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Typesetting:Camera-readybytheauthors/editor Dataconversion:PTP-BerlinProtago-TEX-ProductionGmbH Coverdesign:design&production,Heidelberg Printedonacid-freepaper 54/3141/ts-543210 Preface The concepts and methods of topology and geometry are an indispensable part of theoretical physics today. They have led to a deeper understanding of many crucial aspects in condensed matter physics, cosmology, gravity, and particle physics. Moreover, several intriguing connections between only apparently dis- connected phenomena have been revealed based on these mathematical tools. Topological and geometrical considerations will continue to play a central role intheoreticalphysics.Wehavehighhopesandexpectnewinsightsrangingfrom an understanding of high-temperature superconductivity up to future progress in the construction of quantum gravity. This book can be considered an advanced textbook on modern applications oftopologyandgeometryinphysics.Withemphasisonapedagogicaltreatment alsoofrecentdevelopments,itismeanttobringgraduateandpostgraduatestu- dents familiar with quantum field theory (and general relativity) to the frontier of active research in theoretical physics. The book consists of five lectures written by internationally well known ex- perts with outstanding pedagogical skills. It is based on lectures delivered by theseauthorsattheautumnschool“TopologyandGeometryinPhysics”heldat the beautiful baroque monastery in Rot an der Rot, Germany, in the year 2001. This school was organized by the graduate students of the Graduiertenkolleg “PhysicalSystemswithManyDegreesofFreedom”oftheInstituteforTheoret- ical Physics at the University of Heidelberg. As this Graduiertenkolleg supports graduate students working in various areas of theoretical physics, the topics were chosen in order to optimize overlap with condensed matter physics, parti- cle physics, and cosmology. In the introduction we give a brief overview on the relevance of topology and geometry in physics, describe the outline of the book, and recommend complementary literature. We are extremely thankful to Frieder Lenz, Thomas Schu¨cker, Misha Shif- man, Jan-Willem van Holten, and Jean Zinn-Justin for making our autumn school a very special event, for vivid discussions that helped us to formulate the introduction, and, of course, for writing the lecture notes for this book. For the invaluable help in the proofreading of the lecture notes, we would like to thank Tobias Baier, Kurush Ebrahimi-Fard, Bjo¨rn Feuerbacher, Jo¨rg Ja¨ckel, Filipe Paccetti, Volker Schatz, and Kai Schwenzer. The organization of the autumn school would not have been possible with- out our team. We would like to thank Lala Adueva for designing the poster and the web page, Tobial Baier for proposing the topic, Michael Doran and Volker VI Preface Schatz for organizing the transport of the blackboard, Jo¨rg Ja¨ckel for finan- cial management, Annabella Rauscher for recommending the monastery in Rot an der Rot, and Steffen Weinstock for building and maintaining the web page. Christian Nowak and Kai Schwenzer deserve a special thank for the organiza- tion of the magnificent excursion to Lindau and the boat trip on the Lake of Constance. The timing in coordination with the weather was remarkable. We areverythankfulforthefinancialsupportfromtheGraduiertenkolleg“Physical Systems with Many Degrees of Freedom” and the funds from the Daimler-Benz StiftungprovidedthroughDieterGromes.Finally,wewanttothankFranzWeg- ner, the spokesperson of the Graduiertenkolleg, for help in financial issues and his trust in our organization. Wehopethatthisbookhascapturedsomeofthespiritoftheautumnschool on which it is based. Heidelberg Eike Bick July, 2004 Frank Daniel Steffen Contents Introduction and Overview E. Bick, F.D. Steffen ............................................... 1 1 Topology and Geometry in Physics ............................... 1 2 An Outline of the Book ......................................... 2 3 Complementary Literature....................................... 4 Topological Concepts in Gauge Theories F. Lenz........................................................... 7 1 Introduction ................................................... 7 2 Nielsen–Olesen Vortex .......................................... 9 2.1 Abelian Higgs Model ....................................... 9 2.2 Topological Excitations..................................... 14 3 Homotopy..................................................... 19 3.1 The Fundamental Group.................................... 19 3.2 Higher Homotopy Groups................................... 24 3.3 Quotient Spaces ........................................... 26 3.4 Degree of Maps............................................ 27 3.5 Topological Groups ........................................ 29 3.6 Transformation Groups..................................... 32 3.7 Defects in Ordered Media................................... 34 4 Yang–Mills Theory ............................................. 