Nonperturbative aspects of strings, branes and supersymmetry : proceedings of the Spring School on Nonperturbative Aspects of String Theory and Supersymmetric Gauge Theories, ICTP, Trieste, Italy, 23-31 March 1998 ; proceedings of the Trieste Conference on Super-Five-Branes and Physics in 5+1 Dimensions, ICTP, Trieste, Italy, 1-3 April 1998 edited by B. Greene ... [et al.]. 9789810237851, 9810237855

Table of contents :
Supersymmetry effective actions in four dimensions, P.C. Argyres
supermembranes and M(atrix) Theory, H. Nicolai and R. Helling
etudes on D-branes, C.V. Johnson
two lectures on D-geometry and noncommutative geometry, M.R. Douglas
two lectures on AdS/CFT correspondence, M.R. Douglas and S. Randjbar-Daemi
phenomenological aspects of string theory, J. Louis
conformal symmetry on world volumes of branes, P. Claus et al
non-supersymmetric strings, S. Kachru
self-dual tensors in six-dimensional supergravity, F. Riccioni and A. Sagnotti
Euclidean-signature supergravities and dualities, E. Cremmer et al
AdS gravity and field theories at fixpoints, K. Behrndt
superembeddings, P.S. Howe
on some features of the M-5-brane, D. Sorokin
brane dynamics and four-dimensional quantum field theory, N.D. Lambert and P. West
anti-de-Sitter space, branes, singletons, superconformal field theories and all that, M.J. Duff
aspects of the M-5 brane, E. Sezgin and P. Sundell
matrix theory black holes and the cross-Witten transition, L. Susskind
domain walls and the universe, K.S. Stelle
correlation functions of the global E8 symmetry currents in the heterotic 5-brane theory, M. Krogh
on the construction of SL(2,Z) type IIB 5-branes, J.X. Lu and S. Roy
on the (2,0) theory in six dimensions, M. Rozali
the no-ghost theorem and strings on AdS3, J.M. Evans et al
six-dimensional fixed points from Hanany-Witten setups, A. Karch and I. Brunner.

Citation preview

NONPERTURBATIVE ASPECTS OF STRINGS, BRANES AND SUPERSYMMETRY

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the

abdus salam

international centre for theoretical physics united nations educational, scientnic and cultural organization

international atomic energy agency

~ :tJ/t/\

NONPERTURBATIVE ASPECTS OF STRINGS, BRANES AND SUPERSYMMETRY Proceedings of the Spring School on Nonperturbative Aspects of String Theory and Supersymmetric Gauge Theories ICTP, Trieste, Italy

23-31 March 1998

Proceedings of the Trieste Conference on Super-Five-Branes and Physics in 5 + 1 Dimensions ICTP, Trieste, Italy

1- 3 April 1998

Editors

M. Duff, E. Sezgin and C. Pope Texas A&M University, USA

B. Greene Cornell University, USA

J. Louis Martin-Luther Universitat, Halle, Germany

K. S. Narain, S. Randibar·Daemi and G. Thompson /CTP, Trieste, Italy

'

1 World Scientific p I

Singapore • New Jersey• London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P 0 Box 128, Farrer Road, Singapore 912805 USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661 UK office: 51 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data

Trieste Spring School (1998 : Trieste, Italy) Nonperturbative aspects of strings, branes, and supersymrnetry : proceedings of the Spring School on nonperturbative aspects of string theory and supersymmetric gauge theories, ICTP, Trieste, Italy, 23-31 March 1998: proceedings of the Trieste Conference on Super-Five-Branes and Physics in 5 + 1 Dimensions, ICTP, Trieste, Italy, 1-3 Apri11998/ editors, M. Duff ... [et al.]. p. em. "1998 Trieste Spring School''-- Pref. At head of title: The Abdus Salam International Centre for Theoretical Physics; United Nations Educational, Scientific and Cultural Organization; International Atomic Energy Agency. ISBN 9810237855 (alk. paper) 1. String models -- Congresses. 2. Supersymmetry -- Congresses. I. Duff, M. J. II. Trieste Conference on Super-Five-Branes and Physics in 5 + 1 Dimensions (1998: Trieste, Italy). III. Abdus Salam International Centre for Theoretical Physics. IV. Unesco. V. International Atomic Energy Agency. VI. Title. QC794.6.S85T75 1998 530.14--dc21 99-27206 CIP

British Library Cataloguing-in-PubUcation Data

A catalogue record for this book is available from the British Library.

Copyright @ 1999 by The Abdus Salam International Centre for Theoretical Physics

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

v

PREFACE These proceedings consist of two parts, the 1998 Trieste Spring School and the Trieste Conference on Super-five-branes and Physics in 5+ I Dimensions. There are seperate prefaces for these two activities where more detailed descriptions of each can be found. The Spring School, as always, was devoted to lectures which include a pedagogical element with the emphasis being to build up the required background needed to follow the latest developments. The talks presented during the conference on superfive-branes and physics in 5+ 1 dimensions were devoted to front line research in important areas in the new M-theory revolution. Both meetings took place in the midst of much excitement in string theory and M-theory. Part of this excitement stems from the continued development that comes from duality and p-branes. For this reason the study of p-branes formed a major part of the school and a substantial part of the conference. However, a new impetus comes from a different quarter. One of the most impressive results to come out of the latest revolution is the AdS/CFT correspondence that relates classical supergravity theory on manifolds of the form AdS x X to superconformal Yang-Mills theories on the boundary of the anti-de Sitter space. This theme appeared both in the school and, rather prominently, in the conference. We trust that, in keeping with past proceedings of the Trieste schools and conferences, this volume will prove to be a useful asset to all those wishing to learn about the latest results in string theory and M-theory.

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·vii

CONTENTS

Preface

v

Spring School on Non-Perturbative Aspects of String Theory and Supersymmetric Gauge Theories

Preface Supersymmetric Effective Actions in Four Dimensions P. C. Argyres

xi 1

Supennembranes and M(atrix) Theory H. Nicolai and R. Helling

29

Etudes on D-Branes C. V. Johnson

75

Two Lectures on D-Geometry and Noncommutative Geometry M. R. Douglas

131

Two Lectures on AdS/CFf Correspondence M. R. Douglas and S. Randjbar-Daemi

157

Phenomenological Aspects of String Theory J. Louis

178

Trieste Conference on Super-five-branes and Physics in 5+1 Dimensions

Preface

209

Confonnal Symmetry on World Volumes of Branes P. Claus, R. Kallosh and A. Van Proeyen

211

Nonsupersymmetric Strings S. Kachru

231

viii

Self-Dual Tensors in Six-Dimensional Supergravity F. Riccioni and A. Sagnotti

241

Massive Branes E. A. Bergshoeff

254

Euclidean-Signature Supergravities and Dualities E. Cremmer, /. V. Lavrinenko, H. Lii, C. N. Pope, K. S. Stelle and T. A. Tran

264

AdS Gravity and Field Theories at Fixpoints K. Behrndt

291

Superembeddings P. S. Howe

301

On Some Features of the M-5-Brane D. Sorokin

312

Brane Dynamics and Four-Dimensional Quantum Field Theory N. D. Lambert and P. C. West

324

Anti-de Sitter Space, Branes, Singletons, Superconformal Field Theories and All That M.J. Duff

334

Aspects of the M5-Brane E. Sezgin and P. Sundell

369

Matrix Theory Black Holes and the Gross-Witten Transition L. Susskind

390

Domain Walls and the Universe K. S. Stelle

396

Correlation Functions of the Global E 8 Symmetry Currents in the Heterotic 5-Brane Theory M. Krogh

407

ix

On the Construction of SL(2, Z) Type liB 5-Branes J. X. Lu and S. Roy

417

On the (2, 0) Theory in Six Dimensions M. Rozali

427

The No-Ghost Theorem and Strings on AdS 3 J. M. Evans, M. R. Gaberdiel and M. J. Perry

435

Six-Dimensional Fixed Points from Hanany-Witten Setups A. Karch and/. Brunner

445

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xi

PREFACE The focus of the spring school was on the recent developments in understanding non-perturbative aspects of string theory and supersymmetric field theories. The lectures by P. Argyres discuss duality in four-dimensional supersymmetric field theories. H. Nicolai reviewed some of the older work on supermembranes of the late '80s and nicely developed the connection with the recently proposed M(atrix) Theory. The lectures by C. Johnson center around D-branes, which are the decisive new element of string theory. D-branes and their non-commutative properties are also the topic of the lectures by M.R. Douglas. An exciting new duality between string theory and conformally invariant gauge theories in the Iarge-N limit was the subject matter of the lectures by M.R. Douglas and S. Randjbar-Daemi. Finally, the lectures by J. Louis focus on some phenomenological implications of the recent advances in understanding the non-perturbative structure of string theory.

B. Greene J. Louis K.S. Narain S. Randjbar-Daemi

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SUPERSYMMETRIC EFFECTIVE ACTIONS IN FOUR DIMENSIONS PHILIP C. ARGYRES Newman Laboratory, Cornell University, Ithaca NY 14853, USA argyrestmail.lns.cornell.edu

=

I review the construction of N 0, 1, and 2 supersymmetric low energy effective actions in four space-time dimensions.

