpotential to guarantee that the one-loop diagram of fig. 2b does not exist. The diagram of fig. 3 induces the operator 1 161l' 2M2M lr

!4

- 2[ X t X ] , d BHHD

(62)

where MN is the mass parameter in the superpotential term MN N 5. Notice that the internal line in fig. 3 is an (Sst) propagator which, at small momenta, behaves like jj2 / Mlr 46. Eq. (62) leads to a j.t parameter

(63)

359

for MN = O(VP). BI' can be induced at the two-loop level if the superpotential contains a coupling of the form N 2 S. One contribution comes from a diagram analogous to the one shown in fig. 2b, but where the effective coupling between N and X arises at two loops, as shown in fig. 4. Another contribution comes from the diagram of fig. 5, which induces the effective operator 1

(161r 2 )2M4M&

/

4

-

t

-2

d OHHX XD D

2 [

t

]

X X .

(64)

Both contributions generate a BI' parameter of the correct magnitude

BI' ,...., (1611r2 )

2 A2.

(65)

In addition to the gauge-induced contributions given in eq. (43), the soft masses m'h and mh are also generated at two loops by the diagrams in fig. 4. A Model

Let us now see how to realize this mechanism in an explicit example of ref 47 In this paper two things are done. First, it is a simple model of dynamical gaugemediated SUSY breaking. This model incorporates a mechanism in which the It-problem is solved by a dynamical relaxation mechanism discussed above. (The reason why we call it so will become clear below). It involves no light fields beyond those of the MSSM. In the second part, following ref 47 we show how this mechanism is generalized and give two more examples in which it is incorporated. Since the original pioneering work 3 , there has been significant progress in model building with dynamical GM breaking (see e.g. refs 4-13 New mechanisms of dynamical supersymmetry breaking in the strongly coupled gauge theories have been explored, in particular, 14 based on the quantum-modified moduli spaces 15. It has also been shown, that a class of the simple strongly coupled gauge theories, that usually do not exhibit supersymmetry breaking, in combination with the stringy anomalous U(1) break global SUSY dynamically 16. Recently a class of simple theories has been proposed in which SUSY is unbroken in the global minimum, nevertheles the soft terms are dynamically generated because we live in a metastable SUSY-breaking plateau 10 e. Unfortunatelly the most elegant of 10 theories had negative mass 2 scalars 17. In eOne of the models of 10 has a significant ideological overlap with ref 9 • An interesting model along the similar lines was recently proposed in 11

360

this lecture we will review a simple class of theories, considered in 47, with such metastable vacua which do not suffer from any phenomenological problem. These theories provide for a simple solution to the J.l-problem based on a dynamical relaxation mechanism 18; furthermore they do not require the introduction of any input mass parameters. In the second part of the paper we show how to generalize this mechanism and give two more examples incorporating it. According to the general mechanism of ref 10 our gauge group includes three factors SU(5)s 0 SU(5)w 0 CB. Here SU(5)w is the ordinary (weakly coupled) grand unified group which accommodates quarks and leptons in three families of (1, 10, 1) + (1,5,1) representations. For our discussion it is inessential whether SU(5)w is actually realized as a symmetry of the theory above the GUT scale. In particular, the SU(3)c 0 SU(2) x U(l)y subgroup can suffices. Nevertheless, for convenience we will classify fields by SU(5)w representations. C B is a weakly coupled 'balancing group', whose role is to stabilize the supersymmetry-breaking minimum by the mechanism similar to the Witten's 'inverse hierarchy' scenariJ9. Its nature and the field content is very general: any asymptotically free gauge group with some vector-like matter content can do the job, e.g. we can think of C being an SU(2)B with a single flavour 1/;, -/i;. Finally, SU(5)s is the strongly coupled group with five flavors Q;ti, where i = 1, .. 5 is a flavour index. This index we will identify with an SU(5)w index. Thus the 'quarks' of the strongly-coupled group are in the representations (5,5,1) + (5,5,1). We will denote the strong scale of SU(5)s by A. To complete the matter content we introduce four additional gauge singlets Y, I, S, and N. Their role will become clear later. Our classical superpotential is lW = aYTrM + I((3y2 - -r1/;-/i;) + S(>'lH fI + >'2N2/2 - >'TrM) +standard MSSM (GUT) superpotential

(66)

where M;k = Q;{Jk are the 'mesons' of the strongly coupled group and H, fI are the MSSM Higgs doublets. The role of the various terms in the superpotential can be briefly summarized as follows: 1) After accounting for the non-perturbative dynamics, the first term produces a supersymmetry breaking plateau along which Fy '" A2 and Y is undetermined; 2) The second term is the main improvement with respect to models of ref 10 . It relates the VEV of the singlet Y with the VEV that breaks CB and thus forces the balancing group to be broken ('Higgsed') along the plateau and at one-loop creates a stable minimum at Y >> ,fFY via the Witten's inverse hierarchy mechanism. In this minimum Q, Q automatically play the role of the messengers of the gauge-mediated supersymmetry

361

breaking. Note that without the second term, the only way to stabilize the SUSY breaking minimum is with gauge-non-singlet Y as it was done in ref 10. This would make more difficult to solve the J.l problem, and may also lead to negative values of the squark/slepton masses 17. 3) The last term solves the J.l problem via the mechanism of ref 18: the J.l and BJ.l terms are induced from the original SUSY-breaking scale Fy as oneand two-loop effects respectively. Let us now discuss this model in more detail. First we study its classical vacuum manifold (the moduli space). It includes a flat direction that can be parameterized by the VEV of Y. Along this direction Q, Q states get masses "" Y. For f3 "y 0 there would be the two other independent flat directions on the moduli space. The trivial one,~en by the free field I, and another, parameterized by the invariant

