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Topology and Geometry in Physics 123 Editors Eike Bick d-fine GmbH Opernplatz 60313 Frankfurt Germany Frank Daniel Steffen DESY Theory Group Notkestraße 85 22603 Hamburg Germany E Bick, F.D Steffen (Eds.), Topology and Geometry in Physics, Lect Notes Phys 659 (Springer, Berlin Heidelberg 2005), DOI 10.1007/b100632 Library of Congress Control Number: 2004116345 ISSN 0075-8450 ISBN 3-540-23125-0 Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: Camera-ready by the authors/editor Data conversion: PTP-Berlin Protago-TEX-Production GmbH Cover design: design & production, Heidelberg Printed on acid-free paper 54/3141/ts - 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 disconnected phenomena have been revealed based on these mathematical tools Topological and geometrical considerations will continue to play a central role in theoretical physics We have high hopes and expect new insights ranging from 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 of topology and geometry in physics With emphasis on a pedagogical treatment also of recent developments, it is meant to bring graduate and postgraduate students 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 experts with outstanding pedagogical skills It is based on lectures delivered by these authors at the autumn school “Topology and Geometry in Physics” held at 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 “Physical Systems with Many Degrees of Freedom” of the Institute for Theoretical 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, particle 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 Schă ucker, Misha Shifman, 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, Bjă orn Feuerbacher, Jă org Jă ackel, Filipe Paccetti, Volker Schatz, and Kai Schwenzer The organization of the autumn school would not have been possible without 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, Jă org Jă ackel for nancial 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 organization 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 are very thankful for the financial support from the Graduiertenkolleg “Physical Systems with Many Degrees of Freedom” and the funds from the Daimler-Benz Stiftung provided through Dieter Gromes Finally, we want to thank Franz Wegner, the spokesperson of the Graduiertenkolleg, for help in financial issues and his trust in our organization We hope that this book has captured some of the spirit of the autumn school on which it is based Heidelberg July, 2004 Eike Bick Frank Daniel Steffen Contents Introduction and Overview E Bick, F.D Steffen Topology and Geometry in Physics An Outline of the Book Complementary Literature 1 Topological Concepts in Gauge Theories F Lenz Introduction Nielsen–Olesen Vortex 2.1 Abelian Higgs Model 2.2 Topological Excitations Homotopy 3.1 The Fundamental Group 3.2 Higher Homotopy Groups 3.3 Quotient Spaces 3.4 Degree of Maps 3.5 Topological Groups 3.6 Transformation Groups 3.7 Defects in Ordered Media Yang–Mills Theory ’t Hooft–Polyakov Monopole 5.1 Non-Abelian Higgs Model 5.2 The Higgs Phase 5.3 Topological Excitations Quantization of Yang–Mills Theory Instantons 7.1 Vacuum Degeneracy 7.2 Tunneling 7.3 Fermions in Topologically Non-trivial Gauge Fields 7.4 Instanton Gas 7.5 Topological Charge and Link Invariants Center Symmetry and Confinement 8.1 Gauge Fields at Finite Temperature and Finite Extension 8.2 Residual Gauge Symmetries in QED 8.3 Center Symmetry in SU(2) Yang–Mills Theory 7 9 14 19 19 24 26 27 29 32 34 38 43 43 45 47 51 55 55 56 58 60 62 64 65 66 69 VIII Contents 8.4 Center Vortices 8.5 The Spectrum of the SU(2) Yang–Mills Theory QCD in Axial Gauge 9.1 Gauge Fixing 9.2 Perturbation Theory in the Center-Symmetric Phase 9.3 Polyakov Loops in the Plasma Phase 9.4 Monopoles 9.5 Monopoles and Instantons 9.6 Elements of Monopole Dynamics 9.7 Monopoles in Diagonalization Gauges 10 Conclusions 71 74 76 76 79 83 86 89 90 91 93 Aspects of BRST Quantization J.W van Holten Symmetries and Constraints 1.1 Dynamical Systems with Constraints 1.2 Symmetries and Noether’s Theorems 1.3 Canonical Formalism 1.4 Quantum Dynamics 1.5 The Relativistic Particle 1.6 The Electro-magnetic Field 1.7 Yang–Mills Theory 1.8 The Relativistic String Canonical BRST Construction 2.1 Grassmann Variables 2.2 Classical BRST Transformations 2.3 Examples 2.4 Quantum BRST Cohomology 2.5 BRST-Hodge Decomposition