Differential geometry, in the classical sense, is developed through the theory of smooth manifolds. Modern differential geometry from the author’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. Volume 1 focused on Maxwell fields. Continuing in Volume II, the author extends the application of his sheaf-theoretic approach to Yang–Mills fields in general. The important topics include: cohomological classification of Yang–Mills fields, the geometry of Yang–Mills A-connections and moduli space of a vector sheaf, as well as Einstein's equation in vacuum.
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.
Author(s): Anastasios Mallios
Series: Progress in Mathematical Physics
Edition: 1
Publisher: Birkhäuser Boston
Year: 2010
Language: English
Pages: 244
0817643796......Page 1
Modern Differential Geometry in Gauge Theories: YangMills Fields, Volume II (Progress in Mathematical Physics)......Page 4
Contents......Page 6
General Preface......Page 9
Preface to Volume II......Page 11
Acknowledgments......Page 14
Contents of Volume I......Page 17
Part II
Yang–Mills Theory:
General Theory......Page 18
1 The Differential Setting......Page 19
1.1 Vectorization of the Abstract de Rham Complex (Prolongations)......Page 21
2 The Dual Differential Setting......Page 23
2.1 Dual Differential Operators......Page 26
3 The Abstract Laplace–Beltrami Operators......Page 32
3.1 Positivity of the Laplacian and the Green’s Formula......Page 35
4 The Abstract Yang–Mills Equations......Page 38
4.1 Yang–Mills Fields......Page 40
4.2 The Yang–Mills Category......Page 42
4.3 Gauge Equivalent Yang–Mills Fields......Page 45
4.4 Yang–Mills Equations......Page 50
4.5 Self-Dual Gauge Fields......Page 53
5 Yang–Mills Functional......Page 57
5.1 Group of Gauge Transformations......Page 60
5.2 Gauge Invariance of the Yang–Mills Functional......Page 63
6 First Variational Formula......Page 65
6.1 Variation of the Field Strength, Caused by a Variation of theGauge Potential......Page 67
6.2 Covariant Differential Operators (Prolongations) for EEnnddEE......Page 71
7 Volume Element......Page 74
7.1 A Topological (C-)Algebra (Structure) sheafA......Page 78
8 Yang–Mills Functional (continued): The Variation Formula......Page 81
8.1 Lagrangian Density and Its Variation......Page 84
9 Cohomological Classification of Yang–Mills Fields......Page 86
9.1 Local Characterization of Yang–Mills Fields......Page 89
9.2 The Map (9.1)......Page 92
1 Preliminaries: The Group of Gauge Transformations or Groupof Internal Symmetries......Page 94
1.1 The Internal Symmetry Group, as the Group of Gauge Transformations......Page 99
2 Moduli Space of A-Connections......Page 102
2.1 The Orbit Space ofA-Connections......Page 107
2.2 The Orbit Space of a Maxwell Field......Page 111
3 Moduli Space of A-Connections of a Yang–Mills Field......Page 112
3.1 Moduli Space of Yang–Mills A-Connections......Page 113
4 Moduli Space of Self-Dual A-Connections......Page 115
5 Quantized Moduli Spaces......Page 117
5.1 Morita Equivalence, as Applied to Second Quantization......Page 121
Geometry of Yang–Mills A-Connections......Page 123
1 Abstract Differential-Geometric Jargon in the Moduli Space of A-Connections......Page 124
2 Tangent Spaces......Page 129
3 Geometrical Meaning of T(ConnA(E), D)......Page 130
4 Ω1(End E), as a Topological (C-)Vector Space Sheaf......Page 138
4.1 Vector Sheaves, Locally Topological Modules......Page 140
5 Geometric Meaning of T(ConnA(E), D) (continued)......Page 142
6 Tangent Space of the Orbit of an A-Connection, T(OD, D)......Page 144
7 The Moduli Space of A-Connections as an Affine Space. Gribov’s Ambiguity (`a la Singer)......Page 148
Part III
General Relativity......Page 154
General Relativity, as a Gauge Theory. Singularities......Page 155
1 Abstract Differential-Geometric Setup......Page 157
1.1 Curvature Operators......Page 159
1.2 Scalar Curvature......Page 164
1.3 Semi-Riemannian A-Modules......Page 171
2 Lorentz A-Metrics......Page 172
2.1 Lorentz A-Modules......Page 174
2.2 Lorentz Yang–Mills Fields......Page 177
3 Einstein Field Equations......Page 182
3.1 The Classical Counterpart......Page 185
3.2 Einstein Algebra Sheaves......Page 186
3.3 Einstein–Riemannian Algebra Sheaves......Page 187
4.1 First Variational Formula of the Einstein–Hilbert Functional......Page 189
5 Rosinger’s Algebra Sheaf......Page 192
5.1 Basic Definitions......Page 193
5.2 The Differential Triad, Based on Rosinger’s Algebra Sheaf......Page 196
5.3 And-Metrics......Page 200
5.4 And as a Topological Algebra Sheaf. Radon-Like Measures......Page 201
6 Rosinger’s Algebra Sheaf (continued): Multifoam Algebra Sheaves......Page 202
6.2 A Differential Triad Related to Rosinger’s Multifoam Algebra Sheaf......Page 205
7 Singularities......Page 206
8 Eddington–Finkelstein Coordinates......Page 210
9 Singularities (continued)......Page 211
9.1 ”Singularities” of the Metric......Page 212
10 Quantum Gravity......Page 213
11 Final Remark......Page 223
11.1 On Einstein’s Equation (continued)......Page 225
References......Page 228
Index of Notation......Page 238
Index......Page 241