1

Introduction

The aim of these lectures is to introduce some of the arguments that have been used successfully in the last five years to obtain exact information about strongly coupled field theories. I will focus on four-dimensional field theories (without gravity), although the techniques described here have been applied to theories in other dimensions and to stringiM theory as well. The basic notion is that of a low energy (or Wilsonian) effective action. This is simply a local action describing a theory's degrees of freedom at energies below a given scale E. An example is the low energy effective action for QCD, chiral perturbation theory describing the interactions of pions at energies E < AQCD· In such a theory particles heavier than AQcD are included in the pion theory as classical sources. Other examples are the various ten and eleven-dimensional supergravity theories, which appear as effective actions for stringiM theory at energies below their Planck scales. The effective action is obtained by averaging over (integrating out) the short distance fluctuations of the theory. If there is a sufficiently small ratio E I A between the cutoff energy scale E and the energy scale A characteristic of the dynamics of the degrees of freedom being averaged over, renormalization group arguments imply that the effective action can be systematically expanded as a power series in E I A-essentially an expansion in the number of derivatives of the fields. We will use low energy effective actions to analyze four dimensional field theories by taking the limit as the cutoff energy scale E goes to zero, or equivalently, by just keeping the leading terms (up to two derivatives) in the low energy fields. I will call such E -+ 0 low energy effective actions infrared effective actions (IREAs). The idea is to guess an IR effective field content for the microscopic (UV) theory in question and write down all possible IREAs built from these fields consistent with the global symmetries of the UV theory. For a "generic" UV theory this is no better than doing chiral perturbation theory

2

for QCD, and would seem to give little advantage for obtaining exact results. However, if the theory has a continuous set of inequivalent vacua, it turns out that selection rules from global symmetries of the UV theory can sometimes constrain the IREA sufficiently to deduce exact results. There are a number of reviews deriving these exact results 1 assuming the constraints from supersymmetry. The purpose of these lectures will be to deduce and explain these constraints in a relatively non-technical way. We will start with properties of general IREAs and then progressively 2 supersymmetry. The con1 and then N specialize to those with N straints on the IREAs become progressively more restrictive as the number of supersymmetries is increased; in the N = 2 case they are strong enough to allow quite general and restrictive properties of the moduli space of vacua of gauge theories to be deduced. Important topics omitted include the properties of interacting IREAs-the representation theory of superconformal algebras 2 and their use in analyzing IREAsf instead these lectures concentrate on IR free effective actions. Also missing (as much as possible) are details of supersymmetry algebras and the construction of their representations-many good texts and review articles cover this material 4 -or the application of the ideas presented here to theories in other dimensions.5 Since an IREA describes physics only for arbitrarily low energies, it is, by definition, scale invariant: we simply take the cutoff scale E below any finite scale in the theory. Scale invariant theories and therefore IREAs can therefore fall into one of the following categories: Trivial theories in which all fields are massive, so there are no propagating degrees of freedom in the far IR. Free theories in which all massless fields are non-interacting in the far IR. (They can still couple to massive sources, but these sources should not be treated dynamically in the IREA.) An example is QED, in which the IREA describes free photons when the lightest charged particle is massive. Interacting theories of massless degrees of freedom which are usually assumed to be conformal field theories.6 We generally have no effective description of interacting conformal field theories in four dimensions a so we must limit ourselves to free or trivial theories in the IR. A large class of these is given by the Coleman-Gross theorem 7 which states that for small enough couplings any theory of scalars, spinors, and U(l) vectors in four dimensions flows in the IR to a free theory. We will therefore focus on IREAs with this field content. Note that other IR free theories are known, for example non-Abelian gauge theories with sufficiently many massless

=

=

asee however lectures in this volume on the anti de Sitter/conformal field theory correspondence.

3

charged scalars and spinors. They will not play as important a role as the U(l) theories, since even within supersymmetric theories they can be destabilized by adding mass terms.

2

IREAs with No Supersymmetry

We thus take the field content of our IREA to be a collection of real scalars ,\0_1, where o_l is some charge -1 operator. Say we are interested in the appearance of a given operator 0 _ 10 of charge Q(O-lo) = -10 among the quantum corrections. Normally, one would say that this operator can appear only at tenth and higher orders in r ,_ A,100 -10 + A\ll\0 . theory.. uT per t ur ba t 1on A -10 + . . . + A'10 e-1/I-XI 2 o -10 + ... , (assuming that there is a regular ,\ -t 0 limit, so that no negative powers of,\ are allowed), where I've also indicated potential non-perturbative contributions as well. However, by the above argument we learn that only the tenth-order term is allowed, all the higher-order pieces, including the non-perturbative ones, are disallowed since they necessarily depend on ,\ non-holomorphically. Even more importantly, any operator of positive charge under the U(l) symmetry is completely disallowed, since it would necessarily have to have inverse powers of ,\ as its coefficient. But since we assumed the ,\ -t 0 weakcoupling limit was smooth (i.e. the physics is under control there), such singular coefficients are disallowed. Note that this is again special to supersymmetry, for if non-holomorphic couplings were allowed, one could always include such operators with positive powers of Xinstead. This argument can be summarized prescriptively as follows~ 5 The effective (macroscopic) TJ J is constrained by

(1) holomorphy in the (microscopic) coupling constants, (2) "ordinary" selection rules from symmetries under which the coupling

constants may transform or from electric-magnetic duality, and

(3) smoothness of the physics in various weak-coupling limits. Much of the progress in understanding the non-perturbative dynamics of supersymmetric gauge theories of the past half decade years has resulted from the systematic application ofthe above argument. The most important application of this argument is to AF gauge theories. Consider an AF gauge theory with complex strong coupling scale

(42) so that the one loop complex coupling at a scale p. is

r=

;;i

log (;) .

(43)

The coefficient of the one-loop beta function is given by bo =

~T(adj)-! ET(.R;) 2

2 I.

(44)

18

in an N = 1 supersymmetric gauge theory, where the sum is over each chiral multiplet (¢i, t{i) transforming in the representation R, of the gauge group. This follows from Eq. 27 if we recall that the vector multiplet includes one Weyl fermion in the adjoint representation, and each chiral multiplet has a complex scalar and Weyl fermion in the representation R.. As in Eq. 30, the effective Tat scale 1-' will have the functional form

(A) + f(A

r(A, ¢;';. !-') = 2bo 1ri log ;

b0

,

¢;';· !-'),

(45)

where f is now an arbitrary holomorphic function of its arguments. The important point is that T can only depend holomorphically on A and ¢;'. Since we are dealing with an AF theory, the A -+ 0 limit corresponds to the weak coupling limit, in which the effective couplings should not diverge. Thus we have T

=~log(~) + 21rz

1-'

f:

Abonan(¢1 ; p),

(46)

n=l

(i.e. inverse powers of Abo do not appear). By comparing this expression to the perturbative expansion, where logA"' lfg 2 (a one loop perturbative contribution) while Abon "'e-Smr 2 /g 2 (an n-instanton contribution), we see that the gauge coupling T in the Wilsonian effective action only gets one loop corrections in perturbation theory, t.'&ough non-perturbative corrections are allowed. As in the discussion of Sec. 2.2, these considerations also apply to the U(l)n couplings TJJ of the IREA of an AF theory Higgsed with a sufficiently large vev. In the case of a single U(l) and a single chiral multiplet vev ¢;,the IREA coupling has the form (47)

Here ¢; is some gauge invariant combination of the scalar Higgs fields in the UV theory, which can be determined classically; the power a with which it appears is determined by dimensional considerations. Now, as we make a large circle in the ¢; plane, T undergoes the monodromy T-+ T -

a,

(48)

which should be an element of Sp(2, Z), implying that a is an integer. In any given AF gauge theory Higgsed to U(l)'s this can indeed be checked to be the case, and is a reflection of the Witten effect? The combination of the Sp(2n, Z)

19

and holomorphic properties of TJJ in some cases is sufficient to determine it exactly.12 •16 We should emphasize the main limitation of this "not much renormalization" theorem: it is only derived for weakly coupled theories where the description in terms of the microscopic degrees of freedom is good. As we run the RG down to the IR, the theory will become strongly coupled, and our description in terms of the tPi and :F fields may break down. For example, the above non-renormalization theorem can be sharpened in an important way by using the selection rules of other global symmetries in the theory. An important new element is the treatment of the selection rules stemming from anomalous symmetries, and leads to exact non-perturbative expressions for the Wilsonian beta-function in N = 1 theories.17 However, for low enough scales, these exact beta functions have singular behavior indicative of the breakdown of the description of the low energy physics in terms of the assumed degrees of freedom. (These exact beta functions can nevertheless be used to show the existence of exactly marginal operators in many interesting cases.18 ) Finally, it is important to note that the statements of this and other non-renormalization theorems only hold in a renormalization scheme which preserves the supersymmetric selection rules.19 For instance, the scalar field strength renormalizations depend on the UV parameters as well as their complex conjugates since the Kahler potential does. So, if one worked in a scheme in which one insisted on the canonical normalization of the scalar kinetic terms, one would have to rescale the holomorphic TJ J couplings by the nonholomorphic field strength renormalizations, thus invalidating the supersymmetric selection rule.