A as a result of two-loop renormalization. In this minimum the masses of messengers are = pX and they have the fermibose mass splitting due to couplings with Fx. As a result all the conditions of the standard gauge-mediation are satisfied and supersymmetry breaking is transmitted to the MSSM sector after integrating the Q, Q particles. Now let us show how the J.t and BJ.t terms are generated after supersymmetry breaking. This is arranged by the second term in (66) through the mechanism of 18. Integrating the heavy mesons and substituting TrM = A2 has the following important impact on the S, N, H, fJ sector: 1) It generates an effective superpotential

(73) which, to zeroth order, induces the VEV for the N field N 2 = ~~ A2 • Other VEVs (S, H, fJ) stay zero to this order f

=

'For S 0 there is a continuous degeneracy of the vacuum parameterised by holomorphic invariants N 2 and HH subject to the constraint Fs O. However, since at one-loop S =1= 0 (see the text below), the minimum is fixed at H H 0 for a certain range of parameters 18.

= = =

363

2) At one-loop the tadpole term for the scalar component of S superfield in the potential is inducecP- 8 25 Ap3 4 -S-1A j(X) 67r 2

+ h.c.

(74)

This plays a crucial role, since it generates the VEV S = _;g;~A2j(X) and thus the J-l term! The BJ-l term is also generated, since for S f:. 0 the cancelation

of Fs term is impossible and one gets BJ-l AI F S -?,- Ug;~r A4j(X)2. 3) At the two-loop there is an additional contribution to the soft mass of N, which shifts the one-loop value of the BJ-l so that finally the two-loop expressions for J-l and B J-l are 18

=

=

(75) 1

Generalizing the Vacuum Relaxation Mechanism for J-l

The typical problem in GM theories is that J-l and BJ-l are induced at the same loop order resulting in

(76) we have shown in the previous section for a concrete GM model how the dynamical relaxation mechanism can avoid this problem. In this section we will generalize this mechanism and derive the sufficient condition for it. The key point is that both J-l and BJ-l depend on dynamical fields which, after supersymmetry breaking, will adjust to the right vacuum expectation values due to energy minimization. We note however, that all these additional fields must have masses of order Fx, so that below this scale the low energy spectrum is just that of the MSSM (plus possibly some decoupled light states, e.g. X that only interacts through (X)-1 suppressed couplings). This is the crucial difference from the conventional singlet models in which J-l is induced as a VEV of the light singlet field (an example of the later is the 'sliding singlet' mechanism for generating the J-l term in GM theories 20). The necessary ingredient of our approach is the gauge-singlet field S that couples to the Higgs doublets in the superpotential (with coupling constant of the order of one, to be neglected for the time being). (77) W = SHH

364

The expectation value of 5 then will set a fL parameter, whereas the VEV of Fs will set a value of BfL respectively. Following the notations of the previous section, let X be a superfield that couples to the messengers (Q, Q) and breaks SUSY, through the nonzero Fx term. Then, to be in the right ball park the fL and thus the VEV of 5 should be induced as a one-loop effect from the scale For this to be the case, S must have a tree-level couplings with the messenger fields in the superpotential (just as in the model of the previous section). Now, the issue is to prevent Fs from getting the VEV at the same order. For this we note, that since Fs contributes to the vacuum energy, it can appear to be small just due to energetic reasons. The sufficient condition for this is that it is a function of some field(s) that get no potential apart from FsFs at one-loop order and thus slide and readjust themselves in order to compensate VEV of Fs. In our model such a field was N. In this case Fs and thus B can only appear as higher loop-effects and BfL is acceptably small. Thus, the compensator field N must have no superpotential in the limit 5 = 0 and we are lead to the following general structure of the superpotential (again we omit factors of order one)

fir.

W = 5(HH + I(N)

+ QQ) + XQQ

(78)

In the model of the previous section we had I(N) = A2N2j2-AA2, where the dynamical scale is coming from the condensate of strongly coupled messengers QQ. Now we will study what is the possible general form of the f- function. For the time being we will be assuming a single fundamental scale Fx ____ X2, although this is not in any respect necessary for our mechanism to work as it was clear from our model in which X2 >> Fx. The key idea is most transparent in units (Fx) = 1. In these units, as we will now show, the function 1 must satisfy the following necessary conditions in the minimum to zeroth-loop order:

1=0,1'----1,1"----1

(79)

where prime denotes derivative with respect to the compensator field N. Now since 5 is the only field that gets a correction to the potential from one-loop one-particle-irreducible diagrams, the effective potential to this order can be written as (both Higgs doublets and messengers are put to zero as it should be)

(80) where VI is a one-loop effective potential for 5. Since the 5-VEV in any case will be stabilized by the second term, which gives a curvature ---- 1, it is sufficient to keep only a tadpole part linear in S in VI, which is of order i5. Throughout the paper i will denote a one-loop suppression factor.

365

f,

Now, we minimize with respect to Sand N remembering that Bp = Fa = up to factors of order one, and we get the following equations

(81) since derivatives of fare'" 1 in the zeroth order, the same should hold at one-loop, and, thus, we immediately get that p '" S '" f and Bp '" (2. The simplest explicit model of the dynamical relaxation is based orr- 8 f = N 2 - A 2 and we have shown in the previous section the scale A can be indeed generated dynamically by the condensate of the strongly coupled messenger mesons. What if the messengers are not transforming under a strong gauge group? It is interesting that in this case, the scale A in f can be induced through the kinetic mixing of X and S superfields in the Kahler potential. A priory such a mixed term is not forbidden by any symmetry and even if not present at the tree-level will be induced through the loops with Q-particles. The resulting mixed term in the Kahler potential has the form Ll To? 1\.

