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Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields 1st ed. 2017 [Kõva köide]

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  • Formaat: Hardback, 830 pages, kõrgus x laius: 235x155 mm, 2 Illustrations, color; 13 Illustrations, black and white; XVI, 830 p. 15 illus., 2 illus. in color., 1 Hardback
  • Sari: Theoretical and Mathematical Physics
  • Ilmumisaeg: 29-Mar-2017
  • Kirjastus: Springer
  • ISBN-10: 9402409580
  • ISBN-13: 9789402409581
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Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields 1st ed. 2017Suurem pilt
  • Formaat: Hardback, 830 pages, kõrgus x laius: 235x155 mm, 2 Illustrations, color; 13 Illustrations, black and white; XVI, 830 p. 15 illus., 2 illus. in color., 1 Hardback
  • Sari: Theoretical and Mathematical Physics
  • Ilmumisaeg: 29-Mar-2017
  • Kirjastus: Springer
  • ISBN-10: 9402409580
  • ISBN-13: 9789402409581
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9789402409581.html
Märksõnad:
The book is devoted to the study of the geometrical and topological structure of gauge theories. It consists of the following three building blocks:
- Geometry and topology of fibre bundles,
- Clifford algebras, spin structures and Dirac operators,
- Gauge theory.
Written in the style of a mathematical textbook, it combines a comprehensive presentation of the mathematical foundations with a discussion of a variety of advanced topics in gauge theory.
The first building block includes a number of specific topics, like invariant connections, universal connections, H-structures and the Postnikov approximation of classifying spaces.
Given the great importance of Dirac operators in gauge theory, a complete proof of the Atiyah-Singer Index Theorem is presented. The gauge theory part contains the study of Yang-Mills equations (including the theory of instantons and the classical stability analysis), the discussion of various models with matter fields (including magnetic monopoles, the Seiberg-Witten model and dimensional reduction) and the investigation of the structure of the gauge orbit space. The final chapter is devoted to elements of quantum gauge theory including the discussion of the Gribov problem, anomalies and the implementation of the non-generic gauge orbit strata in the framework of Hamiltonian lattice gauge theory.
The book is addressed both to physicists and mathematicians. It is intended to be accessible to students starting from a graduate level.

Arvustused

The monograph is based on a very vast literature containing nearly 700 items, from very classical articles and books to very recent ones. The seven appendices are helpful in reading it. There are several examples and exercises scattered throughout the book. the book is well organized and clear. The book offers an illuminating and satisfying look at many aspects of modern differential geometry and modern physics and reveals fascinating interactions between pure mathematics and theoretical physics. (Tomasz Rybicki, Mathematical Reviews, January, 2018)

This volume starts with a crash course in differential geometry in the Cartan dialect, emphasizing differential forms, followed by a monograph in itself concerning the algebraic topology ... of fibre bundles. Summing Up: Recommended. Upper-division undergraduates and above; researchers and faculty. (D. V. Feldman, Choice, Vol. 55 (2), October, 2017)

Part II of Differential Geometry and Mathematical Physics isa very important pedagogical contribution and a worthy complement to Part I. It presents fine scholarship at a high level, presented clearly and thoroughly, and teaches the reader a great deal of hugely important differential geometry as it informs physics (and that covers a titanic proportion of both fields). Additionally, Gerd Rudolph and Matthias Schmidt do a fabulous job presenting physics is a manner that mathematicians will not find unheimlich. Ausgezeichnet. (Michael Berg, MAA Reviews, maa.org, May, 2017)

This book is the second part of a two-volume series on differential geometry and mathematical physics. The book is addressed to scholars and researchers in differential geometry and mathematical physics, as well as to advanced graduate students who have studied the material covered in the first part of the series. The reader will benefit from remarks and examples in the text, and from the substantial number of exercises at the end of each section. (Pascal Philip, zbMATH 1364.53001, 2017)