3.2 N = 1 Supersymmetric Effective Action Potential Terms So far we have dealt only with the effective action in the far IR limit which only massless neutral scalars and U(l) gauge bosons survive. This begs the question of whether a non-trivial moduli space exists for a given theory. To answer this question we need to examine possible N = 1 supersymmetric potential terms for both neutral and charged scalar fields. We can not do this in as easy and direct manner as we did for the kinetic terms since inclusion of potential terms necessarily takes us off-shell-by definition the potentials vanish on constant scalar field configurations satisfying the equations of motion. (The supersymmetric constraints on possible scalar potential terms can, of course, be deduced from a direct but fairly technical construction of supersymmetric actions~)

We will deduce the supersymmetric form of the potential terms by the

20 following indirect argument. Denote the (Kahler) manifold of scalar vevs in the IREA of Eq. 41 by Mo = {¢i}. Now think of this action as an effective action at some finite scale E and imagine turning on some relevant operators (a potential) at a scale much less than E. The resulting IREA will again be of the form of Eq. 41, though the set of fields will be smaller (i.e. just those minimizing the potential). In particular the moduli space M of the new IREA must also be a Kabler manifold. Since this construction takes place at arbitrarily weak coupling (since the IREA is IR free), the same set of low energy degrees of freedom can be used to describe the effective action at all scales below E, and so M must be some Kahler submanifold of Mo. It now only remains to write the most general potential whose minimization picks out such a submanifold and also preserves the invariance of the effective action holomorphic reparameterizations (field redefinitions). One way of singling out a Kahler submanifold M C Mo is by specifying a set of holomorphic conditions {F;(¢-f) = 0}. Then it is straightforward to check that M = Mo/{F; = 0} is not only a complex submanifold but also is necessarily istelf Kahler with Kahler potential simply the restriction of the Kahler potential on Mo. The potential giving rise to these complex conditions must itself be real, though, suggesting it must be of the form V = Ei F;Fr. This, however, is not a reparametrization invariant formula. The correct formula must use the Kahler metric on Mo to contract the indices of the F;, implying an F term potential

(49) where giJ is the inverse of 9iJ· Postive definiteness of g (from unitarity of the effective action) implies that Vp takes its minimum value of 0 when the F terms vanish individually: (50) F; =0. Note that Vp = 0 is also the condition for supersymmetry to be unbroken in the vacuum, since if not the supersymmetry algebra Eq. 33 implies that ({Q, Q}) =f. 0 and therefore for some component of Q we would have QIO) =f. 0. There is a further constraint (besides holomorphicity) on the F;. Claiming that Vp is reparametrization invariant assumes that Fi transforms as a vector under reparametrizations. The only way (without some extra structure on Mo) to form such an object out of functions of the scalar vevs 4>i is as the derivative of a holomorphic function on Mo: F; = W,;.

(51)

W is an arbitrary (gauge invariant) holomorphic function on Mo called the superpotential.

21

A general W gives rise to an independent condition F; = 0 for each complex coordinate direction i, and thus generically one expects M to be a single point. It would seem that we are no closer to getting a noJ!-trivial moduli space of vacua with N = 1 supersymmetry than we were without it. But the holomorphicity of W gives rise to non-renormalization theorems (in the same way that the holomorphicity of TfJ does) which allow one (in favorable cases) to specify UV couplings which lead to special low energy superpotentials which admit non-trivial moduli spaces.20 Another way of putting it is that even though non-trivial moduli spaces of inequivalent vacua are still "accidental" in N = 1 supersymmetric theories, our knowledge of their RG flows allow us to arrange the necessary accident. When Mo has isometries, there is an additional structure that one can use to construct potential terms. Suppose Mo has a U(l)n group of isometries generated by ¢; -t ¢; + {} for I = 1, ... , n. Then we can gauge those isometries with the low energy U(1)n gauge group; equivalently, and perhaps more descriptively, we turn on an electric charge under the U(l)I gauge group for the scalars which are shifted under the action of the U(l)I isometry. (Recall the relation between isometries and low energy gauge groups mentioned in Sec. 2 above.) For example, if the U(l) isometries are realized linearly, then e} = qn¢; (no sum on i) where qn is the charge of the complex scalar ¢i under the U{l)I gauge group. Appropriately minimally coupling the charged scalars leads to an effective action as in Eq. 5. Letting the charged ¢;'s get vevs induces a potential (by the Higgs mechanism) ofthe form

(52) where (Imr) 1 J is the matrix inverse oflmriJ and the D terms are given by

(53) where {~ are real constants called Fayet-Jlliopoulos terms. Since lmriJ is postive definite by unitarity of the effective action, VD takes its minimum value VD = 0 when the D terms vanish individually:

(54) which is also a condition for supersymmetry to be unbroken. From the expression for the D term it follows that when all the fields are neutral any non-zero Fayet-Illiopoulos term spontaneously breaks supersymmetry. Henceforth we ignore Fayet-Illiopoulos terms: they can be shown to obey a stringent non-renormalization theorem which prevents them from being generated in an effective action if they are not generated at one loop in perturbation theory.

22

One may worry that since the D-term equations are not holomorphic that upon solving them one finds a moduli space M' Mo/{Dr 0} which is not Kabler, in contradiction to our supersymmetric selection rule for the IREA. Actually, since some scalars are charged, to find the moduli space we must also M'/U{l)n. It turns out that divide by the action of the gauge group: M this process always leads to a Kiihler M, and is known 21 as a Kahler quotient construction, and can be described in a holomorphic way as division of Mo by the natural action of the complexified gauge group: M Mo/U(1)~; The end result is that the D terms always have a solution which is the Kiibler submanifold parametrized by the holomorphic gauge neutral combinations of scalars.

=

=

=

=

4

N

= 2 Supersymmetric IREAs

The basic (no central charges) N of the last section,

= 2 superalgebra is, in the indexless notation m,n = 1,2.

(55)

This is just two copies of the N = 1 alge}?ra, Eq. 33; in particular, it has 1 subalgebras generated by Q1 and Q 2. Note that the N 2 two N algebra has an SU(2)R group of automorphisms under which Qm transforms as a doublet. (Global symmetries under which the supercharges transform are called R symmetries.) On shell irreducible representations of Eq. 55 are easy to construct. Because of the Q1 subalgebra, any N 2 representation must be made up of N = 1 representations. Suppose that one of these N = 1 representations is a chiral multiplet, as in Eq. 34. Under the action of the Q2 generators this 1 representation; but it is easy to see that multiplet must also form an N there is no way to do this consistent with the N 2 algebra. If we replace the initial N 1 chiral multiplet by an N 1 vector multiplet, one also finds no solution. So N 2 representations are formed by combining at least two N 1 multiplets. It is not hard to go through the various possibilities to find that the two solutions are the hypermultiplet, made from two N 1 chiral multiplets (¢, t/J) and (¢, ~) which satisfy

=

=

=

=

=

=

=

=

=

=

=

Qnt/Jm tn.mt/J, Q,.t/J =0, Qn'¢ =OnmPt/lm,

-=

Qnt/Jm = OnmtP, Qn!_ =enmP¢m, Q,.~ =0,

(56)

23

where I have defined 4Jn = ('J',, 4J); and the vector multiplet, made from one N = 1 chiral multiplet (4J, 1/J) and one N = 1 vector multiplet (.X, F) which satisfy =0, =8nmAm, = f.nmF, (57) =8nmP4J,

= -f.nmPAm,

=PXn,

where I have defined An= (1/J, .X). Important distinguishing features of the hypermultiplet are that its scalars form a complex SU(2)R doublet and that this SU(2)R action mixes anN= 1 chiral multiplet scalar with an anti-chiral multiplet partner. This has the immediate consequence that when coupling hypermultiplets in N = 2 gauge theories, they always transform in a real representation Ri$Ri (where Ri is the representation of the N = 1 chiral multiplet and so Ri is the representation of its anti-chiral partner). The bosonic degrees of freedom in a vector multiplet, by contrast, are a single complex scalar and a (real) vector field, both transforming in the adjoint of the gauge group, and both singlets under SU(2)R· In particular, in the case of U(1)n gauge group, which we are interested in for describing IREAs, the vector multiplet scalars are necessarily neutral. In anN= 2 !REA with gauge group U(l)n and neutral hypermultiplets, the general action (following from, say, the Q1 N 1 supersymmetry) would be just as in Eq. 41, where the i,j indices run over all the complex bosons (whether in hyper or vector multiplets). To take into account the N = 2 structure, let us now reserve the i,j indices for the complex (doublet) scalars 4J~ of hypermultiplets, and label the complex (singlet) scalars of the vector multiplets 4JI.

=

The first N = 2 selection rule we wish to derive is that no 811 4JIIJP"f kinetic terms can occur. To see this, suppose there was a term C,..., IC,r.P4JI · P"t,.. Then QmC :::> IC,rrP4JI · P;;;8nm· But it is easy to see, refering to Eq. 57, that there is no term whose Qm variation can could cancel this, implying that we must have /C,r. = 0. This in turn implies that the Ki:i.hler potential splits into the sum of two pieces depending on the hypermultiplet vevs and the vector multiplet vevs separately: i

~

I -I

IC = 1Cn(4Jn, 4Jn) + 1Cv(4J , 4J ).

