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Introduction to Superstring Theory

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Introduction to Superstring Theory CERN-TH/97-218 hep-th/9709062 INTRODUCTION TO SUPERSTRING THEORY Elias Kiritsis∗ Theory Division, CERN, CH-1211, Geneva 23, SWITZERLAND Abstract In these lecture notes, an introduction to superstring theory is presented. Classi- cal strings, covariant and light-cone quantization, supersymmetric strings, anomaly cancelation, compactification, T-duality, supersymmetry breaking, and threshold corrections to low-energy couplings are discussed. A brief introduction to non- perturbative duality symmetries is also included. Lectures presented at the Catholic University of Leuven and at the University of Padova during the academic year 1996-97. To be published by Leuven University Press. CERN-TH/97-218 March 1997 ∗e-mail: [email protected] Contents 1 Introduction 2 2 Historical perspective 3 3 Classical string theory 6 3.1Thepointparticle................................ 7 3.2Relativisticstrings............................... 10 3.3 Oscillator expansions .............................. 16 4 Quantization of the bosonic string 20 4.1Covariantcanonicalquantization....................... 20 4.2Light-conequantization............................. 23 4.3Spectrumofthebosonicstring......................... 23 4.4Pathintegralquantization........................... 25 4.5Topologicallynon-trivialworld-sheets..................... 27 4.6BRSTprimer.................................. 28 4.7BRSTinstringtheoryandthephysicalspectrum.............. 30 5 Interactions and loop amplitudes 33 6 Conformal field theory 35 6.1Conformaltransformations........................... 35 6.2Conformallyinvariantfieldtheory....................... 38 6.3Radialquantization............................... 40 6.4Example:thefreeboson............................ 43 6.5Thecentralcharge............................... 45 6.6Thefreefermion................................ 46 6.7Modeexpansions................................ 47 6.8TheHilbertspace................................ 49 6.9Representationsoftheconformalalgebra................... 51 6.10Affinealgebras................................. 54 6.11 Free fermions and O(N) affine symmetry ................... 57 1 6.12 N=1 superconformal symmetry ........................ 63 6.13 N=2 superconformal symmetry ........................ 65 6.14 N=4 superconformal symmetry ........................ 67 6.15TheCFTofghosts............................... 68 7 CFT on the torus 72 7.1Compactscalars................................. 75 7.2 Enhanced symmetry and the string Higgs effect ............... 81 7.3T-duality.................................... 82 7.4Freefermionsonthetorus........................... 84 7.5Bosonization................................... 86 7.6Orbifolds..................................... 88 7.7CFTonhigher-genusRiemannsurfaces.................... 94 8 Scattering amplitudes and vertex operators of bosonic strings 95 9 Strings in background fields and low-energy effective actions 99 10 Superstrings and supersymmetry 101 10.1Closed(type-II)superstrings.......................... 103 10.2 Massless R-R states............................... 107 10.3Type-Isuperstrings............................... 109 10.4Heteroticsuperstrings............................. 111 10.5Superstringvertexoperators.......................... 114 10.6 Supersymmetric effective actions ........................ 116 11 Anomalies 119 12 Compactification and supersymmetry breaking 127 12.1Toroidalcompactifications........................... 128 12.2Compactificationonnon-trivialmanifolds.................. 132 12.3 World-sheet versus spacetime supersymmetry ................ 137 12.4 Heterotic orbifold compactifications with N=2 supersymmetry ....... 142 12.5 Spontaneous supersymmetry breaking . .................... 150 2 12.6HeteroticN=1theoriesandchiralityinfourdimensions........... 152 12.7Orbifoldcompactificationsofthetype-IIstring................ 154 13 Loop corrections to effective couplings in string theory 156 13.1Calculationofgaugethresholds........................ 157 13.2On-shellinfraredregularization........................ 163 13.3Gravitationalthresholds............................ 166 13.4AnomalousU(1)’s................................ 167 13.5N=1,2examplesofthresholdcorrections................... 168 13.6N=2universalityofthresholds......................... 172 13.7Unification.................................... 175 14 Non-perturbative string dualities: a foreword 176 14.1 Antisymmetric tensors and p-branes . .................... 179 14.2 BPS states and bounds ............................. 180 14.3Heterotic/type-Idualityintendimensions................... 183 14.4Type-IIAversusM-theory............................ 190 14.5 M-theory and the E8×E8 heteroticstring................... 