'"

n5)..PI

1671'2 og

(M~) SX* XX*

(82)

where n is a number of messengers coupled to S. After substituting Fx i 0 VEV and solving the equation of motion for F s, the effect of this term is reduced to the shift of Fs, which from the point of view of the minimization is just equivalent to the shift

(83) The resulting p '" ..fl3ii term that is generated through the vacuum relaxation mechanism is suppressed by "'log factor with respect to the original supersymmetry-breaking scale. Finally we will discuss one more logical possibility which can dynamically introduce the scale of order Fx in the function f and satisfy conditions (79) necessary for the dynamical relaxation mechanism. This is to have a SUSY breaking sector communicating with compensators through the D-term of some gauge U(l). Assume that the supersymmetry breaking VEVs develop along the U(l) - D-non-flat direction. This can induce a non-zero expectation value of the D-term D '" 9 (in Fx units) where 9 is a U(l) gauge-coupling constant. Let us introduce a pair of compensator superfields N _ and N + with charges -1 and + 1 under the above U (1). The function f can be chosen then as f = N _ N +. In zeroth-loop order N -fields have F -flat potential and one of them, (say N_) can slide and compensate the D-term picking up the VEV of

366

order one. The conditions (79) are then automatically satisfied. The J1. term is induced as a one-loop effect through the usual tadpole term, whereas BfJ. is zero in this order and is only induced by higher loop corrections. I would like to thank the organizers of the ICTt> Summer School in High Energy Physics and Cosmology. References

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369

x+ H

H

b)

a)

c)

Fig. 1: Superfield Feynman diagrams for generating one-loop contributions to (a) p" (b) Bp" and (c) miI,

miI.

N

N

H

H

x x

H

H a)

b)

Fig. 2: Superfield Feynman diagrams for generating one-loop contributions to (a) p, and (b) B p, • The encircled cross denotes the VEV of the N scalar component.

370

H

s+

s H

x

Fig. 3: Superfield Feynman diagram for generating a one-loop contribution to po

x+

x

N,H,H Fig. 4: Superfield Feynman diagrams for generating two-loop contributions 2 2 2 t o mN,mH' an d miio

H

s N

H

x x+

Fig. 5: Superfield Feynman diagram for generating a two-loop contribution to BJ.I"

371

NEUTRINOS S. PAKVASA Department of Physics & Astronomy, University of Hawaii Honolulu, HI 96822 USA

1

Introduction

We can start with the question: why are neutrino properties especially interesting? Recall that in the minimal standard model there are no right handed neutrinos and furthermore lepton number is conserved so neutrinos can have neither Dirac nor Majorana masses. Furthermore, with zero masses, the mixing angles in the charged weak current are all zero. Any evidence for non-zero masses or mixing angles is evidence for physics beyond the standard model and hence potentially a powerful tool. Besides, the masses and mixing angles are fundamental parameters which will have to be explained by the eventual theory of fermion masses. Massless neutrino can be guaranteed by imposing chiral symmetry. Since chiral symmetry and just masslessness are difficult to distinguish at present, one can ask if there are any other fundamental reasons for m ll to be zero, such as gauge invariance for the photon and graviton masses. The only other massless particles we know are Nambu-Goldstone particles due to spontaneous breaking of global symmetries. If v was such a particle it would obey soft-v theorems analogues of soft 'IT theorems I. Hence any amplitude A with v as an external particle of momentum q should satisfy lim A(v) = 0 ql-' ~O

i.e. A should vanish linearly as q. This additional dependence of amplitudes on the neutrino momentum, for example, would distort Kurie plots for beta decay grossly from linearity. Since no such deviations are observed we can rest assured that neutrino is not a Nambu-Goldstone particle. Hence the neutrino is neither a gauge particle nor a Nambu-Goldstone particle and is not required to be massless. Once neutrinos have masses, by analogy we expect that the only distinguishing feature between flavors is different masses and mixings. Hence we would expect v e , vI-' and Vr (or the corresponding mass eigenstates VI, v2 and V3) to have different masses in general.

372

Let me review briefly the kinds of neutrino masses that can arise. Let 'IjJ L and O)ve in matter has higher energy than ve and vI-' has higher energy than vI-" Hence if there are lepton-number violating couplings to say, a massless Majoron, then decays such asl 5

(15) can occur in matter but not in vacuum. Matter effects are important for solar neutrinos (if 6m 2 '" 10- 4 to 10- 7 ev 2 ), for upcoming atmospheric neutrinos and can be for supernova neutrinos. v- Decays: For neutrinos below 1 MeV, the only decay modes possible are

(1) v'" (2) v'" (3) v'"

---+ ---+ ---+

v{3 v{3 V{3

+I + ViVi + M.

(16)

The decay rate for the radiative mode, at one loop level, in the standard model is given by

r 1 = G} m~ 1287r4

(~) ~ 16 7r

I" (!!!2..)2 L.J

•

mw

u-.",.{3 W 12

(17)

assuming mOl > > m{3. Cosmological and SN1987A limits on this mode are of order r / BR 1024 s. The rate for the 3v mode depends on whether it is one loop induced or whether there is GIM violation at tree level

(18)

377

It is not easy to arrange for this decay rate to be significant. The decay rate for Majoron or familon decay mode depends on the unknown Yukawa coupling, g, Magnetic Dipole Moment: A Majorana particle cannot have a non-zero magnetic dipole moment. Hence for a non-zero magnetic dipole moment either neutrino is a Dirac particle or the dipole moment is a transition moment between two different Majorana states (e.g. VeL and V~R). In the standard model, the one loop calculation yieldlf6 3m e GF

J-Lv.