(58)

Thus the kinetic terms for the scalars also split as

(59)

24

Coulomb Branch

=

2 moduli space. The Higgs and mixed branches intersect Figure 2: Cartoon of a classical N along a Higgs submanifold A, while the mixed branch intersects the Coulomb branch along a Coulomb submanifold B.

implying that the moduli space has a natural (local) product structure M =MH x Mv;

(60)

MH is the subspace of M along which only the hypermultiplet vevs vary while the vector multiplet vevs remain fixed, and vice versa for M v. In cases where M v is trivial (a point), M = M H is called a Higgs branch of the moduli space; when MH is trivial Mv is called the Coulomb branch (since there are always the massless U(l) vector bosons from the vector multiplets). Cases where both MH and Mv are non-trivial are called mixed branches. In general the total moduli space of a given theory need not be a smooth manifold-it may have "jumps" where submanifolds of different dimensions meet. Classically this occurs as a result of the Higgs mechanism: a charged scalar vev Higgses some vector multiplets, typically lifting them (making them massive). But at the special point where the charged vev is zero, the vector multiplets become massless, leading to extra flat directions and a jump in the dimensionality of the moduli space. Hence, at least classically, the general picture of an N = 2 moduli space is a collection of intersecting manifolds, which can be Higgs, Coulomb, or mixed branchesf 2 •23 see Fig. 2. This classical picture is, of course, modified quantum mechanically. How2 supersymmetric selection rule relating the metric on ever, a further N M v to the generalized coupling TJ J greatly restricts the possible form of these modifications. To see this, consider the U(l)n kinetic term £ "' TIJ:F 1 ;:J. Then Q£ ::> TIJ:F 1 PXJ. To cancel this variation then requires a fermion ki-

=

25 netic term .C' "' TJJ >l P)/. Then Q.C' :::> TJJ Pql P>:'. Finally, to cancel this variation requires a scalar kinetic term .C" "' TIJ Pql P(u- u') {Oa(u), Op(u')}

2 >(u- u'), = 4 ~r!p8< w(u)

(3.6)

it is now straightforward to verify that the constraint has vanishing canonical brackets with the Hamiltonian, and can thus be used to reduce the number of X 4 fields from nine to eight. Let us also mention that for topologically non-trivial membranes, there are extra consistency conditions corresponding to the non-contractible cycles on the membrane. The fact that the constraint commutes with the Hamiltonian implies the existence of a residual gauge symmetry of the lightcone Hamiltonian. This is the invariance under area preserving diffeomorphisms, which we will discuss in detail in section 5.

4

Some Properties of the Lightcone Hamiltonian

A lot about the qualitative features of supermembrane theory can be learnt by studying some general properties of the Hamiltonian (3.1). A first, and rather

38

obvious, observation is that the zero modes

Xo

=Jd2 u~X(u)

Do

=Jd uy'w{O)O(u) 2

do not appear in this Hamiltonian. Likewise, the center of mass momenta

P0 :=-

j d u1l 2

decouple from the non-zero mode degrees of freedom of the membrane. By subtracting this contribution from the Hamiltonian, we arrive at the mass formula

by substituting the expression (3.2) for P0 in terms of the transverse degrees of freedom. The prime indicates that the zero modes are to be omitted. This formula contains all the non-trivial dynamics of the membrane, whereas the center of mass motion is governed by the kinematics of a free relativistic particle. Similarly, the fact that the fermionic zero modes decouple will be used later to show that, if there exists a massless state, it will give rise to precisely one massless supermultiplet of d 11 supergravity. The bosonic part of the Hamiltonian (4.1) is ofthe standard form

=

H=M 2 =T+V with the kinetic energy T. The potential energy V is the integral of the density

9 = det(or.X · a,x) = V'orX 11 o,xb) 2 r,• The potential density vanishes if the surface degenerates, which happens when the X's only depend on one linear combination of the ur•s, i.e. when the membrane grows infinitely thin (i.e. stringlike) spikes, see Fig. 2. This instability has important consequences. We will come back to it below to show that

39

Fig. 2:A membrane growing tubes

this degeneracy must be interpreted as manifestation of the second quantized nature of the quantum supermembrane. The bosonic part of the above formula (4.1) can be generalized to arbitrary p, with the result that the degeneracies of the potential persist for higher p (generally, the zero energy configurations of a p-brane are those where the brane 1, i.e. degenerates to an object of dimension less than p). It is only for p string theory, that the potential is confining when the zero mode contribution is removed. This is also the only case where the potential is quadratic (X'(cr)) 2 • Then the Hamiltonian describes a free system, and we recover the well known result that the (super)string is an infinite set of (supersymmetric) harmonic oscillators. Thus, string theory is "easy" because it is a free theory, so that we can not only find a complete set of solutions to its equations of motion subject to various boundary conditions, but also quantize the theory straightforwardly. By contrast, the membrane and the higher dimensional p-branes are non-linear theories (with potentials oc X2"). So, for instance, only a very limited number of special solutions to the classical membrane equations of motion are known. A related observation is that, at a purely kinematical level, the type IIA string can be derived from the supermembrane by a "double dimensional reduction" .1 9

=

40

For this purpose, one compactifies the X 9 -dimension on a circle, letting the membrane wind around this dimension by making the identification (12

=

x9.

When the circle is shrunk to a point, the q 2 is gone as well and one is left with a string theory in ten dimensions. The resulting Hamiltonian is just the well known Green-Schwarz Hamiltonian

.c = --4-(.P(o? + x'(o?)- or-r9 o'(q) 2P 0

This is as expected if the supermembrane theory is a part of M-Theory, one of whose defining properties is that it must reduce to type IIA string theory upon compactification on a small circle. We would like to stress, however, that these considerations are by no means sufficient to establish that superstrings are contained in the supermembrane at the dynamical level. To recover the full dynamics of superstring scattering amplitudes, and in particular the full multistring vertex operators, is quite a different, and far more complicated task (see e.g. H. Verlinde's lectures on matrix string theory at this School). While these properties are for the most part almost self-evident, and also in accord with everything we know from superstring theory, we now return to the degeneracies of the Hamiltonian pointed out above. These constitute a very important physical property of the membrane, whose full significance has come to be fully appreciated only more recently. Namely, the spikes described above causing the instabilities need not have "free ends". It can also happen that a membrane quenches and separates into several parts that are connected by infinitesimally thin tubes, see Fig. 3. Because these tubes do not carry any energy, such a configuration is physically indistinguishable from the multi-membrane configuration obtained by removing the strings connecting the various pieces. In this way, membranes are allowed to split and merge, and there is no conserved "membrane number". In fact, the generic configuration occurring in a path integral formulation will be such that we cannot even assign a definite "membrane number" to a given configuration. Similar remarks apply to the treatment of different membrane topologies. For instance, one can build a toroidal membrane from a spherical membrane by connecting two points by a stringlike tube without any additional cost in energy. Hence it does not really make sense to talk about membranes of fixed topology , see Fig. 4. This new interpretation represents an important change of viewpoint vis-a-vis the one widely accepted a few years ago, when people were trying to construct a first quantized version of the supermembrane. The futility of these attempts

41

Fig. 3:Three membranes connected by tubes

became glaringly obvious when it was shown that the spectrum of the (SU(N) version of the) Hamiltonian (4.1) is continuous.20 This looked like (and in fact was) a disaster for the interpretation of the quantum supermembrane as a first quantized theory, and prompted several attempts to argue the continuous spectrum away. We note that, of course, the instabilities would go away if the effective action of the quantum supermembrane contained terms depending on the world volume curvature, which would suppress spikes and stringlike tubes. However, such terms presumably would not be compatible with the (desired) renormalizability of the theory, but would also spoil the correspondence with the matrix model. To summarize: the old objection that membrane theories "can't be first quantized" - meaning that the Hamiltonian is not quadratic unlike the superstring Hamiltonian- acquires a new and much deeper significance: the theory cannot be first quantized because the quantum supermembrane is a second quantized theory from the very beginning! This shows why membrane theory is so hard: we are dealing with a theory where the very notion of a one- (or multi-) particle state can only be extracted in certain asymptotic regimes, if it makes sense at

all.

42

Fig. 4:Equivalent membranes with different topologies 5

Area Preserving Diffeomorphisms

We now return to the canonical constraint (3.5) which expresses the invariance of the lightcone supermembrane under area preserving diffeomorphisms (or just APD, for short). To further analyze this residual symmetry, we introduce the following bracket on the space of functions A, B of the membrane coordinates

This is the Poisson bracket associated with the symplectic form

which, as any two form in two dimensions, is closed. The bracket is manifestly antisymmetric {A, B} -{B, A}

=

and obeys the Jacobi identity

{A, {B,C}} + {C, {A,B}} + {B, {C,A}}

=0.

and therefore has all the requisite properties of a Lie bracket. Consequently the functions on the membrane naturally form an infinite dimensional Lie algebra with the above bracket. The dimension p = 2 of the membrane is essential here, as these statements have no analog for p > 2 (at least not within the framework of ordinary Lie algebra theory). By means of this bracket we can rewrite the potential density as

43

Similarly, the non-zero mode part of the Hamiltonian (4.1) can be cast into the form

M2

=I d2uL~ [.P (u)]' + Vwfo1(~{X",Xb} 2 2

+ or-r"{X",O})] (5.1)

Again, the prime indicates that the zero modes have been left out. The advantage of rewriting the previous formulas in this peculiar way is that the similarities with Yang Mills theories become quite evident: the potential energy vanishes whenever {X", Xb} = 0, which is equivalent to the statement that the X" belong to the Cartan subalgebra of the Lie algebra introduced above. When we truncate this infinite dimensional Lie algebra to a finite dimensional matrix Lie algebra, the zero energy configurations just correspond to the diagonal matrices. Because {X" Xb} = tr• 8 X"8 Xb

'

yiW{o:)

r

•

is nothing but the volume (or rather: area) element of the membrane pulled back into space time, the residual invariance consists precisely of those diffeomorphisms which leave the area density invariant. These correspond to diffeomorphisms ur to-t ur + er (u) generated by divergence free vector fields

This equation is solved locally by

For topologically non-trivial membranes, there will be further vector fields which cannot be expressed in this fashion, and which correspond to the harmonic one forms in the standard Hodge decomposition. As a little exercise readers may check that the commutator of two APD vector fields is in one-to-one correspondence with the above Lie bracket in the sense that [e~ar,e~a.]

