Modern differential geometry in gauge theories. Maxwell fields

Contents......Page 4
General Preface......Page 8
Preface to Volume I......Page 10
Acknowledgements......Page 13
Part I Maxwell Fields: General Theory......Page 17
1 The Differential Setting......Page 18
2 A-Connections......Page 23
3 Induced A-Connections......Page 35
4 Existence of A-Connections. Criteria of Existence......Page 42
5 The Space of A-Connections......Page 45
6 Related A-Connections. Moduli Space of A-Connections......Page 48
7 Curvature......Page 56
8 Fundamental Identities of the Curvature (Continued). Torsion......Page 63
9 A-Connections Compatible with A-Metrics......Page 69
10 The Hodge *-Operator. Volume Form......Page 79
1 Preliminaries. Basic Notions......Page 83
2 Classification of Elementary Particles, Through Vector Sheaves, According to Their Spin-Structures......Page 85
3 Quantum State Modules......Page 88
4 Free Bosons and Fermions in Terms of Finitely Generated Projective Modules......Page 93
5 Finitely Generated Projective Modules and Vector Bundles (Serre–Swan Theory)......Page 95
6 Vector Sheaves and Elementary Particles (Continued: Selesnick\'s Correspondence)......Page 98
7 Cohomological Classification of Elementary Particles......Page 105
8 Elementary Particles as Principal Sheaves......Page 112
9 Vector Sheaves Associated with Principal Sheaves and Physical Interpretation......Page 116
3 Electromagnetism......Page 126
1 The Electromagnetic Field. The Maxwell Category......Page 127
2 Characterization of the Maxwell Group Through Local Data......Page 131
3 A Natural Fibration......Page 141
4 The Fibration τ as a Group Morphism......Page 160
5 Action of H[sup(1)](X, C[sup(·)]) on the Maxwell Group Φ[sup(1)][sub(A)](X)[sup(∇)]......Page 165
6 The Hermitian Counterpart......Page 181
7 Equivariant Actions of H[sup(1)](X, C[sup(·)]) (Continued)......Page 194
8 The Maxwell Group Φ[sup(1)][sub(A)](X)[sup(∇)] as a Central Extension (Continued)......Page 205
1 Hypercohomology with Respect to a (Differential) A-Complex......Page 209
2 Čech Hypercohomology......Page 218
3 Čech Hypercohomology Relative to a Two-Term A-Complex......Page 222
4 Čech Hypercohomology, with Respect to the Two-Term Z-Complex [equation omitted]......Page 228
5 Cohomological Wording of the Maxwell Group......Page 233
6 Abstract Maxwell Equations......Page 239
7 The Hermitian Analogue......Page 242
1 Symplectic Sheaves......Page 245
2 Prequantizable Symplectic Sheaves......Page 249
3 The Hermitian Framework......Page 254
4 Cohomological Classification of (Abstract) Geometric Prequantizations of Hermitian Maxwell Fields with a Given Field Strength......Page 258
5 Prequantization of Elementary Particles......Page 262
References......Page 285
Index of Notation......Page 292
C......Page 299
G......Page 300
P......Page 301
S......Page 302
W......Page 303