38 5 ’t Hooft–Polyakov Monopole..................................... 43 5.1 Non-Abelian Higgs Model................................... 43 5.2 The Higgs Phase........................................... 45 5.3 Topological Excitations..................................... 47 6 Quantization of Yang–Mills Theory............................... 51 7 Instantons..................................................... 55 7.1 Vacuum Degeneracy ....................................... 55 7.2 Tunneling................................................. 56 7.3 Fermions in Topologically Non-trivial Gauge Fields ............ 58 7.4 Instanton Gas ............................................. 60 7.5 Topological Charge and Link Invariants....................... 62 8 Center Symmetry and Confinement............................... 64 8.1 Gauge Fields at Finite Temperature and Finite Extension....... 65 8.2 Residual Gauge Symmetries in QED ......................... 66 8.3 Center Symmetry in SU(2) Yang–Mills Theory ................ 69 VIII Contents 8.4 Center Vortices............................................ 71 8.5 The Spectrum of the SU(2) Yang–Mills Theory ................ 74 9 QCD in Axial Gauge ........................................... 76 9.1 Gauge Fixing ............................................. 76 9.2 Perturbation Theory in the Center-Symmetric Phase ........... 79 9.3 Polyakov Loops in the Plasma Phase ......................... 83 9.4 Monopoles ................................................ 86 9.5 Monopoles and Instantons .................................. 89 9.6 Elements of Monopole Dynamics............................. 90 9.7 Monopoles in Diagonalization Gauges ........................ 91 10 Conclusions.................................................... 93 Aspects of BRST Quantization J.W. van Holten ................................................... 99 1 Symmetries and Constraints .................................... 99 1.1 Dynamical Systems with Constraints ........................ 100 1.2 Symmetries and Noether’s Theorems ........................ 105 1.3 Canonical Formalism ...................................... 109 1.4 Quantum Dynamics ....................................... 113 1.5 The Relativistic Particle ................................... 115 1.6 The Electro-magnetic Field ................................. 119 1.7 Yang–Mills Theory ........................................ 121 1.8 The Relativistic String ..................................... 124 2 Canonical BRST Construction .................................. 126 2.1 Grassmann Variables ...................................... 127 2.2 Classical BRST Transformations ............................ 130 2.3 Examples ................................................ 133 2.4 Quantum BRST Cohomology ............................... 135 2.5 BRST-Hodge Decomposition of States ....................... 138 2.6 BRST Operator Cohomology ............................... 142 2.7 Lie-Algebra Cohomology ................................... 143 3 Action Formalism .............................................. 146 3.1 BRST Invariance from Hamilton’s Principle .................. 146 3.2 Examples ................................................ 147 3.3 Lagrangean BRST Formalism ............................... 148 3.4 The Master Equation ...................................... 152 3.5 Path-Integral Quantization ................................. 154 4 Applications of BRST Methods .................................. 156 4.1 BRST Field Theory ....................................... 156 4.2 Anomalies and BRST Cohomology .......................... 158 Appendix. Conventions ............................................. 165 Chiral Anomalies and Topology J. Zinn-Justin ..................................................... 167 1 Symmetries, Regularization, Anomalies............................ 167 2 Momentum Cut-Off Regularization ............................... 170 Contents IX 2.1 Matter Fields: Propagator Modification....................... 170 2.2 Regulator Fields........................................... 173 2.3 Abelian Gauge Theory ..................................... 174 2.4 Non-Abelian Gauge Theories................................ 177 3 Other Regularization Schemes ................................... 178 3.1 Dimensional Regularization ................................. 179 3.2 Lattice Regularization...................................... 180 3.3 Boson Field Theories....................................... 181 3.4 Fermions and the Doubling Problem ......................... 182 4 The Abelian Anomaly .......................................... 184 4.1 Abelian Axial Current and Abelian Vector Gauge Fields ........ 184 4.2 Explicit Calculation........................................ 188 4.3 Two Dimensions........................................... 194 4.4 Non-Abelian Vector Gauge Fields and Abelian Axial Current.... 195 4.5 Anomaly and Eigenvalues of the Dirac Operator ............... 