Modern differential geometry in gauge theories /
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| Author / Creator: | Mallios, Anastasios. |
|---|---|
| Imprint: | Boston : Birkhäuser, c2006- |
| Description: | v. : ill. ; 24 cm. |
| Language: | English |
| Series: | Progress in mathematical physics ; v. 41-42 |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/5822008 |
Table of Contents:
- General Preface
- Preface to Volume I
- Acknowledgements
- Part I. Maxwell Fields: General Theory
- 1. The Rudiments of Abstract Differential Geometry
- 1. The Differential Setting
- 1.1. Logarithmic Derivation
- 2. A-Connections
- 2.1. The Classical Case
- 2.2. Local Definition of an A-Connection
- 2.3. Gauge Transformation
- 3. Induced A-Connections
- 4. Existence of A-Connections. Criteria of Existence
- 5. The Space of A-Connections
- 6. Related A-Connections. Moduli Space of A-Connections
- 6.1. Moduli Space
- 7. Curvature
- 7.1. Local Form of the Curvature
- 7.2. Transformation Law of Field Strength (Curvature)
- 8. Fundamental Identities of the Curvature (Continued). Torsion
- 8.1. Pullback of Curvature
- 8.2. Torsion
- 9. A-Connections Compatible with A-Metrics
- 9.1. Hermitian A-Connections
- 9.2. Matrices of A-Metrics
- 9.3. Kahler A-Metrics
- 9.4. Einstein A-Metrics
- 9.5. Lorentz A-Metrics
- 10. The Hodge *-Operator. Volume Form
- 2. Elementary Particles: Sheaf-Theoretic Classification, by Spin-Structure, According to Selesnick's Correspondence Principle
- 1. Preliminaries. Basic Notions
- 2. Classification of Elementary Particles, Through Vector Sheaves, According to Their Spin-Structures
- 2.1. Standard Classification of Elementary Particles by Spin Number
- 2.2. Classification of Elementary Particles Through Module-Structures (a la Selesnick)
- 3. Quantum State Modules
- 4. Free Bosons and Fermions in Terms of Finitely Generated Projective Modules
- 5. Finitely Generated Projective Modules and Vector Bundles (Serre-Swan Theory)
- 6. Vector Sheaves and Elementary Particles (Continued: Selesnick's Correspondence)
- 6.1. Smooth (C[superscript infinity]-) Case
- 7. Cohomological Classification of Elementary Particles
- 7.1. Vector Sheaves
- 7.2. Line Sheaves
- 7.3. Elementary Particles
- 8. Elementary Particles as Principal Sheaves
- 8.1. Principal Sheaves
- 9. Vector Sheaves Associated with Principal Sheaves and Physical Interpretation
- 9.1. Physical Applications
- 9.2. Interacting Particles
- 3. Electromagnetism
- 1. The Electromagnetic Field. The Maxwell Category
- 2. Characterization of the Maxwell Group Through Local Data
- 2.1. Local Characterization of Maxwell Fields
- 2.2. Local Characterization of (Gauge) Equivalent Maxwell Fields
- 3. A Natural Fibration
- 3.1. The Image of (the Natural Fibration) [tau]
- 3.2. Weil's Integrality Theorem (Again)
- 3.3. The Image of the Map [tau] (Continued)
- 3.4. Cohomology Class Associated with the Field Strength of a Maxwell Field (Continued)
- 4. The Fibration [tau] as a Group Morphism
- 5. Action of H[superscript 1] (X, C[Characters not reproducible]) on the Maxwell Group [Phi][Characters not reproducible](X)[superscript nabla]
- 5.1. Freeness of the Action of H[superscript 1](X, C[Characters not reproducible]) on the Maxwell Group
- 5.2. Transitivity of the Action of H[superscript 1](X, C[Characters not reproducible]) on the Maxwell Group
- 5.3. [Phi][Characters not reproducible](X)[Characters not reproducible] as a Principal Homogeneous Space
- 6. The Hermitian Counterpart
- 6.1. Action of H[superscript 1] (X, S[superscript 1]) on [Phi][Characters not reproducible](X)[superscript nabla]
- 6.2. Hermitian Maxwell Fields
- 6.3. Hermitian Light Bundles
- 6.4. Hermitian Light Bundles over Path-Connected Spaces
- 7. Equivariant Actions of H[superscript 1](X, C[Characters not reproducible]) (Continued)
- 7.1. The Kernel of the Map [tau]
- 7.2. Hermitian Counterpart (Continued)
- 8. The Maxwell Group [Phi][Characters not reproducible](X)[superscript nabla] as a Central Extension (Continued)
- 8.1. The Hermitian Counterpart (Continued)
- 4. Cohomological Classification of Maxwell and Hermitian Maxwell Fields
- 1. Hypercohomology with Respect to a (Differential) A-Complex
- 1.1. Sheaf Cohomology
- 1.2. Hypercohomology
- 2. Cech Hypercohomology
- 3. Cech Hypercohomology Relative to a Two-Term A-Complex
- 3.1. Identification of H[superscript 1] (X,[epsilon superscript 0] [Characters not reproducible] [epsilon superscript 1])
- 4. Cech Hypercohomology, with Respect to the Two-Term Z-Complex A[Characters not reproducible] [Characters not reproducible] [Omega superscript 1]
- 4.1. Characterization of the (Abelian) Cech Hypercohomology Group H[superscript 1] (X, A[Characters not reproducible] [Characters not reproducible] [Omega superscript 1])
- 5. Cohomological Wording of the Maxwell Group
- 6. Abstract Maxwell Equations
- 7. The Hermitian Analogue
- 5. Geometric Prequantization
- 1. Symplectic Sheaves
- 2. Prequantizable Symplectic Sheaves
- 3. The Hermitian Framework
- 4. Cohomological Classification of (Abstract) Geometric Prequantizations of Hermitian Maxwell Fields with a Given Field Strength
- 5. Prequantization of Elementary Particles
- 5.1. Bosonic Case
- 5.2. The Chern Isomorphism (Continued), and Consequences
- 5.3. Geometric Prequantization of Bosons (Continued)
- 5.4. Fermionic Case
- 5.5. Pull-Back of Maxwell Fields
- 5.6. Geometric Prequantization of Fermions (Continued)
- References
- Index of Notation
- Index
