This page intentionally left blank STRING THEORY AND M-THEORY A MODERN INTRODUCTION String theory is one of the most exciting and challenging areas of modern theoretical physics. This book guides the reader from the basics of string theorytoveryrecentdevelopmentsatthefrontierofstringtheoryresearch. The book begins with the basics of perturbative string theory, world-sheet supersymmetry, space-time supersymmetry, conformal field theory and the heterotic string, and moves on to describe modern developments, including D-branes, string dualities and M-theory. It then covers string geometry (including Calabi–Yau compactifications) and flux compactifications, and applications to cosmology and particle physics. One chapter is dedicated to black holes in string theory and M-theory, and the microscopic origin of black-hole entropy. The book concludes by presenting matrix theory, AdS/CFTdualityanditsgeneralizations. Thisbookisidealforgraduatestudentsstudyingmodernstringtheory,andit willmakeanexcellenttextbookfora1-yearcourseonstringtheory.Itwillalso be useful for researchers interested in learning about developments in modern stringtheory.Thebookcontainsabout120solvedexercises,aswellasabout200 homework problems, solutions of which are available for lecturers on a pass- wordprotectedwebsiteatwww.cambridge.org/9780521860697. KATRIN BECKER isaProfessorofphysicsatTexasA&MUniversity.She was awarded the Radcliffe Fellowship from Harvard University in 2006 and receivedtheAlfredSloanFellowshipin2003. MELANIE BECKER isaProfessorofphysicsatTexasA&MUniversity.In 2006shewasawardedanEdward,FrancesandShirleyB.DanielsFellowship from the Radcliffe Institute for Advanced Studies at Harvard University. In 2001shereceivedtheAlfredSloanFellowship. JOHN H. SCHWARZ is the Harold Brown Professor of Theoretical Physics at the California Institute of Technology. He is a MacArthur Fellow and a memberoftheNationalAcademyofSciences. Thisisthefirstcomprehensivetextbookonstringtheorytoalsoofferanup-to- datepictureofthemostimportanttheoreticaldevelopmentsofthelastdecade, including the AdS/CFT correspondence and flux compactifications, which haveplayedacrucialroleinmoderneffortstomakecontactwithexperiment. An excellent resource for graduate students as well as researchers in high- energyphysicsandcosmology. NimaArkani-Hamed,HarvardUniversity An exceptional introduction to string theory that contains a comprehensive treatmentofallaspectsofthetheory,includingrecentdevelopments.Theclear pedagogicalstyleandthemanyexcellentexercisesshouldprovidetheinterested studentorresearcherastraightforwardpathtothefrontiersofcurrentresearch. DavidGross,DirectoroftheKavliInstituteforTheoreticalPhysics,Universityof California,SantaBarbaraandwinneroftheNobelPrizeforPhysicsin2004 Masterfullywrittenbypioneersofthesubject,comprehensive,up-to-dateand replete with illuminating problem sets and their solutions, String Theory and M-theory: A Modern Introduction provides an ideal preparation for research on the current forefront of the fundamental laws of nature. It is destined to becomethestandardtextbookinthesubject. AndrewStrominger,HarvardUniversity This book is a magnificient resource for students and researchers alike in the rapidly evolving field of string theory. It is unique in that it is targeted for students without any knowledge of string theory and at the same time it includestheverylatestdevelopmentsofthefield,allpresentedinaveryfluid andsimpleform.Theluciddescriptionisnicelycomplementedbyveryinstruc- tiveproblems.Ihighlyrecommendthisbooktoallresearchersinterestedinthe beautifulfieldofstringtheory. CumrunVafa,HarvardUniversity Thiselegantlywrittenbookwillbeavaluableresourceforstudentslookingfor anentry-waytothevastandexcitingtopicofstringtheory.Theauthorshave skillfully made a selection of topics aimed at helping the beginner get up to speed.Iamsureitwillbewidelyread. EdwardWitten,InstituteforAdvancedStudy,Princeton, winneroftheFieldsMedalin1990 