Theoretical and Mathematical Physics Gerd Rudolph Matthias Schmidt Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields Differential Geometry and Mathematical Physics Theoretical and Mathematical Physics Theseriesfoundedin1975andformerly(until2005)entitledTextsandMonographsin Physics (TMP) publishes high-level monographs in theoretical and mathematical physics.ThechangeoftitletoTheoreticalandMathematicalPhysics(TMP)signalsthat theseriesisasuitablepublicationplatformforboththemathematicalandthetheoretical physicist. Thewider scope of the series is reflected by the composition ofthe editorial board, comprising both physicists and mathematicians. The books, written in a didactic style and containing a certain amount of elementary background material, bridge the gap between advanced textbooks and research monographs. They can thus serve as basis for advanced studies, not only for lectures and seminars at graduate level, but also for scientists entering a field of research. Editorial Board W.Beiglböck,InstituteofAppliedMathematics,UniversityofHeidelberg,Heidelberg, Germany P. Chrusciel, Gravitational Physics, University of Vienna, Vienna, Austria J.-P. Eckmann, Département de Physique Théorique, Université de Genéve, Geneve, Switzerland H. Grosse, Institute of Theoretical Physics, University of Vienna, Vienna, Austria A. Kupiainen, Department of Mathematics, University of Helsinki, Helsinki, Finland H. Löwen, Institute of Theoretical Physics, Heinrich-Heine-University of Düsseldorf, Düsseldorf, Germany M. Loss, School of Mathematics, Georgia Institute of Technology, Atlanta, USA N.A. Nekrasov, Simons Center for Geometry and Physics, State University of New York, Stony Brook, USA M. Ohya, Tokyo University of Science, Noda, Japan M. Salmhofer, Institute of Theoretical Physics, University of Heidelberg, Heidelberg, Germany S. Smirnov, Mathematics Section, University of Geneva, Geneva, Switzerland L.Takhtajan,DepartmentofMathematics,StonyBrookUniversity,StonyBrook,USA J. Yngvason, Institute of Theoretical Physics, University of Vienna, Vienna, Austria More information about this series at http://www.springer.com/series/720 Gerd Rudolph Matthias Schmidt (cid:129) Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields 123 Gerd Rudolph Matthias Schmidt Institute for Theoretical Physics Institute for Theoretical Physics University of Leipzig University of Leipzig Leipzig Leipzig Germany Germany ISSN 1864-5879 ISSN 1864-5887 (electronic) Theoretical andMathematical Physics ISBN978-94-024-0958-1 ISBN978-94-024-0959-8 (eBook) DOI 10.1007/978-94-024-0959-8 LibraryofCongressControlNumber:2016950402 ©SpringerScience+BusinessMediaDordrecht2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerScience+BusinessMediaB.V. Theregisteredcompanyaddressis:VanGodewijckstraat30,3311GXDordrecht,TheNetherlands Acknowledgements Itisapleasuretothankallourfriendsandcollaboratorsforfruitfuljointworkover so many years. Notably, we wish to thank Szymon Charzyński, Jochen Dittmann, Christian Fleischhack, Hendrik Grundling, Alexander Hertsch, Johannes Huebschmann,PeterJarvis,JerzyKijowski,Yu.A.Kubyshin,RainerMatthes,José Mourão, Olaf Richter†, Torsten Tok, Igor P. Volobuev and Raimar Wulkenhaar. Moreover, we are grateful to Detlev Buchholz, Bernd Crell, Heinz-Dietrich Doebner, Klaus Fredenhagen, Thomas Friedrich, Krzysztof Gawędzki, Bodo Geyer, Arkadiusz Z. Jadczyk, Hartmann Römer, Manfred Salmhofer, Konrad Schmüdgen, Klaus Sibold, Andrzej Trautman, Armin Uhlmann, Rainer Verch, Stanisław L. Woronowicz and Eberhard Zeidler† for helpful and inspiring dis- cussions. We are especially indebted to Tobias Diez and Peter Jarvis for reading partsofthemanuscript.Wealsowishtothankourlibrarian,Mrs.GabrieleMenge, for her permanent support. v Contents 1 Fibre Bundles and Connections . .... .... .... .... .... ..... .... 1 1.1 Principal Bundles.... ..... .... .... .... .... .... ..... .... 1 1.2 Associated Bundles... ..... .... .... .... .... .... ..... .... 14 1.3 Connections .... .... ..... .... .... .... .... .... ..... .... 24 1.4 Covariant Exterior Derivative and Curvature .... .... ..... .... 37 1.5 The Koszul Calculus . ..... .... .... .... .... .... ..... .... 45 1.6 Bundle Reduction.... ..... .... .... .... .... .... ..... .... 