Modern Differential Geometry in Gauge Theories

Differential geometry, in the classical sense, is developed through the theory of smooth manifolds. Modern differential geometry from the authora??s perspective is used in this work to describe physical theories of a geometric character without using any notion of calculus (smoothness). Instead, an axiomatic treatment of differential geometry is presented via sheaf theory (geometry) and sheaf cohomology (analysis). Using vector sheaves, in place of bundles, based on arbitrary topological spaces, this unique approach in general furthers new perspectives and calculations that generate unexpected potential applications. Modern Differential Geometry in Gauge Theories is a two-volume research monograph that systematically applies a sheaf-theoretic approach to such physical theories as gauge theory. Beginning with Volume 1, the focus is on Maxwell fields. All the basic concepts of this mathematical approach are formulated and used thereafter to describe elementary particles, electromagnetism, and geometric prequantization. Maxwell fields are fully examined and classified in the language of sheaf theory and sheaf cohomology. Continuing in Volume 2, this sheaf-theoretic approach is applied to Yanga??Mills fields in general. The text contains a wealth of detailed and rigorous computations and will appeal to mathematicians and physicists, along with advanced undergraduate and graduate students, interested in applications of differential geometry to physical theories such as general relativity, elementary particle physics and quantum gravity.

General Preface	p. ix
Preface to Volume I	p. xi
Acknowledgements	p. xv
Maxwell Fields: General Theory
The Rudiments of Abstract Differential Geometry	p. 3
The Differential Setting	p. 3
Logarithmic Derivation	p. 7
A-Connections	p. 8
The Classical Case	p. 10
Local Definition of an A-Connection	p. 13
Gauge Transformation	p. 18
Induced A-Connections	p. 20
Existence of A-Connections. Criteria of Existence	p. 27
The Space of A-Connections	p. 30
Related A-Connections. Moduli Space of A-Connections	p. 33
Moduli Space	p. 34
Curvature	p. 41
Local Form of the Curvature	p. 44
Transformation Law of Field Strength (Curvature)	p. 46
Fundamental Identities of the Curvature (Continued). Torsion	p. 48
Pullback of Curvature	p. 52
Torsion	p. 53
A-Connections Compatible with A-Metrics	p. 54
Hermitian A-Connections	p. 56
Matrices of A-Metrics	p. 58
Kahler A-Metrics	p. 61
Einstein A-Metrics	p. 62
Lorentz A-Metrics	p. 63
The Hodge *-Operator. Volume Form	p. 64
Elementary Particles: Sheaf-Theoretic Classification, by Spin-Structure, According to Selesnick's Correspondence Principle	p. 69
Preliminaries. Basic Notions	p. 69
Classification of Elementary Particles, Through Vector Sheaves, According to Their Spin-Structures	p. 71
Standard Classification of Elementary Particles by Spin Number	p. 71
Classification of Elementary Particles Through Module-Structures (a la Selesnick)	p. 73
Quantum State Modules	p. 74
Free Bosons and Fermions in Terms of Finitely Generated Projective Modules	p. 79
Finitely Generated Projective Modules and Vector Bundles (Serre-Swan Theory)	p. 81
Vector Sheaves and Elementary Particles (Continued: Selesnick's Correspondence)	p. 84
Smooth (C[superscript infinity]-) Case	p. 89
Cohomological Classification of Elementary Particles	p. 91
Vector Sheaves	p. 92
Line Sheaves	p. 95
Elementary Particles	p. 97
Elementary Particles as Principal Sheaves	p. 98
Principal Sheaves	p. 100
Vector Sheaves Associated with Principal Sheaves and Physical Interpretation	p. 102
Physical Applications	p. 109
Interacting Particles	p. 110
Electromagnetism	p. 113
The Electromagnetic Field. The Maxwell Category	p. 114
Characterization of the Maxwell Group Through Local Data	p. 118
Local Characterization of Maxwell Fields	p. 120
Local Characterization of (Gauge) Equivalent Maxwell Fields	p. 125
A Natural Fibration	p. 128
The Image of (the Natural Fibration) [tau]	p. 130
Weil's Integrality Theorem (Again)	p. 132
The Image of the Map [tau] (Continued)	p. 138
Cohomology Class Associated with the Field Strength of a Maxwell Field (Continued)	p. 142
The Fibration [tau] as a Group Morphism	p. 147
Action of H[superscript 1] (X, C[Characters not reproducible]) on the Maxwell Group [Phi][Characters not reproducible](X)[superscript nabla]	p. 152
Freeness of the Action of H[superscript 1](X, C[Characters not reproducible]) on the Maxwell Group	p. 154
Transitivity of the Action of H[superscript 1](X, C[Characters not reproducible]) on the Maxwell Group	p. 157
[Phi][Characters not reproducible](X)[Characters not reproducible] as a Principal Homogeneous Space	p. 164
The Hermitian Counterpart	p. 168
Action of H[superscript 1] (X, S[superscript 1]) on [Phi][Characters not reproducible](X)[superscript nabla]	p. 171
Hermitian Maxwell Fields	p. 172
Hermitian Light Bundles	p. 175
Hermitian Light Bundles over Path-Connected Spaces	p. 178
Equivariant Actions of H[superscript 1](X, C[Characters not reproducible]) (Continued)	p. 181
The Kernel of the Map [tau]	p. 186
Hermitian Counterpart (Continued)	p. 191
The Maxwell Group [Phi][Characters not reproducible](X)[superscript nabla] as a Central Extension (Continued)	p. 192
The Hermitian Counterpart (Continued)	p. 194
Cohomological Classification of Maxwell and Hermitian Maxwell Fields	p. 197
Hypercohomology with Respect to a (Differential) A-Complex	p. 197
Sheaf Cohomology	p. 197
Hypercohomology	p. 202
Cech Hypercohomology	p. 206
Cech Hypercohomology Relative to a Two-Term A-Complex	p. 210
Identification of H[superscript 1] (X,[epsilon superscript 0] [Characters not reproducible] [epsilon superscript 1])	p. 212
Cech Hypercohomology, with Respect to the Two-Term Z-Complex A[Characters not reproducible] [Characters not reproducible] [Omega superscript 1]	p. 216
Characterization of the (Abelian) Cech Hypercohomology Group H[superscript 1] (X, A[Characters not reproducible] [Characters not reproducible] [Omega superscript 1])	p. 218
Cohomological Wording of the Maxwell Group	p. 221
Abstract Maxwell Equations	p. 227
The Hermitian Analogue	p. 230
Geometric Prequantization	p. 233
Symplectic Sheaves	p. 233
Prequantizable Symplectic Sheaves	p. 237
The Hermitian Framework	p. 242
Cohomological Classification of (Abstract) Geometric Prequantizations of Hermitian Maxwell Fields with a Given Field Strength	p. 246
Prequantization of Elementary Particles	p. 250
Bosonic Case	p. 251
The Chern Isomorphism (Continued), and Consequences	p. 252
Geometric Prequantization of Bosons (Continued)	p. 256
Fermionic Case	p. 257
Pull-Back of Maxwell Fields	p. 259
Geometric Prequantization of Fermions (Continued)	p. 267
References	p. 273
Index of Notation	p. 281
Index	p. 289
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Modern Differential Geometry in Gauge Theories

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