1 Fibre Bundles and Connections
1(92)
1.1 Principal Bundles
1(13)
1.2 Associated Bundles
14(10)
1.3 Connections
24(13)
1.4 Covariant Exterior Derivative and Curvature
37(8)
1.5 The Koszul Calculus
45(7)
1.6 Bundle Reduction
52(7)
1.7 Parallel Transport and Holonomy
59(13)
1.8 Automorphisms
72(4)
1.9 Invariant Connections
76(17)
2 Linear Connections and Riemannian Geometry
93(96)
2.1 Linear Connections
94(14)
2.2 H-Structures and Compatible Connections
108(19)
2.3 Curvature and Holonomy
127(11)
2.4 Sectional Curvature
138(3)
2.5 Symmetric Spaces
141(17)
2.6 Compatible Connections on Vector Bundles
158(6)
2.7 Hodge Theory. The Weitzenboeck Formula
164(17)
2.8 Four-Dimensional Riemannian Geometry. Self-duality
181(8)
3 Homotopy Theory of Principal Fibre Bundles. Classification
189(68)
3.1 Basics
190(10)
3.2 Fibrations
200(12)
3.3 The Covering Homotopy Theorem
212(5)
3.4 Universal Principal Bundles
217(13)
3.5 The Milnor Construction
230(5)
3.6 Classification of Smooth Principal Bundles
235(5)
3.7 Classifying Mappings Associated with Lie Group Homomorphisms
240(4)
3.8 Universal Connections
244(13)
4 Cohomology Theory of Fibre Bundles. Characteristic Classes
257(96)
4.1 Basics
258(9)
4.2 Characteristic Classes for the Classical Groups
267(17)
4.3 Whitney Sum Formula and Splitting Principle
284(13)
4.4 Field Restriction and Field Extension
297(11)
4.5 Characteristic Classes for Manifolds
308(3)
4.6 The Weil Homomorphism
311(25)
4.7 Genera
336(9)
4.8 The Postnikov Tower and Bundle Classification
345(8)
5 Clifford Algebras, Spin Structures and Dirac Operators
353(108)
5.1 Clifford Algebras
354(11)
5.2 Spinor Groups
365(12)
5.3 Representations
377(16)
5.4 Spin Structures and Spinc-Structures
393(7)
5.5 Clifford Modules and Dirac Operators
400(10)
5.6 Weitzenboeck Formulae
410(6)
5.7 Elliptic Complexes. The Hodge Theorem
416(17)
5.8 The Atiyah-Singer Index Theorem
433(21)
5.9 Applications
454(7)
6 The Yang-Mills Equation
461(84)
6.1 Gauge Fields. The Configuration Space
461(10)
6.2 The Yang-Mills Equation. Self-dual Connections
471(6)
6.3 The BPST Instanton Family
477(12)
6.4 The ADHM Construction
489(19)
6.5 The Instanton Moduli Space
508(18)
6.6 Instantons and Smooth 4-manifolds
526(4)
6.7 Stability
530(8)
6.8 Non-minimal Solutions
538(7)
7 Matter Fields and Model Building
545(90)
7.1 Matter Fields
545(4)
7.2 Yang-Mills-Higgs Systems
549(14)
7.3 The Higgs Mechanism
563(8)
7.4 Magnetic Monopoles
571(10)
7.5 The Bogomolnyi-Prasad-Sommerfield Model
581(5)
7.6 The Seiberg-Witten Model
586(18)
7.7 The Standard Model of Elementary Particle Physics
604(13)
7.8 Dimensional Reduction. Basics
617(8)
7.9 Dimensional Reduction. Model Building
625(10)
8 The Gauge Orbit Space
635(58)
8.1 Introduction
635(2)
8.2 Gauge Orbit Types
637(6)
8.3 The Gauge Orbit Stratification
643(9)
8.4 Geometry of Strata
652(12)
8.5 Classification of Howe Subgroups
664(5)
8.6 Classification of Howe Subbundles
669(12)
8.7 Enumeration of Gauge Orbit Types
681(3)
8.8 Partial Ordering
684(9)
9 Elements of Quantum Gauge Theory
693(66)
9.1 Path Integral Quantization
694(6)
9.2 The Gribov Problem
700(6)
9.3 Anomalies
706(19)
9.4 Hamiltonian Quantum Gauge Theory on the Lattice
725(7)
9.5 Field Algebra and Observable Algebra
732(9)
9.6 Including the Nongeneric Strata
741(9)
9.7 A Toy Model
750(9)
Appendix A Field Restriction and Field Extension 759(6)
Appendix B The Conformal Group of the 4-Sphere 765(6)
Appendix C Simple Lie Algebras. Root Diagrams 771(4)
Appendix D ζ-Function Regularization 775(2)
Appendix E K-Theory and Index Bundles 777(4)
Appendix F Determinant Line Bundles 781(6)
Appendix G Eilenberg-MacLane Spaces 787(2)
References 789(26)
Index 815