52 1.7 Parallel Transport and Holonomy. .... .... .... .... ..... .... 59 1.8 Automorphisms . .... ..... .... .... .... .... .... ..... .... 72 1.9 Invariant Connections. ..... .... .... .... .... .... ..... .... 76 2 Linear Connections and Riemannian Geometry.... .... ..... .... 93 2.1 Linear Connections... ..... .... .... .... .... .... ..... .... 94 2.2 H-Structures and Compatible Connections.. .... .... ..... .... 108 2.3 Curvature and Holonomy... .... .... .... .... .... ..... .... 127 2.4 Sectional Curvature .. ..... .... .... .... .... .... ..... .... 138 2.5 Symmetric Spaces.... ..... .... .... .... .... .... ..... .... 141 2.6 Compatible Connections on Vector Bundles. .... .... ..... .... 158 2.7 Hodge Theory. The Weitzenboeck Formula. .... .... ..... .... 164 2.8 Four-Dimensional Riemannian Geometry. Self-duality. ..... .... 181 3 Homotopy Theory of Principal Fibre Bundles. Classification... .... 189 3.1 Basics. .... .... .... ..... .... .... .... .... .... ..... .... 190 3.2 Fibrations.. .... .... ..... .... .... .... .... .... ..... .... 200 3.3 The Covering Homotopy Theorem.... .... .... .... ..... .... 212 3.4 Universal Principal Bundles. .... .... .... .... .... ..... .... 217 3.5 The Milnor Construction ... .... .... .... .... .... ..... .... 230 3.6 Classification of Smooth Principal Bundles . .... .... ..... .... 235 3.7 Classifying Mappings Associated with Lie Group Homomorphisms. .... ..... .... .... .... .... .... ..... .... 240 3.8 Universal Connections ..... .... .... .... .... .... ..... .... 244 vii viii Contents 4 Cohomology Theory of Fibre Bundles. Characteristic Classes.. .... 257 4.1 Basics. .... .... .... ..... .... .... .... .... .... ..... .... 258 4.2 Characteristic Classes for the Classical Groups... .... ..... .... 267 4.3 Whitney Sum Formula and Splitting Principle ... .... ..... .... 284 4.4 Field Restriction and Field Extension.. .... .... .... ..... .... 297 4.5 Characteristic Classes for Manifolds... .... .... .... ..... .... 308 4.6 The Weil Homomorphism .. .... .... .... .... .... ..... .... 311 4.7 Genera .... .... .... ..... .... .... .... .... .... ..... .... 336 4.8 The Postnikov Tower and Bundle Classification.. .... ..... .... 345 5 Clifford Algebras, Spin Structures and Dirac Operators. ..... .... 353 5.1 Clifford Algebras .... ..... .... .... .... .... .... ..... .... 354 5.2 Spinor Groups .. .... ..... .... .... .... .... .... ..... .... 365 5.3 Representations.. .... ..... .... .... .... .... .... ..... .... 377 5.4 Spin Structures and Spinc-Structures... .... .... .... ..... .... 393 5.5 Clifford Modules and Dirac Operators . .... .... .... ..... .... 400 5.6 Weitzenboeck Formulae.... .... .... .... .... .... ..... .... 410 5.7 Elliptic Complexes. The Hodge Theorem... .... .... ..... .... 416 5.8 The Atiyah–Singer Index Theorem.... .... .... .... ..... .... 433 5.9 Applications.... .... ..... .... .... .... .... .... ..... .... 454 6 The Yang–Mills Equation. ..... .... .... .... .... .... ..... .... 461 6.1 Gauge Fields. The Configuration Space .... .... .... ..... .... 461 6.2 The Yang–Mills Equation. Self-dual Connections .... ..... .... 471 6.3 The BPST Instanton Family. .... .... .... .... .... ..... .... 477 6.4 The ADHM Construction... .... .... .... .... .... ..... .... 489 6.5 The Instanton Moduli Space. .... .... .... .... .... ..... .... 508 6.6 Instantons and Smooth 4-manifolds ... .... .... .... ..... .... 526 6.7 Stability ... .... .... ..... .... .... .... .... .... ..... .... 530 6.8 Non-minimal Solutions..... .... .... .... .... .... ..... .... 538 7 Matter Fields and Model Building... .... .... .... .... ..... .... 545 7.1 Matter Fields ... .... ..... .... .... .... .... .... ..... .... 545 7.2 Yang–Mills–Higgs Systems . .... .... .... .... .... ..... .... 549 7.3 The Higgs Mechanism ..... .... .... .... .... .... ..... .... 563 7.4 Magnetic Monopoles . ..... .... .... .... .... .... ..... .... 571 7.5 The Bogomolnyi–Prasad–Sommerfield Model ... .... ..... .... 581 7.6 The Seiberg–Witten Model.. .... .... .... .... .... ..... .... 586 7.7 The Standard Model of Elementary Particle Physics... ..... .... 604 7.8 Dimensional Reduction. Basics... .... .... .... .... ..... .... 