= 4v'271'2

mv.J-LB

(19)

which is 3.1O- 19 (m v /eV)J-LB. With mixing to a heavy lepton of mL '" 100 GeV and mixing Ve-VL of 0.1, this can be enhanced to 1O- 14 J-LB. The simplest modifications of standard of model which can yield large magnetic dipole moment for Ve are ones with extra scalar fields. Double Beta Decay: Of particular interest for neutrino properties is the neutrino-less variety: (20) This can only happen if VC == V and mv =f. O( or Ve mixes with a massive Majorana particle). The decay rate depends on the mv and the nuclear matrix element 17.

r~

2 1 ~5 [MGT]2 FvGc . 0 yr

(21)

where G c is the Coulumb correction factor, Fv= m~/m~ f(€e/me) and MGT is the nuclear matrix element.

MGT "'< f

m

L n r;t; r;t

Um .

Unr~; i >

(22)

1

This is in the limit of small mv. The strongest limits on are for Ge of almost 10-24 yr.-I. Using calculated nuclear matrix elements this places a limit on a Majorana mass for Ve of m~. and the value>. '" 10- 12 required if the CMB anisotropies induced by the model are to have their observed values, it follows that (for (1 < mpt)

(26) This would imply no GUT baryogenesis, no GUT-scale defect production, and no gravitino problems in supersymmetric models with m3/2 > TR, where m3/2 is the gravitino mass. As we shall see, these conclusions change radically if we adopt an improved analysis of reheating.

Modem Theory of Reheating As was first realized in 29, the above analysis misses an essential point. To see this, we focus on the equation of motion for the matter field X coupled to the inflaton 'P via the interaction Lagrangian .cr of (22). Taking into account for the moment only the cubic interaction term, the equation of motion for the Fourier modes Xk becomes (27) where kp is the time-dependent physical wavenumber. Let us for the moment neglect the expansion of the Universe and assume the case of narrow resonance (g2(1'PO < w2). In this case, the friction term in (27) drops out and kp is time-independent, and Equation (27) becomes a harmonic oscillator equation with a time-dependent mass determined by the dynamics of 'P. In the reheating phase, 'P is undergoing oscillations. Thus, the mass in (27) is varying periodically. In the mathematics literature, this equation is called the Mathieu equation. It is well known that there is an one loop order, the cubic interaction term will contribute to >. by an amout .0.>' ~ >. smaller than g2 needs to be finely tuned at each order in perturbation theory, which is "unnatural". a At

g2. A renormalized value of

422

instability. In physics, the effect is known as parametric resonance (see e.g. Wn corresponding to half integer multiples of the frequency W of the variation of the mass, i.e. 30). At frequencies

(28) there are instability bands with widths ~wn. For values of instability band, the value of Xk increases exponentially: Xk "" e

I't

.

WIt

h

Wk

within the

2

9 mpo JL "" - - ,

(29)

W

with , with a coefficient tensor which depends on the background dynamics. In the slow-rolling approximation we obtain 82 (91) and (92) This demonstrates that the effective energy-momentum tensor of long-wavelength cosmological perturbations has the same form as a negative cosmological constant.

Acknowledgments I wish to thank the organizers for putting together an excellent school, and for inviting me to speak. I also thank my collaborators, in particular Raul Abramo and Slava Mukhanov, for collaboration on some of the work reported on at the end of these lecture notes. This work has been supported in part by the U.S. Department of Energy under Contract DE-FG029lER40688, Task A; and by an NSF collaborative research award NSF-INT-9312335.

References 1. S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972).

441

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Formation of Topological Defects Tanmay Vachaspati Physics Department, Case Western Reserve University, Cleveland OH 44106-7079, USA. In these lectures, I review cosmological phase transitions and the topological aspects of spontaneous symmetry breaking. I then discuss the formation of walls, strings and monopoles during phase transitions including lattice based studies of defect formation and recent attempts to go beyond the lattice. The close connection of defect formation with percolation is described. Open problems and possible future directions are mentioned.

1

Motivation and Introduction

An exciting development in cosmology has been the realization that the universe may behave very much like a condensed matter system. After all, the cosmos is the arena where very high energy particle physics is relevant and this is described by quantum field theory which is also the very tool used in condensed matter physics. In condensed matter systems we have seen a rich array of phenomenon and we expect that the cosmos has seen an equally rich past. A routine observation in condensed matter (and daily life!) is that of symmetry breaking which is the basis of all schemes in particle physics to achieve unification of forces. Symmetry breaking would lead to phase transitions in the early universe, making their study crucial to our understanding of the cosmos. There are many cosmic theories that hinge on processes happening during phase transitions. These include a large number of inflationary models, baryogenesis and structure formation. Many of the concepts that have gone into the explosion of cosmology in the last two decades are linked to each other. As an example, consider the birth of the inflationary idea. The success of the electroweak modelled particle physicists to attempt to unify the strong force with the electroweak forces by postulating Grand Unified Theories (GUTs). In such theories, a number of phase transitions occur and, based on mathematical results on the topology of group manifolds, it is known that GUTs always contain topological defects known as magnetic monopoles. The occurence of phase transitions in cosmology then tells us that monopoles must have formed in the early universe 1. In fact, standard cosmology then predicts an over-abundance of magnetic monopoles in our present universe that is clearly in conflict with observations 3. This head-on confrontation of particle physics and cosmology led Guth 2 to come