+--+ {6,6}

This means that the above Lie algebra can be identified with the Lie algebra of divergence free vector fields which is the Lie algebra of area preserving

44

diffeomorphisms. Using this correspondence, we can calculate the variation of any function f under an area preserving diffeomorphism as Jf

= -er orf = {!,, !}.

It is now straightforward to verify that the mass M commutes indeed with the constraint generator (3.5). The latter can be reexpressed by means of the above Lie bracket as

={fi,x}- {o,r-o} ~ o

¢(o-)

This should be compared to closed string theory after the light cone gauge has been imposed: In that case, the length preserving diffeomorphisms are just the constant shifts O" 1-t O" + const. This constraint implies that the oscillator levels of the left and the right movers are equal on physical states: NL=NR As is well known, already this single constraint has many non-trivial consequences! 6

Supercharges and Superalgebra

The supercharges expessing the global supersymmetry of the model are obtained by integrating the supercurrent (2.3) over the membrane:

Q

=I

d2o-Jo

=Q+ + Q-

The chiral supercharges are given by

=I =~r-r+Q J

Q+:: Q-

~r+r-Q

d2 o- (2P 4 fa

=

+ ~{X 4 ,X 11 }fa 11 ) 0

d2 o-S = 2r-oo

We see that Q- acts only on the fermionic zero modes. The zero mode part

ofQ+ Qt = 2Pofa0o is also conserved. Evidently, matics.

Qci

=Q+- Qt

is only relevant to the center of mass kine-

45

With the help of the canonical Dirac bracket (3.6) one can now determine the full superalgebra. This calculation, including possible surface contributions which could give rise to central charges was already performed in 17 • The most general d = 11 superalgebra has the form 21

- fJ } = {Qa' Q

r"a{J P#J + 21r"v z 1 r"vptrT zIJVPtTT • a{J IJV + 5! a{J

The corresponding Lie-superalgebra, including the bosonic commutation relations of the bosonic operators, is known as 0Sp(lj32) in mathematical terminology. We recognize that besides the membrane charge Z,.v the algebra also admits a 5-brane charge Z,.vptrT· However, a recent investigation of the lightcone superalgebra for the supermembranes with winding 22 •23 •24 •25 •26 has shown that this 5-brane charge is, in fact, absent despite the fact that d = 11 supergravity admits both 2-brane and 5-brane-like solutions~ 7 • 28 This little puzzle should not come as a total surprise in view of the following fact. While the 2-brane couples to the 3-index field of lld supergravity, the 5-brane charge would couple to a "dual" 6-index field; one would therefore expect the 5-brane charge to be related to another version of 11d supergravity with a 6-index field. However, for all we know such a version of 11d supergravity does not exist; 9 but see 30 for more recent references and a reformulation containing both a 3-index and a 6-index tensor field). The remaining part Qt of q+ obtained upon removing all zero-mode contributions contains the non-trivial information about the supermembrane dynamics. It gives rise to the superalgebra

{Qta,Qtp} = rtpM 2 The existence of massless states at threshold is equivalent to finding a normalizable state 'llio obeying M 2 '1lio = 0. Once such a state is found, a whole massless multiplet of d = 11 supergravity is generated by acting with the zero mode supercharges given above. If M 2 does not annihilate the state, one gets many more states, and the supermultiplet is a massive (long) supermultiplet of d = 11 supersymmetry. If d = 11 supergravity is to emerge as a low energy limit from the supermembrane there must be normalizable states that are annihilated by the quantum operator for M 2 • Let us also note that the winding states of minimal mass are BPS states, and hence belong to short multiplets, too 24,25. 7

Supermembranes and Matrix Models

We are now ready to establish the connection between the supermembrane Hamiltonian and the large N limit of supersymmetric SU (N) matrix model,17

46

which also underlies the recent proposal for a concrete formulation of MTheory~1 To this aim, we need to truncate the supermembrane theory to a supersymmetric matrix model with finitely many degrees of freedom. The idea is then to define the quantum supermembrane as the limit where the truncation is removed, taking into account possible renormalizations. The truncated model can be alternatively obtained by dimensional reduction of the maximally supersymmetric SU(N) Yang Mills theory from 1+9 to 1+0 dimensions, a reduction which had been originally investigated in 32 •33 •3 4). Upon quantization it becomes a model of supersymmetric quantum mechanics with extended (N 16) supersymmetry. To proceed, we expand all superspa.ce coordinates in terms of some complete orthonormal set of functions YA(u) on the membrane

=

X(o') = Xo + L:.XAYA(u) A

We next define a metric on this function space by

j d2i, their right moving superpartners e- in the Be and the 32 left moving fermions .Att. This is simply the content of the 80{32) heterotic string in static gauge where the .Af are the current algebra fermions. The action for this theory is simply the light-cone gauge Green-Schwarz action (29) for the heterotic string with a current algebra term added. In the case of the multiple D 1-branes, we have a non-abelian generalisation of that model:

t

Here, g2 "'9s fa:' is the effective gauge coupling of the (1 +1)-dimensional theory, and S'{,=.(e~,e+Jg), withJP=.(V+, V_). In models such as this, the "extra" terms may be written as a combination of commutators between the various fields, their precise form determined by gauge invariance and supersymmetry. In this example, one such term is g2 [tj>i, cf>i] 2 and a similar term for the 8 0 ., and a Yukawa term coupling the and tJ>i. Such terms constitute the "scalar potential" of the model.

e±

109

Obbligato: The supersymmetric vacua of such a gauge theory are those for which the "scalar potential" is identically zero. We can immediately study the classical solutions of this condition by just treating the vanishing of those terms as an algebra problem. The space of gauge inequivalent solutions of this condition is grandly termed the ·"moduli space of classical vacua". In general, quantum corrections can modify our answer, but with the right type or amount of supersymmetry (for example), the classical analysis is equivalent to the quantum analysis. This moduli space is the space of allowed values that the fields can take. Given that we have already realized that the fields on the world~volume of the branes are in one-to-one correspondence with the geometry that the brane encounters -both the embedding space and the shape that it can take in that embedding space- evaluating the moduli space of vacua is equivalent to discovering this geometry. This is the key to many relationships between geometry and field theory. Turning to the moduli space of vacua of this model, we see that the point with gauge symmetry O(N) is a special point of enhanced gauge symmetry. All of the scalar fields have zero vevs, and so the commutators (and hence the scalar potential) vanishes identically. This corresponds to all of the D1-branes being at the same point, which we have taken to be the origin of the eightdimensional space R 8 parameterized by 4>i, i=2, ... , 9. Generically, the fields can have non-zero vevs, but we wish to still consider supersymmetric solutions, which is to say we want the potential still to vanish. A solution is to make the i's non-zero, but all commute. In this way, we break the gauge symmetry down to the "maximal torus" (the largest Abelian subgroup) of S0(2N), which is U(1)f. This corresponds to separating out N D1-branes pairs. This situation is further reducible however (in contrast to a similar situation for D5-branes 78 •89 ), and we may split the D1-brane pairs. The resulting gauge group is {0} as we have seen, and there is one eight component scalar 4>i left for each of theN D1-branes, representing their transverse positions. (Indeed, that we can Higgs the gauge group away leaving N scalars follows from the fact that the difference between the dimensions N (N -1) /2 of the adjoint and the N(N +1)/2 of the symmetric is N.) As they all commute, we may find a basis where we simultaneously diagonalize the 4>i matrices, putting their eigenvalues down the diagonal: each eigenvalue represents an individual D1-brane. Notice that the Weyl group is still a gauge symmetry here, acting to permute the eigenvalues. This translates into the fact that the theory does not care if we rearrange the N D 1-branes, as they are identical. Therefore the classical moduli space of vacua is not (R8)N, but (R8)N /SN. The action of this SN will have important conse-

110

quences shortly. Notice that we can get special points of O(n) enhanced gauge symmetry whenever n D 1-branes coincide, which corresponds to having n simultaneous eigenvalues in the eight ¢i•s. We have not quite finished the job yet, as we have not discussed the allowed vacua of the superpartners ~ at all. However, this is not necessary, as we have sought supersymmetric solutions here. Therefore, their allowed values are determined by the unbroken supersymmetries. There remains to be determined the allowed values of the left-moving current algebra fermions ~f. Up to subtleties we will mention later, this is simply parameterized by the fact that they are fundamentals of the D9--brane gauge group 80(32), and hence parameterize the vector space V3 2 ::::::R32 that it acts on. So the full moduli space is schematically (R8 xV32 )N /8N.

