String Theory in a Nutshell. Elias Kiritsis

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1 String Theory in a Nutshell Elias Kiritsis P R I N C E T O N U N I V E R S I T Y P R E S S P R I N C E T O N A N D O X F O R D

2 Contents Preface Abbreviations xv xvii Introduction Prehistory The Case for String Theory A Stringy Historical Perspective Conventions 8 Bibliography 9 Classical String Theory The Point Particle Relativistic Strings Oscillator Expansions Closed strings Open strings The Virasoro constraints 24 Bibliography 26 Exercises 26 Quantization of Bosonic Strings Covariant Canonical Quantization Light-cone Quantization Spectrum of the Bosonic String Unoriented Strings Open strings and Chan-Paton factors Path Integral Quantization 37

3 viii Contents 3.6 Topologically Nontrivial World-sheets 3.7 BRST Primer 3.8 BRST in String Theory and the Physical Spectrum Bibliography Exercises Conformal Field Theory 4.1 Conformal Transformations The case of two dimensions 4.2 Conformally Invariant Field Theory 4.3 Radial Quantization 4.4 Mode Expansions 4.5 The Virasoro Algebra and the Central Charge 4.6 The Hilbert Space 4.7 The Free Boson 4.8 The Free Fermion 4.9 The Conformal Anomaly 4.10 Representations of the Conformal Algebra 4.11 Affine Current Algebras 4.12 Free Fermions and O(N) Affine Symmetry 4.13 Superconformal Symmetry M = (l,0) 2 superconformal symmetry M = (2,0) 2 superconformal symmetry M = (4,0) 2 superconformal symmetry 4.14 Scalars with Background Charge 4.15 The CFT of Ghosts 4.16 CFT on the Disk Free massless bosons on the disk Free massless fermions on the disk The projective plane 4.17 CFT on the Torus 4.18 Compact Scalars Modular invariance Decompactification The torus propagator Marginal deformations Multiple compact scalars Enhanced symmetry and the string Brout-Englert-Higgs effect T-duality 4.19 Free Fermions on the Torus 4.20 Bosonizarion "Bosonization" of bosonic ghost system 4.21 Orbifolds

4 Contents ix 4.22 CFT on Other Surfaces of Euler Number Zero CFT on Higher-genus Riemann Surfaces 116 Bibliography 117 Exercises 118 Scattering Amplitudes and Vertex Operators Physical Vertex Operators Calculation of Tree-level Tachyon Amplitudes The closed string The open string The One-loop Vacuum Amplitudes The torus The cylinder The Klein bottle The Mobius strip Tadpole cancellation UV structure and UV-IR correspondence 140 Bibliography 141 Exercises 142 Strings in Background Fields The Nonlinear a -model Approach The Quest for Conformal Invariance Linear Dilaton and Strings in D < 26 Dimensions T-duality in Nontrivial Backgrounds 151 Bibliography 151 Exercises 152 Superstrings and Supersymmetry Af = (1,1) 2 World-sheet Superconformal Symmetry Closed (Type-II) Superstrings Massless R-R states Type-I Superstrings Heterotic Superstrings Superstring Vertex Operators One-loop Superstring Vacuum Amplitudes The type-iia/b superstring The heterotic superstring The type-i superstring Closed Superstrings and T-duality The type-11 string theories The heterotic string Supersymmetric Effective Actions 175

5 x I Contents 7.9 Anomalies 176 Bibliography 182 Exercises 183 D-branes Antisymmetric Tensors and p-branes Open Strings and T-duality D-branes D-branes and R-R Charges D-instantons D-brane Effective Actions The Dirac-Bom-Infeld action Anomaly-related terms Multiple Branes and Nonabelian Symmetry T-duality and Orientifolds D-branes as Supergravity Solitons The supergravity solutions Horizons and singularities The extremal branes and their near-horizon geometry NS 5 -branes 211 Bibliography 213 Exercises 213 Compactifications and Supersymmetry Breaking Narain Compactifications World-sheet versus Space-time Supersymmetry M = I4 space-time supersymmetry J\f = 24 space-time supersymmetry Orbifold Reduction of Supersymmetry A Heterotic Orbifold with J\f = 2± Supersymmetry Spontaneous Supersymmetry Breaking A Heterotic jv = I4 Orbifold and Chirality in Four Dimensions Calabi-Yau Manifolds Holonomy Consequences of SU(3) holonomy The CY moduli space M = I4 Heterotic Compactifications The low-energy M = I4 heterotic spectrum K3 Compactification of the Type-II String M = 2 6 Orbifolds of the Type-II String CY Compactifications of Type-II Strings Mirror Symmetry Absence of Continuous Global Symmetries 255