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up with the idea of an inflationary universe - an idea that is now deeply embedded in cosmology as it not only solves the monopole problem but also several other unrelated problems that standard (pre-inflationary) cosmology did not address. Magnetic monopoles are one example of topological defects that can arise during phase transitions. But topological defects can occur in other varieties as well. They can be one dimensional, in which case they are called "strings". If they are two dimensional, they are called "domain walls". Magnetic monopoles and domain walls are two kinds of topological defects that have unpleasant cosmological consequences unless they are kept: very light or are somehow eliminated at some early epoch (for example, by inflation). Cosmic strings are believed to be more benevolent and may even have been responsible for structure formation in the universe if they wfire formed at the GUT phase transition. The study of topological defects is fascinating for several reasons. As in the example of magnetic monopoles and inflation above, they can provide important constraints on particle physics models and cosmology. On the other hand, if a GUT topological defect is found, it would provide a direct window on the universe at about 10- 35 secs. In a manner of speaking, a part of the early universe is trapped in the interior of the defect - much like dinosaur fossils trapped in amber. GUT topological defects would also shed light on the problem of how galaxies were formed. Their discovery would give us important constraints on the symmetry structure of very high energy particle physics - this is non-perturbative information. In addition, it would give us information about phase transitions in the early universe, giving confidence in our understanding of the thermal history of the universe. Also, the topological defects would most likely have tremendous astrophysical impact since they are generally very massive and have unusual gravitational and electromagnetic properties. In the early 80's, people would talk of the marriage of particle physics and cosmology. I actually think there is something wrong with this picture since a third party is also involved. This is condensed matter physics. When it comes to understanding cosmological phase transitions, we are forced to consider the corresponding advances in condensed matter physics since many of the ideas are the same. Topological defects have been studied by condensed matter physicists for many, many years. If we want experimental input in our theories of the very early universe, we must go to the condensed matter laboratory. This is a growing area of research - condensed matter experiments are being inspired by cosmological questions, and, cosmologists are revising their theories based on experimental input.

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In these lectures, my aim is to discuss the formation of topological defects and to bring the reader to a point where the most relevant questions are apparent. The existing work on topological defects paints a certain picture of their formation but there are some limitations. I will describe both the picture and the limitations. I will start out by describing phase transitions and the "effective potential" way of treating them in field theory. Then I will describe why topological defects come about and finally get to the work on their formation in a phase transition. It is impossible to describe every aspect of this subject within these lectures especially since the subject branches out into a large variety of different areas of research. The references provided in the bibliography should help the reader dive deeper into whichever area he/she chooses to pursue. 2 2.1

Phase Transitions Effective Potential: Formalism

In statistical mechanics, the basic quantity one tries to find is the partition function Z where j3 = l/T and the sum is over all states with energy H. partition function one can derive the Helmholtz free energy A by

From the

and the derivatives of A then lead to the thermodynamic functions. To discuss phase transitions, one performs a Legendre transform on the Helmholtz free energy to get the Gibbs free energy C. For a gas, we have C(P,T)=A+PV while for an ensemble of spins, C(M,T) =A+HM where H is the external magnetic field and M is the magnetization of the system. In the magnetic system, we have

ac =H aM

449

and so, in the absence of an external magnetic field, the minima of G describe the various phases of the system. In field theory, these concepts carryover almost word for word 6. For the time being we restrict ourselves to the zero temperature case. Then, in analogy with the partition function, one defines the generating functional in the presence of an external current J (x) (analogous to the external magnetic field)

Z[J] =

f D¢>exp[i 1T dt f d x(.c[¢>] + J(x)¢>(x))] 3

where, the time integration is over a large but finite interval and Lagrangian density for the system. In other words,

Z[J] =< 0le-iHTIO >= exp[-E[J]]

(1)

.c is a suitable (2)

where, 10 > is the vacuum state and H is the Hamiltonian. The energy functional E[J] is the analog ofthe Helmholtz free energy. (Note however that E is not really an energy - for example, it does not have the dimensions of energy.) Now, we find,

JE[J] JJ(x)

J D¢>e i J(c+.74»¢>(x) J D¢>e i J(c+.74»

< OI¢>(x)IO >J -¢>cl(X) which is (minus) the order parameter. Next we get the effective action, which is the analog of the Gibbs free energy, by Legendre transforming the energy functional:

To obtain the value of the order parameter for any given external current, we need to solve:

and, in particular, when the external current vanishes, the phases are described by the extrema of the effective action. So far we have taken the external current to be a function of space but, in practice, the current is taken to be a constant and ¢>cl is also constant. Then, in this restricted case, only non-derivative terms can be present in the effective

450

action and the effective action leads to an effective potential with quantum corrections: 1

VeJj,q(l/Jct) == -VTr[l/Jctl where, VT is the spacetime volume. These issues and, in particular, the correspondence between statistical mechanics and field theory, are described very clearly in the textbooks by Peskin and Schroeder 6 and Rivers 7. 2.2

Effective Potential: Quantum Corrections

The actual calculation of the quantum effective potential is done by using perturbation theory 8,9. The general result is as follows. For the model:

where

OJ.lI/J = (OJ.l -

ieA~Ta)1/J

F;". = oJ.lA~ - o".A~

+ efabcAtA~

Cp = i1j)'YJ.l DJ.l1/J -1jJrj1/Jl/Jj in conventional notation, the one loop correction to the potential is the ColemanWeinberg correction:

Here J-L, M and m are the scalar, vector and spinor masses and u is a renormalization scale. The factors of 1, 3 and 4 in front of the three terms are due to the spin degrees of freedom of a scalar, massive vector and spinor (fermionantifermion) field. Also, the - sign in front of the fermionic contribution is due to Fermi statistics. The renormalization conditions used to derive this form of V1,q(l/J) are the "zero momentum" conditions 7 . As a specific simple example, consider the Lagrangian

