Table Of ContentTheoretical 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
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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
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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