,.t.3 From D1-bmnes to Fundamental Strings So we understand now that the type I supergravity model of the solitonic heterotic string that we were studying previously represents the fields around N coincident D1-branes. The world-volume theory of that soliton has been found more precisely to be our 1+1 dimensional gauge theory. Notice that the coupling of the gauge theory is a function of the type I string coupling. We had promised that as 9s goes to infinity, we would arrive at the heterotic string theory. What does this mean for the 1+1 dimensional model? The 1+1 dimensional coupling gets strong too, and so as a 1+1 dimensional gauge field theory, it should flow to the infra-red, presumably to a fixed point. In the special case of one D1-brane, the conformal field theory that we flow to is clearly the (0,8) supersymmetric (cL,cR) = (24, 12) conformal field theory of the free 80(32) heterotic string, but what of other N? For general N, the model is a non-Abelian gauge theory, and therefore there are potential terms like g2 [¢i, ¢i] 2 • As the string coupling goes large, so does g, and this term becomes very important. Indeed, at infinite g, the only way to find supersymmetric vacua is to force this term to zero by demanding that the 4>i all commute, generically (to set them all to zero is highly nongeneric). So in effect, strong coupling forces us out onto the Abelian (Coulomb) branch again, and the allowed values of the ¢i•s are in (R8 )N /8N. What is the interpretation of this? Given that we identify configurations related by the action of 8N, the permutation of the eigenvalues, it is useful to think of 8N as a sort of discrete gauge symmetry. The usefulness of this comes when we recall that our worldvolume theory arose in representing the dynamics of a stable closed string made by winding it about a circle (x 1 ) with a very large radius R 1 • We have

111

not discussed that feature much so far, but it is crucial. Indeed, this model of N Dl-branes is indistinguishable from a model of one Dl-brane wound N times around the large circle, or any number of Dl-branes with individual windings distributed among them to make total winding N. The moduli space (R8 )N fSN encodes precisely that. The interpretation as N D-branes that can be permuted by S N is therefore a small part of the story. The g.l~ and ljz > > g.l~ = l!u the eleven-dimensional Planck length. As discussed in 17 , one can think of the scale lpu as defining a "size" of the D-brane, a scale where its interactions become strong, and the moduli space picture breaks down. By taking g8 sufficiently small we can explore the substringy regime, l R < < l 8 • In the following we will do this by considering amplitudes from the sphere and disk world-sheets.

134

2.1

General considerations from conformal field theory

Given a conformal field theory defining a superstring compactification, our first problem is to identify the moduli space of boundary states corresponding to DO-branes. The conditions for a boundary state to correspond to a BPS state in space-time are given in 42 , while to decide that it is a DO-brane we should compute its RR charge. D~branes of different p are generally related by dualities and it is interesting to consider the same questions for p "# 0, making contact with mirror symmetry (e.g. see 51 ), but that deserves another lecture. Massless fluctuations then correspond to boundary operators of dimension 1; call a basis for these Oi. In the case that the moduli space is a manifold with local coordinates xi, we can choose the basis such that an insertion of J Oi corresponds to the variation aI axi. Although this will be true in the examples we consider, it is not always true, in CFT terms because the operators Oi can gain anomalous dimensions. A counterexample is the minima of a potential such as V = ¢~¢~ such as would be found in the world-volume Lagrangian of a system of 0-branes and 4-branes. The DO-brane metric will then be the Zamolodchikov metric on the moduli space. We can define this as the two-point function on the disk,

gf}(x)

= (Oi(0)0;(1)},

(2)

considered as a function on the moduli space. The choice of two distinct points (here 0 and 1) on the boundary is conventional. Heuristically the relation between this and the effective action is clear as gf] determines the normalization of a state created by Oi. To justify this from the string theory S-matrix one must consider a four-particle scattering and relate this to the curvature R[gD], but this relation is standard in effective field theory. 2.2 Non-linear sigma models

A simple M to consider is the supersymmetric non-linear sigma model with background metric 9ii (x). This will be a conformal field theory if the beta function for the metric vanishes. This can be computed using the covariant background field method~ 8 One writes the fields as a sum X = Xo + where Xo is a chosen background point and is a quantum fluctuation. Jn Riemann normal coordinates the action has an expansion whose coefficients are tensors constructed from g(xo):

e

s = 1~

J~u u~J>(xo)aeiae; + Ri;kt[u]aeiaekee' +...

e

(a)

135

The leading term in the beta function then comes from a one-loop diagram in which the R interaction vertex has a single self-contraction, producing a divergence proportional to the Ricci tensor. Thus one can obtain a conformal theory to this order by taking g = g< 0 > a Ricci flat metric. Working to higher orders, one obtainiO (4)

with Rf; given in many references such as 2 • The condition is no longer Ricci flatness but it has been shown that by adding finite corrections to g< 0 > at each order in l~ one can obtain a solution g"" of (4)~ 1 Just as in flat space, the allowed boundary conditions on bosons are those with 9i;(X)6Xio,.Xi = 0, either Dirichlet or Neumann. For Dirichlet boundary conditions Xilo-=o =xi, the marginal operators are Oi = u!J>o,.Xi and the Zamolochikov metric at leading order is just (g< 0 >)ii· We can proceed to compute l. corrections using the background field method. These have two origins a priori. One type of correction comes from bulk renormalization (closed string effects). Using the scheme of adding finite corrections to g, these corrections produce the metric g'{;, which is thus a better approximation to gD. In principle there can be additional finite corrections to the correlation function (2), which would naturally be tensors constructed from g""(xo). These are potentially important not just to get the correct result but also for the following reason of principle. The sigma model metric g'{; is not directly observable and is fact ambiguous. In renormalizing the theory we need to make choices for the finite parts of counterterms at each order in l~; a renormalization prescription. A priori these could be any local functionals of g. This corresponds to the possibility to make local field redefinitions in the resulting target space supergravity theory. On the other hand, the D-brane metric gf} is a metric on a moduli space and as such is observable. Its curvature will govern the scattering of small fluctuations on the world-volume. Although there is an ambiguity of field redefinition in its computation as well, this just corresponds to the freedom to choose a coordinate system on the moduli space. Of course if we considered other terms in the world-volume Lagrangian, for example the gauge kinetic term, we would have additional field redefinitions to deal with, but these do not affect gD. Thus any additional finite terms contributing to gD will be scheme dependent, in a way which exactly compensates the scheme dependence of g"". This suggests the possibility that there exists a preferred renormalization scheme

136

in which gD = gtr. The best candidate for this scheme is the usual minimal subtraction scheme (which subtracts only poles in dimensional regularization) and in fact it can be seen that in this scheme finite terms do not appear at order l~ and l! ~ 6 The idea that gD = gtr in a preferred scheme would have a number of interesting consequences. For one thing, it would be determined by an "equation of motion" (4) expressed as a series in local tensors. Using N = 2 perturbation theory, another consequence is that the complex structure of gD would be equal to that of the Ricci flat g(0 ). Even if the preferred scheme does not exist, for any point in the string theory moduli space (for which DO-branes exist), we can compute gD in principle, as a series expansion in tensors formed from gtr. It would be an interesting question whether it was governed by an equation analogous to (4). The nonlinear sigma model approach provides a universal definition of gD; on the other hand we expect any new qualitative features to appear in the substringy regime where the series expansion in l~ / l~ is at present an impractical approach. Rather, our knowledge of gD in the substringy regime comes entirely from examples. Let us discuss a few. ~.3

Orbifolds

D-branes on orbifolds give a large set of substringy examples. This has been a much studied subject (see 22 and papers which cite it for a list of references) and deserves a review of its own. Here we focus on geometric aspects. In general one defines string theory on an orbifold M ;r by first adding to the theory twisted string sectors and then projecting on r invariant states. For closed strings the twisted sectors are closed only up to an element of r, while for open strings the twisted sectors start and end on D-branes which will be identified after we make the projection. This problem is somewhat simpler than that for closed strings as we can derive the complete D-brane world-volume theory as a subsector of the theory of D-branes in M. For M = R n this theory will be the usual maximally supersymmetric super Yang-Mills so the entire construction is summarized in the projection. This can be written "1-t(g) > l. or lR 1 these algebras have no non-trivial homomorphisms to C, so the previous strategy fails; however we clearly want to associate to these algebras both the space XA and the number N as interesting topological data. We can do this by considering projections, elements e E A such that e2 = e. Clearly these will be matrix functions which at every point are diagonal with eigenvalues 0 and 1. The number of unitary equivalence classes of these determines N. Given an e of rank 1, the subalgebra of elements eMe is isomorphic to A, from which we recover XA. Projections are also useful in formulating a generalization of the idea of vector bundle. A vector bundle V over XA is determined by the algebra of its sections Sv. Sv admits an action by multiplication by functions on X; in other words it is a module over A. We can turn this around by considering all modules over A satisfying certain conditions, which will enable them to determine vector bundles. The condition is that Sv is free and projective: it can be obtained by taking n-component vectors with components in A, and applying a projection in Mn(A). One can go quite far in reformulating topological notions in this way. For example, one can generalize the formalism of characteristic classes to this case, and prove index theorems for operators acting on vector bundles over noncommutative spaces defined in this sense~ There has also been much work in formulating geometric notions. What is quite intriguing and useful is that much of this work takes as the central concept gauge theory on a noncommutative space, in other words a theory of connections on projective modules which allows writing an action, a Dirac operator and so forth. Combining this with the idea that in D-brane physics, coordinates are naturally promoted to matrices, we are motivated to study the subject with this physical application in mind.

146

3.1 Deformation quantization It is probably more helpful at this point to give examples than to develop abstract formalism, and a broad class of examples come from trying to deform algebras of commutative functions into noncommutative algebras~ In other words, given functions j,g E C(X), we seek a new multiplication law f * g which is associative and reduces to the original one in some limit: (21)

Here n is just a parameter characterizing the limit and {, } some bilinear operation characterizing the deformation to first order. Imposing associativity will constrain this operation and determine the O(n2 ) and higher terms. The problem is to find all such deformations, and was solved in 35 •27 • One type of deformation which is not so interesting is to make linear redefinitions of our functions. In other words we choose an invertible linear operator 0, say with an expansion in 1i such as (22)

We then define a new multiplication

=

1 * u o- 1 (0(f)O(g)).