(4) where,

~

is real and

451

Next write

cJ?(x) = tP,cI

+ X(x)

where, tP~1 is a constant and equal to 1'2/ A at tree level. Then,

L

= 21 (8/-1X )2 - 21 (3AtPcI2 -

2

2

A

3

I' )X - AtPclX - 4"X

4

1 2 2 + 21' tPcI -

A

4

4"tPcl

The last two terms are the tree level effective action for tPcl. We now want to find the one loop correction to the effective action. This comes about due to two ingredients: (i) the X fluctuations contribute to the energy of the vacuum, and, (ii) the mass ofthe X particles depends on the value of tPcl. Hence the X vacuum fluctuations (loops in Feynman diagram language) contribute to the potential felt by tPcl. There is only one Feynman diagram that contributes at one loop - simply a X particle propagating in a loop. Then the contribution to the effective potential from this one loop is:

V1,q(tPcl)

1 = 2(211")4

where,

m 2(tPcl)

f

d4 kln[k 2

= (3AtP~1 -

+ m 2 (tPcd ]

1'2)

A clear physical meaning can be obtained by performing the integration over ko: 1 (5) V1,q (tPcl ) -- (211")3 d3 k [-2 k + m 2( tPcl )]1/2

f

where an infinite constant has been removed by shifting the zero energy level. The integrand now is the energy of a X particle with momentum k and so the one loop correction to the effective potential is just the energy in all the modes of X - corresponding to the sum over hw /2 in field theory. In elementary applications of quantum field theory, this zero point energy is removed by normal ordering, but here it contains a non-trivial dependence on tPcl which cannot be removed and should be included in the effective potential. The integration in eq. (5) is divergent and needs to be renormalized. This is done by choosing a set of renormalization conditions. Linde's choice 9 is:

dV\ dtP and

-0

=/-1)..-1/2 -

452 where we have dropped subscripts on the effective potential and the order parameter to simplify notation. The final result for the effective potential is:

This result differs from the general form in eq. (3) because of a different choice of renormalization conditions. The physical consequences are, however, independent of the renormalization conditions one chooses 9.

2.3

Effective Potential: Thermal Corrections

Now we look at thermal corrections to the potential. These corrections can be understood as follows: a thermal bath necessarily contains a thermal distribution of particles. The properties of any particle is influenced by its interactions with the particles in the thermal background. This leads to an effective Lagrangian in which the effects of temperature are already included. The derivation of the temperature dependence of the effective potential is closely analogous to the derivation of the quantum corrections. The calculation is now done in Euclidean space with the Euclidean time coordinate being periodic with period 1/ (27rT). This means that the zeroth component, k o, of the four momentum of a particle is now discrete. For bosons, periodic boundary conditions are used and so ko is replaced by n(27rT) while for fermions, antiperiodic boundary conditions are necessary and ko is replaced by (n+ 1/2)27rT where n is any integer. Then integrals over ko get transformed to a sum over n: So n=+oo dko -t 27rT

f

L

n=-oo

For example, in the simple model at the end of the previous section, the temperature dependent correction to the effective potential is:

where, once again, m 2 (4)) = 3>"4>2 -

J.!2 •

The result of doing the sum and the integration and applying the renormalization conditions is:

(6)

453

in the temperature range T > > m to first order in A. We now describe a more intuitive (but equivalent) way of deriving the thermal corrections to the potential. The idea is that if we write

+X there is a thermal background of X particles that contribute to the effective potential for 4>. This contribution can be calculated by taking the ensemble average of the microscopic (bare) potential for 4>. So,

where denotes ensemble averaging. The ensemble average of odd powers of X vanish by symmetry while the average of J-t 2 X2 and X4 simply shift the zero level. The only non-trivial contribution comes from

Replacing X by an expansion in terms of creation and annihilation operators gives: 2 _ 1 d3 k t < X >- (27r)3 2Wk < 2ak ak + 1 >

f

The operator alak is simply the number operator and, since the number distribution of spin zero particles with momentum k in a thermal bath is given by the Bose distribution, we have

where, Wk = (k 2 + m 2 ) 1/2. After subtracting out the constant infinite piece from < X 2 > we find

where we have also rescaled the integration variable by liT to make it dimensionless and in = miT. In the limit of large T, we have in -+ 0 and the integral can be done in closed form. In this limit, the result reduces to eq. (6) without the 4> independent terms. A plot of the effective potential is shown in Fig. 1. At low temperatures, there are two minima and, in a rapid quench, the system would have to transit

454

HighT \

LowT

\ \

\

\ \ \ \ \ \

\

...

...

... ...

Figure 1: Effective Potential.

from the minimum at ¢ = 0 to the minimum at ¢ "# O. Here the transition can take place continuously and is a second-order phase transition. A number of different behaviours for the effective potential have been found. In certain systems, there are two phases with an energy barrier separating them. This leads to first order phase transitions. If the barrier separating the two phases is very large, the transition from one phase to another would have to be somehow activated over the barrier or quantum mechanical tunneling would eventually complete the transition. However, both processes can be very slow to occur and so the system can be trapped in the higher energy phase (false vacuum) for a long time. In other words, the system can supercool. Note that we have been working with examples where cooling leads to spontaneous reduction in symmetry. This is indeed generic. However, many

455

instances are also known where cooling leads to symmetry restoration 10. This has been observed in Rochelle salts. 2.4

Order of the Transition: Ehrenfest Classification

The role of the thermodynamic potentials is played by the effective potential. So one should take derivatives of VeJ J with respect to the temperature and find the order of the lowest derivative which is discontinuous at the critical temperature. This will give the order of the phase transition. As an example, consider the simple model in eq. (4) for which the effective potential is given in eq. (6). For convenience, let us write:

where, we have defined M2 to absorb the lengthy expressions in eq. (6). The critical temperature, Te, is defined by

