Modern Differential Geometry in Gauge Theories (eBook)

Contents 6
General Preface 9
Preface to Volume II 11
Acknowledgments 14
Contents of Volume I 17
Yang–Mills Theory:General Theory 18
Abstract Yang–Mills Theory 19
1 The Differential Setting 19
1.1 Vectorization of the Abstract de Rham Complex (Prolongations) 21
2 The Dual Differential Setting 23
2.1 Dual Differential Operators 26
3 The Abstract Laplace–Beltrami Operators 32
3.1 Positivity of the Laplacian and the Green’s Formula 35
4 The Abstract Yang–Mills Equations 38
4.1 Yang–Mills Fields 40
4.2 The Yang–Mills Category 42
4.3 Gauge Equivalent Yang–Mills Fields 45
4.4 Yang–Mills Equations 50
4.5 Self-Dual Gauge Fields 53
5 Yang–Mills Functional 57
5.1 Group of Gauge Transformations 60
5.2 Gauge Invariance of the Yang–Mills Functional 63
6 First Variational Formula 65
6.1 Variation of the Field Strength, Caused by a Variation of theGauge Potential 67
6.2 Covariant Differential Operators (Prolongations) for EndE 71
7 Volume Element 74
7.1 A Topological (C-)Algebra (Structure) sheafA 78
8 Yang–Mills Functional (continued): The Variation Formula 81
8.1 Lagrangian Density and Its Variation 84
9 Cohomological Classification of Yang–Mills Fields 86
9.1 Local Characterization of Yang–Mills Fields 89
9.2 The Map (9.1) 92
Moduli Spaces of A-Connections of Yang–Mills Fields 94
1 Preliminaries: The Group of Gauge Transformations or Groupof Internal Symmetries 94
1.1 The Internal Symmetry Group, as the Group of Gauge Transformations 99
2 Moduli Space of A-Connections 102
2.1 The Orbit Space ofA-Connections 107
2.2 The Orbit Space of a Maxwell Field 111
3 Moduli Space of A-Connections of a Yang–Mills Field 112
3.1 Moduli Space of Yang–Mills A-Connections 113
4 Moduli Space of Self-Dual A-Connections 115
5 Quantized Moduli Spaces 117
5.1 Morita Equivalence, as Applied to Second Quantization 121
Geometry of Yang–Mills A-Connections 123
1 Abstract Differential-Geometric Jargon in the Moduli Space of A-Connections 124
2 Tangent Spaces 129
3 Geometrical Meaning of T(ConnA(E), D) 130
4 O1(End E), as a Topological (C-)Vector Space Sheaf 138
4.1 Vector Sheaves, Locally Topological Modules 140
5 Geometric Meaning of T(ConnA(E), D) (continued) 142
6 Tangent Space of the Orbit of an A-Connection, T(OD, D) 144
7 The Moduli Space of A-Connections as an Affine Space. Gribov’s Ambiguity (`a la Singer) 148
General Relativity 154
General Relativity, as a Gauge Theory. Singularities 155
1 Abstract Differential-Geometric Setup 157
1.1 Curvature Operators 159
1.2 Scalar Curvature 164
1.3 Semi-Riemannian A-Modules 171
2 Lorentz A-Metrics 172
2.1 Lorentz A-Modules 174
2.2 Lorentz Yang–Mills Fields 177
3 Einstein Field Equations 182
3.1 The Classical Counterpart 185
3.2 Einstein Algebra Sheaves 186
3.3 Einstein–Riemannian Algebra Sheaves 187
4 Einstein–Hilbert Functional and Its First Variation 189
4.1 First Variational Formula of the Einstein–Hilbert Functional 189
5 Rosinger’s Algebra Sheaf 192
5.1 Basic Definitions 193
5.2 The Differential Triad, Based on Rosinger’s Algebra Sheaf 196
5.3 And-Metrics 200
5.4 And as a Topological Algebra Sheaf. Radon-Like Measures 201
6 Rosinger’s Algebra Sheaf (continued): Multifoam Algebra Sheaves 202
6.2 A Differential Triad Related to Rosinger’s Multifoam Algebra Sheaf 205
7 Singularities 206
8 Eddington–Finkelstein Coordinates 210
9 Singularities (continued) 211
9.1 ”Singularities” of the Metric 212
10 Quantum Gravity 213
11 Final Remark 223
11.1 On Einstein’s Equation (continued) 225
References 228
Index of Notation 238
Index 241