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Topology and Geometry in Physics 123 Editors Eike Bick d-fine GmbH Opernplatz 2 60313 Frankfurt Germany Frank Daniel Steffen DESY Theory Group Notkestraße 85 22603 Hamburg Germany E. Bick, F.D. Steffen (Eds.), TopologyandGeometryinPhysics,Lect.NotesPhys.659 (Springer, Berlin Heidelberg 2005), DOI 10.1007/b100632 Library of Congress Control Nu mber: 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.Duplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsof 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 Be rlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, e ven 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-T E X-Production GmbH Cover design: design & production,Heidelberg Printed on acid-free paper 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 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 stu- 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 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 Theoret- 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 Sch¨ucker, 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, Bj¨orn Feuerbacher, J¨org J¨ackel, 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, J¨org J¨ackel 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 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 Weg- ner, 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 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. Sch¨ucker 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 [...]... the rapid change in direction of the velocity field close to the center of a vortex in a fluid However, with the modulus of the Higgs field approaching zero, no in nite energy density is associated with this in nite variation in the phase In the Ginzburg–Landau theory, the core of the vortex contains no Cooper pairs (φ = 0), the system is locally in the ordinary conducting phase containing a magnetic field... devoted to “Supersymmetric Solitons and Topology and, in particular, on critical or BPS-saturated kinks and domain walls His discussion includes minimal N = 1 supersymmetric models of the Landau–Ginzburg type in 1+1 dimensions, the minimal Wess–Zumino model in 3+1 dimensions, and the supersymmetric CP(1) model in 1+1 dimensions, which is a hybrid model (Landau–Ginzburg model on curved target space)... during this process This is devastating for the proof of renormalizability since gauge invariance is needed to constrain the terms appearing in the renormalized theory BRST quantization solves this problem using concepts transferred from algebraic geometry More generally, the BRST formalism provides an elegant framework for dealing with constrained systems, for example, in general relativity or string... restriction to Topological Concepts in Gauge Theories 21 Fig 4 Phase of matter field with winding number n = 0 continuous functions follows from energy considerations Discontinuous changes of fields are in general connected with in nite energies or energy densities For instance, a homotopy of the “spin system” shown in Fig 4 is provided by a spin wave connecting some initial F (x, 0) with some final configuration... C1 in the magnetic field generated by a current flowing along C2 Under continuous deformations of these curves, the value of lk{C1 , C2 }, the Linking Number (“Anzahl der Umschlingungen”), remains unchanged This quantity is a topological invariant It is an integer which counts the (signed) number of F Lenz, Topological Concepts in Gauge Theories, Lect Notes Phys 659, 7–98 (2005) c Springer-Verlag Berlin... Lectures on Functional u Nanostructures Vol.659: E Bick, F.D Steffen (Eds.), Topology and Geometry in Physics Topological Concepts in Gauge Theories F Lenz Institute for Theoretical Physics III, University of Erlangen-N¨ rnberg, u Staudstrasse 7, 91058 Erlangen, Germany Abstract In these lecture notes, an introduction to topological concepts and methods in studies of gauge field theories is presented The three... subject are far reaching and go beyond the standard model: From new aspects of the confinement problem to the understanding of superconductors, from the motivation for cosmic in ation to intriguing phenomena in supersymmetric models Accompanying the progress in quantum field theory, attempts have been made to merge the standard model and general relativity In the setting of noncommutative geometry, it is... http://www.springerlink.com/ 8 F Lenz intersections of the loop C1 with an arbitrary (oriented) surface in R3 whose boundary is the loop C2 (cf [2,3]) In the same note, Gauß deplores the little progress in topology (“Geometria Situs”) since Leibniz’s times who in 1679 postulated “another analysis, purely geometric or linear which also defines the position (situs), as algebra defines magnitude” Leibniz also had in. .. unexpected links between seemingly very different phenomena This common basis in the theoretical description not only refers to obvious topological objects like vortices, which are encountered on almost all scales in physics, it applies also to more abstract concepts “Helicity”, for instance, a topological invariant in inviscid fluids, discovered in 1969 [5], is closely related to the topological charge in gauge... winding number n This winding number counts how often the phase θ winds around the circle when the asymptotic circle (ϕ) is traversed once A formal definition of the winding number is obtained by decomposing a continuous but otherwise arbitrary θ(ϕ) into a strictly periodic and a linear function θn (ϕ) = θperiod (ϕ) + nϕ n = 0, ±1, where θperiod (ϕ + 2π) = θperiod (ϕ) The linear functions can serve . German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Be rlin Heidelberg 2005 Printed in Germany The use. vortices which in space-time are respectively singular on a point, a world-line, or a world-sheet. They are solutions to classical non-linear field equations.