196 5 Instantons, Anomalies, and θ-Vacua .............................. 198 5.1 The Periodic Cosine Potential ............................... 199 5.2 Instantons and Anomaly: CP(N-1) Models .................... 201 5.3 Instantons and Anomaly: Non-Abelian Gauge Theories ......... 206 5.4 Fermions in an Instanton Background ........................ 210 6 Non-Abelian Anomaly .......................................... 212 6.1 General Axial Current...................................... 212 6.2 Obstruction to Gauge Invariance............................. 214 6.3 Wess–Zumino Consistency Conditions ........................ 215 7 Lattice Fermions: Ginsparg–Wilson Relation ....................... 216 7.1 Chiral Symmetry and Index................................. 217 7.2 Explicit Construction: Overlap Fermions...................... 221 8 Supersymmetric Quantum Mechanics and Domain Wall Fermions .... 222 8.1 Supersymmetric Quantum Mechanics......................... 222 8.2 Field Theory in Two Dimensions ............................ 226 8.3 Domain Wall Fermions ..................................... 227 Appendix A. Trace Formula for Periodic Potentials..................... 229 Appendix B. Resolvent of the Hamiltonian in Supersymmetric QM....... 231 Supersymmetric Solitons and Topology M. Shifman ....................................................... 237 1 Introduction ................................................... 237 2 D = 1+1; N =1 .............................................. 238 2.1 Critical (BPS) Kinks ....................................... 242 2.2 The Kink Mass (Classical) .................................. 243 2.3 Interpretation of the BPS Equations. Morse Theory ............ 244 2.4 Quantization. Zero Modes: Bosonic and Fermionic ............. 245 2.5 Cancelation of Nonzero Modes............................... 248 2.6 Anomaly I ................................................ 250 2.7 Anomaly II (Shortening Supermultiplet Down to One State) .... 252 3 Domain Walls in (3+1)-Dimensional Theories ...................... 254 X Contents 3.1 Superspace and Superfields ................................. 254 3.2 Wess–Zumino Models ...................................... 256 3.3 Critical Domain Walls...................................... 258 3.4 Finding the Solution to the BPS Equation .................... 261 3.5 Does the BPS Equation Follow from the Second Order Equation of Motion?................................................ 261 3.6 Living on a Wall........................................... 262 4 Extended Supersymmetry in Two Dimensions: The Supersymmetric CP(1) Model................................ 263 4.1 Twisted Mass ............................................. 266 4.2 BPS Solitons at the Classical Level .......................... 267 4.3 Quantization of the Bosonic Moduli.......................... 269 4.4 The Soliton Mass and Holomorphy........................... 271 4.5 Switching On Fermions..................................... 273 4.6 Combining Bosonic and Fermionic Moduli .................... 274 5 Conclusions.................................................... 275 Appendix A. CP(1) Model = O(3) Model (N =1 Superfields N)......... 275 Appendix B. Getting Started (Supersymmetry for Beginners)............ 277 B.1 Promises of Supersymmetry................................. 280 B.2 Cosmological Term......................................... 281 B.3 Hierarchy Problem......................................... 281 Forces from Connes’ Geometry T. Schu¨cker ....................................................... 285 1 Introduction ................................................... 285 2 Gravity from Riemannian Geometry .............................. 287 2.1 First Stroke: Kinematics.................................... 287 2.2 Second Stroke: Dynamics ................................... 287 3 Slot Machines and the Standard Model............................ 289 3.1 Input .................................................... 290 3.2 Rules .................................................... 292 3.3 The Winner............................................... 296 3.4 Wick Rotation ............................................ 300 4 Connes’ Noncommutative Geometry .............................. 303 4.1 Motivation: Quantum Mechanics............................. 303 4.2 The Calibrating Example: Riemannian Spin Geometry ......... 305 4.3 Spin Groups .............................................. 308 5 The Spectral Action ............................................ 311 5.1 Repeating Einstein’s Derivation in the Commutative Case ...... 311 5.2 Almost Commutative Geometry ............................. 314 5.3 The Minimax Example ..................................... 317 5.4 A Central Extension ....................................... 322 6 Connes’ Do-It-Yourself Kit ...................................... 323 6.1 Input .................................................... 323 6.2 Output................................................... 327 6.3 The Standard Model ....................................... 329