(23)

Clearly this is not going to give us anything essentially new, and we should look for deformations modulo this gauge-like ambiguity. After fixing this ambiguity, it can be shown that associativity requires{,} to be a Poisson bracket, and then determines the higher order terms uniquely. (The explicit expressions require making a choice of connection compatible with the Poisson bracket; this choice is related to the ambiguity). In this sense of a series expansion - formal because we are not guaranteed that the series converges, and in many examples it does not - the problem has been solved. Deformation quantization gives us a noncommutative algebra for each choice of Poisson structure on our manifold. This choice can be parameterized by a bivector e~"(x) satisfying the Jacobi identity, or when this is non-degenerate by a closed two-form w = 1 •

e-

3.2 The noncommutative torus An example where the series converges and this works perfectly is to start with X= Tn then-torus. Let this be coordinatized by xi with 1 $ i $nand

147

0 $xi< 21r. We take constant eii and define the multiplication law as (f * g)(x)

= exp

(-i1r6'. ()xia()yia) l(x)g(y) I 3

y=z

(24}

This is the familiar rule for multiplying Weyl ordered symbols of quantum mechanical operators. We can present the same rule in terms of generators and relations: any function can be written as a sum of monomials in the variables (25} and on these (24) becomes

Ui * U;

= e,..•·e•; e'· as the lowest weight vector, and live in short multiplets of SU(2, 214). This follows simply because KK reduction will only produce states with spins not exceeding 2. Other choices of the lowest weight vectors such as ee ...e10 > will produce a

168

longer spin range and thereby also longer multiplets! Restricting attention to this choice, the multiplets are conveniently characterized by their lowest weights (h, JR, E) under the subgroup 80(4) x 80(2) C 80(4, 2). For example, for each p > 1, the 80(4, 2) representations defined by the lowest weight vector (1, 1, 2p + 4) contain all spin 2 modes in AdSs transforming in the representations characterized by the Dynkin label (O,p- 2, 0) of 80(6). The complete multiplet contains 128" 2 ('i~-l) fermionic and the same number of bosonic states (see Table I of 16 ). For p = 2 the spin 2 mode is the 5-dimensional graviton which is a singlet of 80(6). In addition this multiplet (the massless supergraviton multiplet) contains four complex gravitini, 15 vector fields in the adjoint of 80(6), plus spin zero and spin 1/2 objects for a total of 128 + 128 physical states. The p > 2 representations will correspond to the massive KK towers. In fact, the complete KK spectrum on AdSs x 8 5 is obtained by taking a single copy of each p ~ 2 representation. Again with each SO( 4, 2) lowest weight we can associate a particular mass shell mode in AdSs in a definite 80(6) representation. To close this section we remark that since the supersymmetries transform under 80(6), so will the gravitinos. The presence of vector fields in the adjoint of 80(6) thus means that the 5-dimensional theory is one of the known gauged supergravities 28 • 6

More on AdS

The SO(d- 1, 2) symmetry of AdSd is directly analogous to the SO(d + 1) action on Sd and can be realized linearly in the same way, by embedding as a surface in one higher dimension. Thus we introduce ad+ 1 dimensional flat space with a Lorenzian metric of signature (2, d- 1) and coordinates X A, and represent AdSd as the surface satisfying d-1

x:l +X5- L:xl = R 2 •

(19)

i=l

with the curvature radius R given in (18). This construction also provides global coordinates on AdSd: the relation between the D-brane coordinates (zl.l, r) and the XA is

(20) I For p

= 2 a complete classification of these has been given in 17 •

169

We see that the full metric is non-singular for r::; 0. This region is behind an event horizon from the point of view of an asymptotic observer in the original D-brane metric (6). The global structure of the AdS metric is better visualized by solving the constraint (19) in a different way (take R 1):

=

Xo

cosp

_ sint cosp Xi = Zitanp Li zl = 1

(21)

which leads to the metric in the form 1

ds 2 = - 2- ( -dt 2 + dp 2 ) cos p

+ tan 2 p dDL 2 •

(22)

These are usually called 'global' coordinates as they cover all of AdS as a single copy of the region 0 :S t :S 21T, 0 :S p :S 1T /2 and Zi E Sd- 2 . (For d = 2 we take S0 as the two points z 1 ±1). Furthermore, the metric is conformally flat in the pt plane, so the causal structure is easy to understand, and is depicted in Figure 1. The shaded region is the region described in the original ('Minkowski') coordinates. Note that the coordinate t is periodic and thus AdS space contains closed time-like curves. Often these are problematic and to avoid this we can work instead on the universal cover by simply ignoring the periodicity oft. If we do so, the future boundary of the shaded region is a horizon, but only because the infalling observer is forced to leave through a copy of the original asymptotic region. The physics is quite different from that of a black hole horizon. Rather, the striking feature of AdS geometry is that the limit r -+ oo (equivalently p -+ 1T /2) takes us to a time-like boundary, with geometry (on the cover) sd- 2 X R. The presence of the timelike boundary leads to major differences with physics in Minkowski space. For one thing, we do not have a well-posed initial value problem unless we put boundary conditions there. Of course we always need boundary conditions at infinity, but for Minkowski space the boundary conditions can be decomposed into incoming and outgoing waves, leading to the usual idea of particle and S-matrix. By contrast in AdS a small fluctation can typically be decomposed into normalizable and nonnormalizable components; the correct way to proceed in such situations 25 •2 is to quantize the normalizable fluctuations (each such mode is associated with a creation and annihilation operator) while the nonnormalizable modes are instead considered to be a static background. The two

=

170

Figure 1: Penrose diagram for AdS space-time

171

types of modes are distinguished by their asymptotics as p -t 11"/2 and so the background can be determined from boundary conditions at p = 11"/2 uniquely up to a choice of normalizable component. Thus we can regard all observables in AdSd quantum gravity as functions of a set of d - 1 dimensional fields, in one-to-one correspondence with nonnormalizable modes and thus with fields on AdSd. 7

The AdS/CFT correspondence

As we saw in the previous sections, superstring theory admits a class of BPS solitons- N parallel coincident D3-branes- with two very different descriptions in the near-horizon limit r

1 have

185

the lowest order (tree level) an S-matrix element typically has a pole in the external momentum which corresponds to the exchange of a massless mode L. The finite part is a power series in p 2 I M.2 and corresponds to the exchange of the whole tower of massive H -modes. Cetr is then constructed to reproduce the string S-matrix elements in the limit p2 I M.2 « 1 with S-matrix elements constructed entirely from the effective field theory of the £-modes. In this low energy effective theory the exchange of the H-modes in the string scattering is replaced by an effective interaction of the £-modes. For a four-point amplitude this procedure is schematically sketched in figure 3. The first row denotes the

+t

and u channels

+ t and u channels Figure 3: The S-matrix approach.

string scattering amplitude and its separation in a 'pole piece' (exchange of a massless mode) and the finite piece (exchange of the heavy modes). The second row indicates ordinary field-theoretical Feynman diagrams computed from the effective Lagrangian. The pole piece is reproduced by the same exchange of the massless modes while the finite part is identified with an effective interaction. Using this procedure Cetr can be systematically constructed as a power series in both p 2 I M.2 and g,. The power of p2 counts the number of spacetime derivatives in Cetr; at order (p2 I M.2 ) 0 one finds the effective potential while the order (p2 I M?) corresponds to the two-derivative kinetic terms.h hJnstead of using the S-matrix approach one can alternatively construct the effective action by computing the 11-functions of the two-dimensional u-model and interpreting them as the equations of motion of string theory. The effective action is then constructed to reproduce these equations of motion~

186

The selection criteria 1-4 already significantly reduce the number of string vacua for which the low energy effective theory has to be computed. Further simplification of the S-matrix approach comes from the use of all symmetries a string vacuum might have in that one does not have to compute separately S-matrix elements which are related by a symmetry? One only has to determine those couplings in the effective theory which are not related by general coordinate transformations, gauge transformations and N = 1 supersymmetry. Therefore, let us recall the bosonic terms of the most general gauge invariant supergravity Lagrangian with only chiral and vector multiplets and no more than two derivatives 15 •16

.C =

-v'u( 2 ~ 2 R+GIJDm~JDm4/+V(,~) + I) 4~~ (Fmnpmn)a + 3 ~: 2 (F F)a) a

+

(8)

fermionic terms) ,

where K- 2 = 8rrMpj 2 . Supersymmetry imposes constraints on the couplings of .C in eq. (8). The metric GJJ of the manifold spanned by the complex scalars b T (r) c5ab of the gauge group generators ra in the representation r. In order to make contact with the formulas of the previous section one combines the dilaton with the dual axion a of the antisymmetric tensor into a complex superfield S = e- 2 tf> + i a such that in perturbation theory

=

Ia = kaS + !~ 1 >(T) , aa(T, 'i') = Rej~ 1 >(T) + A(T, T) .

(16)

lc The physical Yukawa couplings, for example, do depend on the Ti (or rather their vaccum expectation values) and thus the computation of the fermion masses requires a mechanism which lifts these flat directions.