The stable phases of the system are defined by dV = 0 dif;

and with the second derivative being positive. Hence, when M2(T) is negative (i. e. high temperatures), the phase is:

if; = 0,

Phase I

and when M2(T) is positive (i.e. low temperatures):

if; =

M$'),

Phase II

Now we find the potential in both phases:

= 0, (T) = _ [M(T)]4 VJ(T)

Vi II

From these expressions, using M(Te)

4,X

= 0, we have

456 Also, dVI

dT

I

= 0 = dVI I

Tc

dT

I Tc

since M2(T) is of the form (J-l2 - aT2) where a is a constant. (Therefore the derivative of M2(T) with respect to T is well behaved and so the derivative of M4 vanishes.) However,

and so we conclude that the phase transition is second order. The above scheme (Ehrenfest classification) for defining the order of a transition in terms of discontinuities in the derivatives of the potentials can fail if any of the derivatives of the potential do not exist. Generally, one says that the phase transition is first order if there is a barrier in the effective potential that separates the two vacuua. If the tunneling probability from the false vacuum to the true vacuum is very small, the phase transition is said to be strongly first order but if the tunneling rate is not too small, it is weakly first order. If there is no barrier between the different phases, the phase transition is said to be second-order. However, this terminology does not imply a definite understanding of how the phase transition proceeds. Only in the case of a strongly first order phase transition does one know that the phase transition proceeds by the growth of bubbles of the lower energy (true vacuum) phase.

2.5

Limitations of the Effective Potential

Even in thermodynamics, the Van der Waal's equation of state for a gas leads to a PV diagram in which the derivative of P with respect to V is positive. But this is unphysical since it is an unstable situation: an increase in the volume leads to an increase in the pressure, which leads to a further increase in the volume and so on. The resolution was found in the assumption used to derive the PV diagram - that the sample is entirely in one phase. More accurately, there will be regions of PV space where the system will consist of an admixture of phases. And the free energy of the coexisting phases can be lower than the free energy of just one phase. In our discussion of phase transitions in field theory we have also treated the order parameter ¢cI as being uniform in space. In reality, ¢cI will vary over space and the two phases will coexist. If one applies a Maxwell construction to the effective potential, the result is a straight line joining the two vacuua. The dynamics of transiting from one phase to the other depends on the rate at which external parameters are varied, the shape and structure of the

457

effective potential, and other factors. Gravitational effects may be important in certain situations too. For example, if the tunneling rate is very small, the universe could start inflating and the phase transition would never complete. To add to these complications is the presence of topological defects that prevent the transition from false to true vacuua from occurring globally. 3

Topological Defects

We first consider the existence of topological defects. In the context of the previous lectures, these may be viewed as obstructions to the completion of a phase transition. In this lecture, however, we will simply view them as classical solutions in a model that exist for topological reasons. I will start out by providing the simplest examples of topological defects and then later discuss their classification via homotopy groups. 3.1

Domain Walls

Consider the Z2 Lagrangian in 1+ 1 dimensions

(7) where ¢ is a real scalar field - also called the order parameter. The Lagrangian is invariant under ¢ -+ -¢ and hence possesses a Z2 symmetry. For this reason, the potential has two minima: ¢ = ±1]. And the "vacuum manifold" has two-fold degeneracy. -00. Consider the possibility that ¢ +1] at x +00 and ¢ -1] at x In this case, the continuous function ¢(x) has to go from -1] to +1] as x is taken from -00 to +00 and so must necessarily pass through ¢ = O. But then there is energy in this field configuration since the potential is non-zero when ¢ = 0. Also, this configuration cannot relax to either of the two vacuum configurations, say ¢(x) = +1], since that involves changing the field over an infinite volume from -1] to +1], which would cost an infinite amount of energy. Another way to see this is to notice the presence of a conserved current:

=

=

=

=

where J1., v = 0,1 and (I'V is the antisymmetric symbol in 2 dimensions. Clearly jl' is conserved and so we have a conserved charge in the model: Q=

f

dxjO = ¢(+oo) - ¢(-oo) .

458

=

=

For the vacuum Q 0 and for the configuration described above Q 1. So the configuration cannot relax into the vacuum - it is in a different topological sector. To get the field configuration with the boundary conditions ¢(±oo) = ±7], one would have to solve the field equation resulting from the Lagrangian (7). This would be a second order differential equation. Instead, one can use the clever method first derived by Bogomolnyi 11 and obtain a first order differential equation. The method uses the energy functional:

E

f f f

dX[(Ot¢)2

+ (ox¢)2 + v(¢)]

dX[(Ot¢)2

+ (ox¢ + Jv(¢)

)2 - 2JV(¢)ox¢]

dX[(Ot¢)2 + (Ox¢ + JV(¢) )2]- 2

((+oo) 1(-00)

d¢'JV(¢')

Then, for fixed values of ¢ at ±oo, the energy is minimized if

and Furthermore, the minimum value of the energy is:

1

(+00)

Emin = 2

d¢'JV(¢') .

( -00)

In our case,

V(¢) = ~(¢2 _ 7]2)2

4 which can be inserted in the above equations to get the "kink" solution:

.J>.TJX) ¢ = 7]tanh ( -2 for which the energy is: 4 "ATJ3 Ekink = 3"V

Note that the energy density is localized in the region where ¢ is not in the vacuum, i.e. in a region of thickness 2(v'XTJ)-1 around x = o.

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We can extend the model in eq. (7) to 3+1 dimensions and consider the case when ¢ only depends on x but not on y and z. We can still obtain the kink solution for every value of y and z and so the kink solution will describe a "domain wall" in the yz-plane. Notice that the existence of the domain wall only depends on the fact that there were discrete vacuua in the theory.