STRING THEORY AND M-THEORY A Modern Introduction KATRIN BECKER, TexasA&MUniversity MELANIE BECKER, TexasA&MUniversity and JOHN H. SCHWARZ CaliforniaInstituteofTechnology cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521860697 ©K.Becker,M.BeckerandJ.H.Schwarz2007 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2006 isbn-13 978-0-511-25653-0eBook (EBL) isbn-10 0-511-25653-1 eBook (EBL) isbn-13 978-0-521-86069-7hardback isbn-10 0-521-86069-5 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. v To our parents vi An Ode to the Unity of Time and Space Time, ah, time, how you go o(cid:11) like this! Physical things, ah, things, so abundant you are! The Ruo’s waters are three thousand, how can they not have the same source? Time and space are one body, mind and things sustain each other. Time, o time, does not time come again? Heaven, o heaven, how many are the appearances of heaven! From ancient days constantly shifting on, black holes (cid:13)aring up. Time and space are one body, is it without end? Great indeed is the riddle of the universe. Beautiful indeed is the source of truth. To quantize space and time the smartest are nothing. To measure the Great Universe with a long thin tube the learning is vast. Shing-Tung Yau Contents Preface page xi 1 Introduction 1 1.1 Historical origins 2 1.2 General features 3 1.3 Basic string theory 6 1.4 Modern developments in superstring theory 9 2 The bosonic string 17 2.1 p-brane actions 17 2.2 The string action 24 2.3 String sigma-model action: the classical theory 30 2.4 Canonical quantization 36 2.5 Light-cone gauge quantization 48 3 Conformal (cid:12)eld theory and string interactions 58 3.1 Conformal (cid:12)eld theory 58 3.2 BRST quantization 75 3.3 Background (cid:12)elds 81 3.4 Vertex operators 85 3.5 The structure of string perturbation theory 89 3.6 The linear-dilaton vacuum and noncritical strings 98 3.7 Witten’s open-string (cid:12)eld theory 100 4 Strings with world-sheet supersymmetry 109 4.1 Ramond{Neveu{Schwarz strings 110 4.2 Global world-sheet supersymmetry 112 4.3 Constraint equations and conformal invariance 118 4.4 Boundary conditions and mode expansions 122 vii viii Contents 4.5 Canonical quantization of the RNS string 124 4.6 Light-cone gauge quantization of the RNS string 130 4.7 SCFT and BRST 140 5 Strings with space-time supersymmetry 148 5.1 The D0-brane action 149 5.2 The supersymmetric string action 155 5.3 Quantization of the GS action 160 5.4 Gauge anomalies and their cancellation 169 6 T-duality and D-branes 187 6.1 The bosonic string and Dp-branes 188 6.2 D-branes in type II superstring theories 203 6.3 Type I superstring theory 220 6.4 T-duality in the presence of background (cid:12)elds 227 6.5 World-volume actions for D-branes 229 7 The heterotic string 249 7.1 Nonabelian gauge symmetry in string theory 250 7.2 Fermionic construction of the heterotic string 252 7.3 Toroidal compacti(cid:12)cation 265 7.4 Bosonic construction of the heterotic string 286 8 M-theory and string duality 296 8.1 Low-energy e(cid:11)ective actions 300 8.2 S-duality 323 8.3 M-theory 329 8.4 M-theory dualities 338 9 String geometry 354 9.1 Orbifolds 358 9.2 Calabi{Yau manifolds: mathematical properties 363 9.3 Examples of Calabi{Yau manifolds 366 9.4 Calabi{Yau compacti(cid:12)cations of the heterotic string 374 9.5 Deformations of Calabi{Yau manifolds 385 9.6 Special geometry 391 9.7 Type IIA and type IIB on Calabi{Yau three-folds 399 9.8 Nonperturbative e(cid:11)ects in Calabi{Yau compacti(cid:12)cations 403 9.9 Mirror symmetry 411 9.10 Heterotic string theory on Calabi{Yau three-folds 415 9.11 K3 compacti(cid:12)cations and more string dualities 418 9.12 Manifolds with G and Spin(7) holonomy 433 2 10 Flux compacti(cid:12)cations 456 10.1 Flux compacti(cid:12)cations and Calabi{Yau four-folds 460 10.2 Flux compacti(cid:12)cations of the type IIB theory 480