192 14.6Self-dualityofthetype-IIBstring....................... 193 14.7 D-branes are the type-II R-R chargedstates................. 195 14.8D-braneactions................................. 198 14.9Heterotic/type-IIdualityinsixandfourdimensions............. 201 15 Outlook 208 Acknowledgments 209 AppendixA:Thetafunctions............................ 210 AppendixB:Toroidallatticesums......................... 213 AppendixC:ToroidalKaluza-Kleinreduction................... 216 AppendixD:N=1,2,4,D=4supergravitycoupledtomatter........... 218 Appendix E: BPS multiplets and helicity supertrace formulae . ......... 221 AppendixF:Modularforms............................. 229 AppendixG:Helicitystringpartitionfunctions.................. 231 3 AppendixH:Electric-MagneticdualityinD=4.................. 237 References 240 4 1 Introduction String theory has been the leading candidate over the past years for a theory that consis- tently unifies all fundamental forces of nature, including gravity. In a sense, the theory predicts gravity and gauge symmetry around flat space. Moreover, the theory is UV- finite. The elementary objects are one-dimensional strings whose vibration modes should correspond to the usual elementary particles. At distances large with respect to the size of the strings, the low-energy excitations can be described by an effective field theory. Thus, contact can be established with quantum field theory, which turned out to be successful in describing the dynamics of the real world at low energy. I will try to explain here the basic structure of string theory, its predictions and prob- lems. In chapter 2 the evolution of string theory is traced, from a theory initially built to describe hadrons to a “theory of everything”. In chapter 3 a description of classical bosonic string theory is given. The oscillation modes of the string are described, preparing the scene for quantization. In chapter 4, the quantization of the bosonic string is described. All three different quantization procedures are presented to varying depth, since in each one some specific properties are more transparent than in others. I thus describe the old covariant quantization, the light-cone quantization and the modern path-integral quantization. In chapter 6 a concise introduction is given, to the central concepts of conformal field theory since it is the basic tool in discussing first quantized string theory. In chapter 8 the calculation of scattering amplitudes is described. In chapter 9 the low-energy effective action for the massless modes is described. In chapter 10 superstrings are introduced. They provide spacetime fermions and real- ize supersymmetry in spacetime and on the world-sheet. I go through quantization again, and describe the different supersymmetric string theories in ten dimensions. In chapter 11 gauge and gravitational anomalies are discussed. In particular it is shown that the super- string theories are anomaly-free. In chapter 12 compactifications of the ten-dimensional superstring theories are described. Supersymmetry breaking is also discussed in this con- text. In chapter 13, I describe how to calculate loop corrections to effective coupling constants. This is very important for comparing string theory predictions at low energy with the real world. In chapter 14 a brief introduction to non-perturbative string con- nections and non-perturbative effects is given. This is a fast-changing subject and I have just included some basics as well as tools, so that the reader orients him(her)self in the web of duality connections. Finally, in chapter 15 a brief outlook and future problems are presented. I have added a number of appendices to make several technical discussions self-contained. 5 In Appendix A useful information on the elliptic ϑ-functions is included. In Appendix B, I rederive the various lattice sums that appear in toroidal compactifications. In Appendix C the Kaluza-Klein ansatz is described, used to obtain actions in lower dimensions after toroidal compactification. In Appendix D some facts are presented about four-dimensional locally supersymmetric theories with N=1,2,4 supersymmetry. In Appendix E, BPS states are described along with their representation theory and helicity supertrace formulae that can be used to trace their appearance in a supersymmetric theory. In Appendix F facts about elliptic modular forms are presented, which are useful in many contexts, notably in the one-loop computation of thresholds and counting of BPS multiplicities. In Ap- pendix G, I present the computation of helicity-generating string partition functions and the associated calculation of BPS multiplicities. Finally, in Appendix H, I briefly review electric–magnetic duality in four dimensions. I have not tried to be complete in my referencing. The focus was to provide, in most cases, appropriate reviews for further reading. Only
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