3.2

Cosmic Strings

Consider the Lagrangian:

(8) where ¢ is now taken to be a complex scalar field. The Lagrangian is invariant under ¢ -+ ¢' = eiO/¢ and hence the model has a U(l) (global) symmetry. The vacuum expectation value of ¢ is 'T}eiO/ where Q can take any value. So the ground state of the model has continuous degeneracy. The degeneracy is labelled by the phase angle Q and hence the vacuum manifold is a circle. Vortices are formed if we consider the model in two spatial dimensions and let Q be such as to wrap around the vacuum manifold. For example, we could take Q 0, the polar angle. Then, since the field is single valued everywhere, there must be at least one point at which ¢ = O. The field carries energy at this point since ¢ = 0 is not on the vacuum manifold. The location of this point may be defined as the location of a vortex. (An example of a vortex is universally encountered by people taking baths or washing dishes. As the water flows down the drain, it circulates. We cannot interpolate the circulating velocity field all the way to the center of the vortex since it would have to become multi-valued. Instead the fluid density in the central region of the vortex vanishes.) If we now take the model in three spatial dimensions, the vortex becomes a line stretching in the third dimension. The vortex line is called a "string". The crucial element in the existence of the vortex was that Q could "wrap around" the vacuum manifold. In other words, vortices exist if the vacuum manifold contains incontractable closed paths. Bogomolnyi's method cannot be applied to construct the vortex solution in model (8). In fact, the energy of an isolated global vortex diverges. If the model (8) is gauged, then Bogomolnyi's method does work for a particular choice of parameters in the model. Note that gauging the mod'el makes no

=

460 difference to the vacuum manifold and so the topological arguments that show the existence of vortices still apply.

3.3

Monopoles

The model:

L = laJJ il 2 - ~(i2 _1]2)2 4

i

where is a triplet of scalar fields contains global monopole solutions. To see this, note that the Lagrangian is invariant under

where, R is a rotation matrix in three dimensions. Hence the model has an 0(3) (global) symmetry which is broken down to 0(2) once gets a vacuum expectation value. For example, if ex i 3 , then the rotations in the unbroken group are the rotations about the C3 axis. A monopole solution is obtained when

i

i

i ex cr = (sinOcos¢>, sinOsin¢>, cosO) where 0 and ¢> are the angular spherical coordinates. For this "hedgehog" configuration of the field, there must be a point in space where = 0 and the energy density is non-vanishing. In fact, in the global symmetry case the energy of the monopole is infinite because of the slow fall off of gradient energy at infinity. If the model is gauged, the field configuration can be accompanied by a gauge field that cancels off the gradient energy at infinity. This then leads to a finite energy solution but with a non-vanishing magnetic flux at infinity the famous "magnetic monopole" of 't Hooft and Polyakov 12 ,13. For the magnetic monopole, the field on the asymptotic two sphere has to be non-trivial. So if the vacuum manifold admits incontractable two spheres, we can have mappings from spatial infinity to the vacuum manifold that cannot be smoothly deformed to the trivial mapping (in which all of space is assigned the same point on the vacuum manifold). And each such mapping would lead to a monopole solution.

i

i

3.4

Textures

A simple model with (global) texture is

(9)

461

where a = 1,2,3,4. Now the vacuum manifold is a three sphere. If our universe is a three sphere, 4>a could wrap around it an integer number of times. Such a solution is a texture - the field does not vanish anywhere but there is still some gradient energy present. In the cosmology literature, the term texture does not necessarily refer to the case when the universe is a three sphere. One can consider a ball in space and find the winding of the field within this ball. Such configurations can have any winding. The term texture is taken to apply to the situation where the winding is unity. Such textures are time-dependent solutions. They collapse to a point where the configuration unwinds and then the energy radiates away. (The scalar field is zero at one point in space-time.) 3.5

Hybrids

In a sequence of symmetry breakings, one can get "hybrid" defects such as walls bounded by strings, or, monopoles connected by strings. For example, in the symmetry breaking sequence:

SU(2) -+ U(l) -+ 1 the first symmetry breaking yields monopoles which get connected by strings in the second stage of symmetry breaking. Similarly one can get domain walls that are bounded by strings. For further discussion of hybrid defects see Vilenkin and Shellard 14.

3.6

Topological Criterion and Homotopy Groups

The criterion for having domain walls in a model in which the symmetry group G is spontaneously broken to H by the vacuum expectation value of a field can now be specified. First, if H is the trivial group, any element of G acting on the order parameter would yield a possibly new value of the order parameter which would still be in the minima of the potential and hence would be a degenerate vacuum. Then the manifold of vacuum states is simply given by the manifold of G. Next, if H is not the trivial group, the non-trivial elements of H leave the order parameter invariant. But since H is a subgroup of G, there are elements of G whose action on the order parameter is identical. In fact, the elements of G can be ascribed to equivalence classes - the action of each element within an equivalence class on the order parameter is identical but the action of two elements belonging to two different equivalence classes can be different. These equivalence classes are nothing but the elements gH of the coset space Gj H - elements of gH acting on the order parameter give the same

462

result as 9 acting on the order parameter since elements of H leave the order parameter invariant. So the vacuum manifold is the manifold corresponding to G j H. The connectivity of a manifold is described by the zeroth homotopy group of the manifold: 7ro(Gj H). Domain walls are present in the model if the vacuum manifold has disconnected components, that is, if 7ro( G j H) is not trivial. Now we can generalize these considerations further. The next step is to consider a vacuum manifold that contains incontractable closed paths. For example, the vacuum manifold could be a one sphere as in the U(l) case when