6 Contents xi 9.14Orientifolds K3 orientifolds The Klein bottle amplitude D-branes on T 4 /Z 2 ' The cylinder amplitude The Mo'bius strip amplitude Tadpole cancellation The open string spectrum D-branes at Orbifold Singularities Magnetized Compactifications and Intersecting Branes Open strings in an internal magnetic field Intersecting branes Intersecting D^-branes Where is the Standard Model? The heterotic string Type-II string theory The type-i string Unification 284 Bibliography 286 Exercises 287 W Loop Corrections to String Effective Couplings Calculation of Heterotic Gauge Thresholds On-shell Infrared Regularization Evaluation of the threshold Heterotic Gravitational Thresholds One-loop Fayet-Iliopoulos Terms M = 1,2 4 Examples of Threshold Corrections AT = 2 4 Universality of Thresholds Unification Revisited 315 Bibliography 317 Exercises 317 Ti~ Duality Connections and Nonperturbative Effects Perturbative Connections BPS States and BPS Bounds Nonrenormalization Theorems and BPS-saturated Couplings Type-IIA versus M-theory Self-duality of the Type-IIB String U-duality of Type-II String Theory U-duality and bound states Heterotic/Type I Duality in Ten Dimensions The type-i Desiring 339

7 xii Contents The type-i D s -brane Further consistency checks M-theory and the E 8 x E 8 Heterotic String Unification at strong heterotic coupling Heterotic/Type II Duality in Six Dimensions Gauge symmetry enhancement and singular K3 surfaces Heterotic /type II duality in four dimensions Conifold Singularities and Conifold Transitions 356 Bibliography 362 Exercises Black Holes and Entropy in String Theory A Brief History The Strategy Black-hole Thermodynamics The Euclidean continuation Hawking evaporation and greybody factors The Information Problem and the Holographic Hypothesis Five-dimensional Extremal Charged Black Holes Five-dimensional Nonextremal RN Black Holes The Near-horizon Region Semiclassical Derivation of the Hawking Rate The Microscopic Realization The world-volume theory of the bound state The low-energy SCFT of the D V D 5 bound state Microscopic calculation of the entropy Microscopic derivation of Hawking evaporation rates Epilogue 396 Bibliography 398 Exercises I The Bulk/Boundary Correspondence Large-N Gauge Theories and String Theory The Decoupling Principle The Near-horizon Limit Elements of the Correspondence Bulk Fields and Boundary Operators Holography Testing the AdS 5 /CFT 4 Correspondence The chiral spectrum ofjsf = 4 gauge theory Matching to the string theory spectrum M = &five-dimensionalgauged supergravity Protected correlation functions and anomalies 422

8 Contents xiii 13.8 Correlation Functions Two-point Junctions Three-point functions The gravitational action and the conformal anomaly Wilson Loops AdS5/CFT 4 Correspondence at Finite Temperature Af 4 super Yang-Mills theory at finite temperature The near-horizon limit of black Dj-branes Finite-volume and large-nphase transitions Thermal holographic physics Spatial Wilson loops in (a version of) QCD The glueball mass spectrum AdS 3 /CFT 2 Correspondence The greybody factors revisited The Holographic Renormalization Group Perturbations of the CFT Domain walls and flow equations A RG flow preserving J\f = 1 supersymmetry The Randall-Sundrum Geometry An alternative to compactiflcation 459 Bibliography 462 Exercises String Theory and Matrix Models M(atrix) Theory Membrane quantization Type-IIA D 0 -branes and DLCQ Gravitons and branes in M(atrix) theory The two-graviton interaction from M(atrix) theory Matrix Models and D = 1 Bosonic String Theory The continuum limit Solving the matrix model The double-scaling limit Thefree-fermionpicture Matrix Description of D = 2 String Theory Matrix quantum mechanics andfreefermions on the line The continuum limit The double-scaling limit D-particles, tachyons, and holography 496 Bibliography 498 Exercises 498 Appendix A Two-dimensional Complex Geometry 503

9 xiv Contents Appendix B Differential Forms 505 Appendix C Theta and Other Elliptic Functions 507 C.I & and Related Functions 507 C.2 The Weierstrass Function 510 C.3 Modular Forms 510 C.4 Poisson Resummation 512 Appendix D Toroidal Lattice Sums 513 Appendix E Toroidal Kaluza-Klein Reduction 516 Appendix F The Reissner-Nordstrom Black Hole 519 Appendix G Electric-magnetic Duality in D = Appendix H Supersymmetric Actions in Ten and Eleven Dimensions 525 H.I The Af = In Supergravity 526 H.2 Type-IIA Supergravity 527 H.3 Type-IIB Supergravity 528 H.4 Type-II Supergravities: The Democratic Formulation 529 H.5 Af = lio Supersymmetry 530 Appendix I A/* = 1,2, Four-dimensional Supergravity Coupled to Matter Af = I4 Supergravity Af = 2 4 Supergravity 535 Appendix j BPS Multiplets in Four Dimensions 537 Appendix K The Geometry of Anti-de Sitter Space 541 K.1 The Minkowski Signature AdS 541 K.2 Euclidean AdS 544 K.3 The Conformal Structure of Flat Space 546 K.4 Fields in AdS 548 K.4.1 The wave equation in Poincare coordinates 549 K.4.2 The bulk-boundary propagator 550 K.4.3 The bulk-to-bulk propagator 551 Bibliography 553 Index 575

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