189

Contrary to naive expectation Aa(T, T) is not a harmonic function of the moduli but due to infrared effects aquires a non-harmonic term A known as the holomorphic anomaly.23,24,25,10 M. is the string scale which is related to MpJ and 9s as in (5) with the precise numerical coefficients 22 (17)

The string scale is roughly one order of magnitude bigger than the phenomenologically preferred GUT-scale MauT RS 3 · 10 16 GeV. At this scale the experimentally measured gauge couplings of the Standard Model unify with RS ~; under the assumption that right above Mweak the parti9buT RS cle spectrum of the Standard Model is replaced by the spectrum of the supersymmetric Standard Model.26 Thus, the perturbative heterotic string does reproduce the experimental situation of a unified gauge coupling. However, the unification occurs not quite at the right scale. The problem is that M. is not an independent parameter in the heterotic string but tied to g8 and Mp1 via (17). The mismatch between M. and MauT needs an explanation but the fact that it comes so close is one of the attractive model independent features of the perturbative heterotic string. In the past a number of attempts to overcome this mismatch have been suggested and we briefly review some of them here. One of the early suggestions has been that maybe the compactification scale of Calabi-Yau manifolds can be chosen lower than M. and therefore serve as MauT· However, within the perturbative heterotic string this suggestion is problematic~ 7 Since this argument partially breaks down in non-perturbative string theory let us go through it in slightly more detail. Compactification of the ten-dimensional effective field theory on a CalabiYau threefold yields a relation between the string coupling g~ 4 ) in the fourdimensional action and the string coupling gpo) in the ten-dimensional action which involves the volume V6 of the Calabi-Yau threefold

g;

(18) where 18 ::: R is the string length. The volume V6 is in principle an independent scale in the problem, the compactification scale. The perturbative decompactification limit sends V6l; 6 -+- oo and demands that the string coupling stays in the perturbative regime, ie. g! 10 ) is kept fixed and small. Eq. (18) then implies in this limit g~ 4) -+- 0. On the other hand, the measured gauge

190

couplings do not allow an arbitrarily small gauge coupling as a consequence of (15). Instead one roughly has to have 2 ( (4))2 ( (10))2 16 16 _231 = gGUT ~ ~ = gs s < _s_ 411" 411" 411" v6 411"V6

for

g(lO) s

' + 2ml-'/14>'

-

J.' 86 . + AK 6(j>i(x) (kJ.'f/>)'(x)

.

(12) where 8 indicates a right derivative. The first terms originate from the Ktransformations contained in Eq. 5 and Eq. 11. In most cases these are sufficient to find the invariance and no (k~-'4>) are necessary. However, e.g. for the theories which one obtains from adS backgrounds, such extra terms are present which depend on the field r, the distance from the brane. In supersymmetric theories, the conformal symmetry implies the presence of a second supersymmetry S, usually denoted as 'special supersymmetry'. Indeed, the commutator of the special conformal transformations, and the ordinary supersymmetry Q implies this S due to [K~-', Q] = -y~-'8. The anticommutator {Q, S} generates also an extra bosonic algebra (sometimes called R symmetry). The whole superalgebra can be represented in a supermatrix, as e.g. (symbolically) Q+S) ( SO(d,2) Q-S R .

(13)

We can consider such superalgebras in general. That is what Nahm 4 did in his classification. The requirements for a superconformal algebra in d or a super-adS algebra in d + 1 are: 1. SO(d, 2} should appear as a factored subgroup of the bosonic part of the superalgebra. For Nahm, this requirement was motivated by the Coleman-Mandula theorem, but here it is imposed in order to have that the bosonic algebra is the isometry algebra of a space which has the adS space as a factor. 2. fermionic generators should sit in a spinorial representation of that group.

215 Table 1: Lie superalgebras of classical type.

Name

Range

Bosonic algebra

8U(m!n)

m~2

8U(m) $ 8U(n) $U(1) no U{1} 8£(m) $ 8£(n) 8U(m- p,p) $ 8U(n- q, q) 8U*(2m) $ 8U*(2n) 80(m) $ 8p(n) {m,n)

m =f: n m=n 8£(m!n) 8U(m- p,pjn- q,q) 8U*(2ml2n) 08p(m!n) m~1 n=2,4, .. 08p(m- p,pln) 08p(m*ln- q,q) D{2, 1,a) 0P.)

(19)

218

are on the adS space distorted to Dadsx~-'

=

•

-A~-'-- A~-'vxv-

A+ +x~-'

-(x2 + R2 z2 )A~-' + + 2x~-'xvAv + DadSZ =

-z(A++-2x~-'A~-'+).

(20)

On the other hand, the p-brane solutions of D-dimensional supergravity have a metric of the form d=p+l; d==.D-p-3

(21) where m = 0, 1, ... p denotes the directions along the brane, and m' = 1, ... Dp-1 those orthogonal to the brane. The solutions describe N coincident branes, and the parameter R is proportional to N 1/d. When the constant 1 is neglected in the expression for the harmonic function in the transverse coordinates H, one obtains an adSd+l metric. Indeed then

(22) and identifying z in Eq. 18 with (~)4 we find that (Xm,r) is a (d +I)dimensional adS space and the remainder is a (d + 1)-sphere. The described limit can be seen in 3 different ways: • We can see the brane solution as an interpolation 11 between the asymptotically flat region where H = 1, and a near horizon anti-de Sitter

geometry where H = ( ~)J. • The limit can also be seen as a large N (many branes solution) limit 12 • This will correspond in the field theory to large N for the SU(N) gauge theory. • there is a special duality transformation that removes the constant 13 . This mechanism applies in various situations. We give in Table 3 cases where the manifold is always of the form adSd+l x sJ+t. The isometry groups are thus SO(d, 2) x SO(d+2). For later convenience we mention also w ==. dfd. E.g. the self-dual string can also be obtained from compactifying a tO-dimensional string theory on adSa X S 3 X E4, where E4 denotes an Euclidean space. The Tangerlini black hole has been discussed, together with related rotating black holes in 5 dimensions in 14 , where the mentioned superalgebra was obtained

219

=

Table 3: Brane solutions with horizon geometry adSd+l x sci+l, where d p + 1. The number w d/d is the Weyl weight of the scalars in the conformal theory.

=

M5 M2 D3 Self-dual string (D1+D5) Magnetic string Tangerlini black hole Reissner-Nordstrom black hole

D 11 11 10 6 5 5 4

d 6 3 4 2 2 1 1

d 3

w 2

6 4 2 1 2 1

2

1

1 1 2 1

2

1

superalgebra 0Sp(8*14) 0Sp(8l4) SU(2,2I4) SU(l, 112} $ SU(1, 112} SU(1, 112} $ SU(1, 1) SU(1, 112} El: SU(2} SU(1, 112}

(the extra SU(2} rotates the supercharges, it is in fact D 2 (2, 1, 0)}. We are not aware of a similar result for the magnetic string, but we conjecture the appearance of the superalgebra in the table. In many cases other manifolds than simple spheres appear in the compactification, see e.g. 15 .

4

Bosonic world-volume theory

The world-volume theory is in general a sum of a Born-Infeld term, a WessZumino term and an extra one in case of the M5 brane. We first concentrate on the bosonic part.

Scl

=

Ssi

=

Ss1 + Swz + S'

-I

a:Juyr-_-d_e_t-(g_}r_~_+_T._I-'11-) ;

g:lld

= al-'xMallxNGMN ,(23}

where G M N is the metric on target space, solution of the supergravity theory. We use for the latter immediately the near-horizon form Eq. 22. Concentrating on the first three and the last case of Table 3, RNorM2 D3

T"'11 = 0 T"'11 = F"'11

S' =0; S' =0;

M5

T"'11 = i?i;11

S'

=I

d!'u?i*"'ll?i"'v .

(24)

In the case of M5 we use the Pasti-Sorokin-Tonin (PST) formulation 16 for describing self-dual tensors. That means that there is an auxiliary field a

220

apart from the antisymmetric tensor Bp.zn and

(25) The field equations of BJi.v and extra gauge invariances imply that a is a gauge degree of freedom and 1ip.vp is self dual. The input which we thus need for the world-volume action is G M N and the Wess-Zumino term. The latter is related to the integral of the (p + 2)-field strength form of the p-brane. Its exact form can vary for different cases, see e.g. M5 01 = dXo ... dXsdH- 1 ; D3 Os = dXo ... dX3 dH- 1 ; (26) F = dX 0 dH- 1 . BH The isometries of the solution lead to rigid symmetries of the action. These are thus the SO(d + 2) rotations and the adS isometries

OadS(~)Xm

=

-~m(X, r) = -~m(X)- (wR)

OadS(~)xm'

=

wAv(X)Xm' = w(>.v- 2XmAKm)Xm',

2 ( :)

2

/w AI(

(27)

In the full theory there is also rigid supersymmetry, determined by the Killing spinors of the metric. Note that in case of d = 1 or 2 infinite dimensional symmetries exist. In adS3 they have been found as asymptotic geometries by Brown and Henneaux 17 . That means that these are not symmetries of the action, but rather between different geometries which have adS3 as near-horizon limit. They form the Virasoro algebra of which 80(2, 2) = SU(1, 1) x SU(1, 1) is the finite dimensional subgroup. However, it has also been found that the action d2 0" J- det(gp.v + Fp.v) has an infinite symmetry group 18 . For any isometry with hM(X) as Killing vector, the action is invariant under

J

hM (X) A( .F) ;

8XM 8Vp.

=

.F = - t::.;;v ;

->.'(.F)vg(1+F2 )t:p.v(8"'XM)hM(X),

(28)

where >.(.F) is an arbitrary function which provides a Kac-Moody extension of the isometry group. Furthermore there are the local symmetries: world-volume diffeomorphisms and its fermionic partner: the ~~:-symmetry, which we will discuss in section 7.

221

Gauge fixing of the diffeomorphisms, e.g. by identifying the first d spacetime fields with the coordinates on the brane X~'(a) = a~' leaves invariant linear combinations of the rigid isometries with local symmetries where the parameters of the latter are determined functions of the rigid parameters of the isometries and local fields.

{29) The result is that these remaining symmetries take the form of conformal transformations on the world-volume. The remaining scalar fields are xm', which are scalars of Weyl weight w in Table 3. There are extra parts in the special conformal transformations

Alf