abdus salam international centre for theoretical physics united nations educational, scientific international atomic and cultural organization energy agency Proceedings of the Trieste 2000 Spring Workshop on UPERSTRINGS AND RELATED MATTERS Editors C. Bachas J. Maldacena K. S. Narain S. Randjbar-Daemi World Scientific s UPERSTRINGS AND RELATED MATTERS the abdus salam international centre for theoretical physics united nations educational, scientific international atomic Jo3?S and cultural organization energy agency Sky" ,<r s Proceedings of the Trieste 2000 Spring Workshop on UPERSTRINGS AND RELATED MATTERS ICTP, Trieste, Italy 27 March - 4 April 2000 Editors C. Bachas Ecole Normale Superieure, France J. Maldacena Harvard University, USA K. S. Narain and S. Randjbar-Daemi W Italy V fe World Scientific »• SSininqgaappoorere • •N Neeww J Jeersrseeyy • 'LL ondon • Hong Kong Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. SUPERSTR1NGS AND RELATED MATTERS Proceedings of the ICTP Spring Workshop Copyright © 2001 by The Abdus Salam International Centre for Theoretical Physics ISBN 981-02-4525-4 Printed in Singapore by World Scientific Printers V PREFACE The Abdus Salam ICTP's Spring Workshop, saw a variety of currently interesting topics in string theory. These proceedings contain four sets of lectures presented at this workshop. GaberdiePs lectures contain a comprehensive introduction to the boundary state approach to the D-branes and, in particular, non-BPS D-branes. Exam ples are given that arise in the context of dualities between string theories. Kachru's lectures deal with warped compactification in 5d theories and the brane world scenarios that emerge. Gauge heirarchy and cosmological con stant problems in this context are discussed in detail. Finally these lectures also include string theory constructions that involve D-branes wrapped on various cycles of Calabi-Yau spaces and the corresponding superpotential computa tions in the world volume gauge theory are described. This is important for both understanding supersymmetry breaking phenomenon as well as under standing the quantum geometry of the Calabi-Yau spaces when the branes are considered as probes. Maldacena's lectures include a description of the holographic correspon dence between field theories and string M theory. In particular, the correspon dence between string/M theory compactification on AdS spaces and conformal field theories living in the boundary, is discussed in detail. The evidences for this correspondence are described in the context of maximally supersymmetric gauge theories in 4d. It is known that the effective world volume theory on D-branes in the pre sence of NS-NS antisymmetric tensor field is a non-commutative gauge theory. Nekrasov's lectures give a pedagogical introduction to this subject and, in particular, focus on the non-perturbative solutions such as monopoles and instantons in the non-commutative gauge theories. vii CONTENTS Preface v Lectures on Non-BPS Dirichlet Branes 1 M. R. Gaberdiel Lectures on Warped Compactifications and Stringy Brane Constructions 43 S. Kachru Large N Field Theories, String Theory and Gravity 74 J. Maldacena Trieste Lectures on Solitons in Noncommutative Gauge Theories 141 N. A. Nekrasov 1 LECTURES ON NON-BPS DIRICHLET BRANES MR. GABERDIEL Department of Mathematics King's College London Strand, London WC2R 2LS, U. K. A comprehensive introduction to the boundary state approach to Dirichlet branes is given. Various examples of BPS and non-BPS Dirichlet branes are discussed. In particular, the non-BPS states in the duality of Type IIA on K3 and the heterotic string on T4 are analysed in detail. 1 Introduction The past few years have seen a tremendous increase in our understanding of the dynamics of superstring theory. In particular it has become apparent that the five ten-dimensional theories, together with an eleven-dimensional theory (M-theory), are different limits in moduli space of some unifying description. A crucial ingredient in understanding the relation between the different per- turbative descriptions has been the realisation that the solitonic objects that define the relevant degrees of freedom at strong coupling are Dirichlet-branes that have an alternative description in terms of open string theory.x'2'3 The D-branes that were first analysed were BPS states that break half the (spacetime) supersymmetry. It has now been realised, however, that because of their description in terms of open strings, D-branes can be constructed and analysed in much more general situations. In fact, D-branes are essentially described by a boundary conformal field theory4'5'6'7,8'9'10'11 (see also Refs. 12- 17 for earlier work in this direction), the consistency conditions of which are not related to spacetime supersymmetry18'19'20 (for an earlier non-supersymmetric orientifold construction see also Ref. 16). In an independent development, D- branes that break supersymmetry have been constructed in terms of bound states of branes and anti-branes by Sen21'22'23'24'25'26 (see also Ref. 27 for a good review). This beautiful construction has been interpreted in terms of K-theory by Witten,28 and this has opened the way for a more mathematical treatment of D-branes. 29'30'31 It has also led the way to new insights into the nature of the instability that is described by the open string tachyon.32 The motivation for studying D-branes that do not preserve spacetime su persymmetry (and that are therefore sometimes called non-BPS D-branes) is at least four-fold. First, in order to understand the strong/weak coupling dualities of supersymmetric string theories in more detail, it is important to 2 analyse how these dualities act on states that are not BPS saturated. After all, the behaviour of the BPS states at arbitrary coupling is essentially deter mined by spacetime supersymmetry (provided that it remains unbroken for all values of the coupling constant), and thus one is not really probing the underlying string theory unless one also understands how non-BPS states be have at strong coupling. The dualities typically map perturbative states to non-perturbative (D-brane type) states, and thus one will naturally encounter non-BPS D-branes in these considerations. The second motivation is related to the question of whether string duality should intrinsically only apply to supersymmetric string theories, or whether also non-supersymmetric theories should be related by duality. This is cer tainly, a priori, an open question" it is conceivable that spacetime super- symmetry is a crucial ingredient without which there is no reason to believe that these dualities should exist, but it is also conceivable that spacetime su persymmetry is just a convenient tool that allows one to use sophisticated arguments and techniques to verify conjectures that are otherwise difficult to check. Dirichlet branes play a central role in the understanding of string du alities, and if one wants to make progress on this question, it is important to develop techniques to analyse and describe Dirichlet branes without reference to spacetime supersymmetry. Thirdly, one of the interesting implications of the Maldacena conjecture39 is that one can obtain non-trivial predictions about field theory from string theory. In the original formulation this was applied to supersymmetric string and field theories, but it is very tempting to believe that similar insights may be gained for non-supersymmetric theories. This line of thought has been developed recently, starting with a series of papers by Klebanov & Tseytlin.20 Finally, non-BPS D-branes offer the intriguing possibility of string com- pactifications in which supersymmetry is preserved in the bulk but broken on the brane. Various orientifold models involving branes and anti-branes have been constructed recently,40,41'42,43 but it is presumably also possible to con struct interesting models involving non-BPS D-branes. (Non-BPS -branes in Type II theories have recently been considered in Refs. 44 and 45.) The fact that at specific points in the moduli space their spectrum is Bose-Fermi de generate may be of significance in this context.46 The main aim of these lectures is to explain the boundary state approach to D-branes, and to give some applications of it, in particular to the construction of non-BPS D-branes. The structure of the lectures is as follows. In section 2 we explain carefully the underpinnings of the boundary state approach and "Recently, some suggestive proposals have however been made.18''