617 7.9 Dimensional Reduction. Model Building ... .... .... ..... .... 625 8 The Gauge Orbit Space .. ..... .... .... .... .... .... ..... .... 635 8.1 Introduction .... .... ..... .... .... .... .... .... ..... .... 635 8.2 Gauge Orbit Types... ..... .... .... .... .... .... ..... .... 637 8.3 The Gauge Orbit Stratification ... .... .... .... .... ..... .... 643 Contents ix 8.4 Geometry of Strata... ..... .... .... .... .... .... ..... .... 652 8.5 Classification of Howe Subgroups .... .... .... .... ..... .... 664 8.6 Classification of Howe Subbundles.... .... .... .... ..... .... 669 8.7 Enumeration of Gauge Orbit Types ... .... .... .... ..... .... 681 8.8 Partial Ordering . .... ..... .... .... .... .... .... ..... .... 684 9 Elements of Quantum Gauge Theory .... .... .... .... ..... .... 693 9.1 Path Integral Quantization .. .... .... .... .... .... ..... .... 694 9.2 The Gribov Problem.. ..... .... .... .... .... .... ..... .... 700 9.3 Anomalies.. .... .... ..... .... .... .... .... .... ..... .... 706 9.4 Hamiltonian Quantum Gauge Theory on the Lattice... ..... .... 725 9.5 Field Algebra and Observable Algebra. .... .... .... ..... .... 732 9.6 Including the Nongeneric Strata.. .... .... .... .... ..... .... 741 9.7 A Toy Model... .... ..... .... .... .... .... .... ..... .... 750 Appendix A: Field Restriction and Field Extension .... .... ..... .... 759 Appendix B: The Conformal Group of the 4-Sphere... .... ..... .... 765 Appendix C: Simple Lie Algebras. Root Diagrams .... .... ..... .... 771 Appendix D: f-Function Regularization . .... .... .... .... ..... .... 775 Appendix E: K-Theory and Index Bundles... .... .... .... ..... .... 777 Appendix F: Determinant Line Bundles . .... .... .... .... ..... .... 781 Appendix G: Eilenberg–MacLane Spaces.... .... .... .... ..... .... 787 References.... .... .... .... ..... .... .... .... .... .... ..... .... 789 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 815 Introduction This is the second part of our book on Differential Geometry and Mathematical Physics.ItisbasedonourteachingofthesesubjectsattheUniversityofLeipzigto students of physics and of mathematics and on our research in gauge field theory over many years. As in Part I, let us start with some historical remarks. The concept of gauge invariance first appeared in the famous papers [660] and [661] of Hermann Weyl from the year 1918.1 In this work, Weyl extended Einstein’s principle of general relativity by postulating that, additionally, the scale of length can vary smoothly frompointtopointinspacetime.Inmoredetail,Weyl’sbasicideawastodevelopa purely infinitesimal geometry. Behind that concept was his belief that ‘a true infinitesimalgeometry should,however,recognizeonlya principle fortransferring themagnitudeofavectortoaninfinitesimallyclosepoint…’,seepage25in[660]. In this context, the notion of connection appeared for the first time in the mathe- matical literature.2 In a modern geometric language, he was led toa generalization of Riemannian geometry characterized by a pair consisting of a conformal Riemannian structure and a connection in a line bundle over spacetime. Weyl proposed to identify the connection form with the electromagnetic gauge potential and, consequently, its curvature with the electromagnetic field tensor. Thus, he obtained a unification of general relativity with electromagnetism. However, it quickly became clear that this model was not compatible with basic physical principles. It was Einstein who observed that if this theory was correct, then the behaviour of clocks would depend on their history. This is in contradiction with empirical evidence.3 Although this model did not survive, the gauge principle did though. In 1929 Weyl proposed to apply it to quantum mechanics. He recognized 1Inthesepapers,theterm‘gaugeinvariance’appearsinGermanas‘Maßstab-Invarianz’. 2Ofcourse,therewere predecessors,notablyChristoffel,Ricci andLevi-Civita. Thelatter hada clearmathematicalunderstandingofparalleltransportandofthecovariantderivativeoperator,but uptoourknowledge,hedidnotinventtheterm‘connection’. 3See the postscript by Einstein in [660] and the author’s reply. This started a long discussion betweenWeylandEinstein.Forfurtherreference,seealso[604]and[496]. xi