of States 2.6 BRST Operator Cohomology 2.7 Lie-Algebra Cohomology Action Formalism 3.1 BRST Invariance from Hamilton’s Principle 3.2 Examples 3.3 Lagrangean BRST Formalism 3.4 The Master Equation 3.5 Path-Integral Quantization Applications of BRST Methods 4.1 BRST Field Theory 4.2 Anomalies and BRST Cohomology Appendix Conventions 99 99 100 105 109 113 115 119 121 124 126 127 130 133 135 138 142 143 146 146 147 148 152 154 156 156 158 165 Chiral Anomalies and Topology J Zinn-Justin 167 Symmetries, Regularization, Anomalies 167 Momentum Cut-Off Regularization 170 Contents IX 2.1 Matter Fields: Propagator Modification 2.2 Regulator Fields 2.3 Abelian Gauge Theory 2.4 Non-Abelian Gauge Theories Other Regularization Schemes 3.1 Dimensional Regularization 3.2 Lattice Regularization 3.3 Boson Field Theories 3.4 Fermions and the Doubling Problem The Abelian Anomaly 4.1 Abelian Axial Current and Abelian Vector Gauge Fields 4.2 Explicit Calculation 4.3 Two Dimensions 4.4 Non-Abelian Vector Gauge Fields and Abelian Axial Current 4.5 Anomaly and Eigenvalues of the Dirac Operator Instantons, Anomalies, and θ-Vacua 5.1 The Periodic Cosine Potential 5.2 Instantons and Anomaly: CP(N-1) Models 5.3 Instantons and Anomaly: Non-Abelian Gauge Theories 5.4 Fermions in an Instanton Background Non-Abelian Anomaly 6.1 General Axial Current 6.2 Obstruction to Gauge Invariance 6.3 Wess–Zumino Consistency Conditions Lattice Fermions: Ginsparg–Wilson Relation 7.1 Chiral Symmetry and Index 7.2 Explicit Construction: Overlap Fermions Supersymmetric Quantum Mechanics and Domain Wall Fermions 8.1 Supersymmetric Quantum Mechanics 8.2 Field Theory in Two Dimensions 8.3 Domain Wall Fermions Appendix A Trace Formula for Periodic Potentials Appendix B Resolvent of the Hamiltonian in Supersymmetric QM 170 173 174 177 178 179 180 181 182 184 184 188 194 195 196 198 199 201 206 210 212 212 214 215 216 217 221 222 222 226 227 229 231 Supersymmetric Solitons and Topology M Shifman Introduction D = 1+1; N = 2.1 Critical (BPS) Kinks 2.2 The Kink Mass (Classical) 2.3 Interpretation of the BPS Equations Morse Theory 2.4 Quantization Zero Modes: Bosonic and Fermionic 2.5 Cancelation of Nonzero Modes 2.6 Anomaly I 2.7 Anomaly II (Shortening Supermultiplet Down to One State) Domain Walls in (3+1)-Dimensional Theories 237 237 238 242 243 244 245 248 250 252 254 X Contents 3.1 3.2 3.3 3.4 3.5 Superspace and Superfields Wess–Zumino Models Critical Domain Walls Finding the Solution to the BPS Equation Does the BPS Equation Follow from the Second Order Equation of Motion? 3.6 Living on a Wall Extended Supersymmetry in Two Dimensions: The Supersymmetric CP(1) Model 4.1 Twisted Mass 4.2 BPS Solitons at the Classical Level 4.3 Quantization of the Bosonic Moduli 4.4 The Soliton Mass and Holomorphy 4.5 Switching On Fermions 4.6 Combining Bosonic and Fermionic Moduli Conclusions Appendix A CP(1) Model = O(3) Model (N = Superfields N ) Appendix B Getting Started (Supersymmetry for Beginners) B.1 Promises of Supersymmetry B.2 Cosmological Term B.3 Hierarchy Problem 254 256 258 261 263 266 267 269 271 273 274 275 275 277 280 281 281 Forces from Connes Geometry T Schă ucker Introduction Gravity from Riemannian Geometry 2.1 First Stroke: Kinematics 2.2 Second Stroke: Dynamics Slot Machines and the Standard Model 3.1 Input 3.2 Rules 3.3 The Winner 3.4 Wick Rotation Connes’ Noncommutative Geometry 4.1 Motivation: Quantum Mechanics 4.2 The Calibrating Example: Riemannian Spin Geometry 4.3 Spin Groups The Spectral Action 5.1 Repeating Einstein’s Derivation in the Commutative Case 5.2 Almost Commutative Geometry 5.3 The Minimax Example 5.4 A Central Extension Connes’ Do-It-Yourself Kit 6.1 Input 6.2 Output 6.3 The Standard Model 285 285 287 287 287 289 290 292 296 300 303 303 305 308 311 311 314 317 322 323 323 327 329 261 262 ... exp{i2nπsτ } (5 4) uc (s) = (5 5) and the constant map The mapping H(s, t) = exp π − i tτ π exp i t(τ cos 2πns + τ sin 2πns) has the desired properties (cf (3 4)) H(s, 0) = 1, H(s, 1) = u2n (s), H(0, t)... continuously deformed into the linear function mϕ The mapping H(ϕ, t) = (1 − t) ? ?(? ?) + t ϕ ? ?(2 π) 2π with the properties H(0, t) = ? ?(0 ) = , H(2π, t) = ? ?(2 π) , (3 5) Topological Concepts in Gauge Theories... Frank Daniel Steffen DESY Theory Group Notkestraße 85 22603 Hamburg Germany E Bick, F.D Steffen (Eds.), Topology and Geometry in Physics, Lect Notes Phys 659 (Springer, Berlin Heidelberg 2005), DOI