Geometry, Topology, and Physics by Mikio Nakahara in 2003
1. Quantum Physics
1.1 Analytical mechanics
1.2 Canonical quantization
1.3 Path integral quantization of a Bose particle
1.4 Harmonic oscillator
1.5 Path integral quantization of a Fermi particle
1.6 Quantization of a scalar field
1.7 Quantization of a Dirac field
1.8 Gauge theories
1.9 Magnetic monopoles
1.10 Instantons
2. Mathematical Preliminaries
3. Homology groups
3.1 Abelian groups
3.2 Simplexes and simplicial complexes
3.3 Homology groups of simplicial complexes
3.4 General properties of homology groups
4. Homotopy groups
4.1 Fundamental groups
4.2 General properties of fundamental groups
4.3 Examples of fundamental groups
4.4 Fundamental groups of polyhedra
4.5 Higher homotopy groups
4.6 General properties of higher homotopy groups
4.7 Examples of higher homotopy groups
4.8 Orders in condensed matter systems
4.9 Defects in nematic liquid crystals
4.10 Textures in superfluid 3He-A
5. Manifolds
5.1 Maniflds
5.2 The calculus of manifolds
5.3 Flows and Lie derivatives
5.6 Lie groups and Lie algebras
5.7 The action of Lie groups on manifolds
6. de Rham Cohomology Groups
6.1 Stokes’ theorem
6.2 de Rham cohomology groups
6.3 Poincare lemma
6.4 Structure of de Rham cohomology groups
7. Riemannian Geometry
7.1 Riemannian manifolds and pseudo-Riemannian manifolds
7.2 Parallel transport, connection and covariant derivative
7.3 Curvature and torsion
7.4 Levi-Civita connections
7.5 Holonomy
7.8 Non-coordinate bases
7.10 Aspects of general relativity
7.11 Bosonic string theory
8. Complex Manifolds
8.1 Complex manifolds
8.2 Calculus on complex manifolds
8.4 Hermitian manifolds and Hermitian differential geometry
8.5 Kahler manifolds and Kahler differential geometry
8.7 Almost complex manifolds
8.8 Orbifolds
9. Fibre bundles
9.1 Tangent bundles
9.2 Fibre bundles
9.3 Vector bundles
10. Connections on Fibre bundles
10.2 Holonomy
10.3 Curvature
10.4 The covariant derivative on associated vector bundles
10.5 Gauge theories
10.6 Berry’s phase
11. Characteristic classes
11.1 Invariant polynomials and the Chern-Weil homomorphism
11.2 Chern classes
11.3 Chern characters
11.4 Pontrjagin and Euler classes
11.6 Stiefel-Whitney classes
12. Index theorems
12.1 Elliptic operators and Fredholm operators
12.2 The Atiyah-Singer index theorem
12.3 The de Rham complex
12.4 The Dolbeault complex
12.5 The signature complex
12.6 Spin complexes
12.7 The heat kernel and generalized \[\zeta\]-functions
12.8 The Atiyah-Patodi-Singer index theorem
12.9 Supersymmetric quantum mechanics
12.10 Supersymmetric proof of index theorem
13. Anomalies in Gauge Field Theories
13.1 Introduction
13.2 Abelian anomalies
13.3 Non-Abelian anomalies
13.4 The Wess-Zumino consistency conditions
13.5 Abelian anomalies versus non-Abelian anomalies
13.6 The parity anomaly in odd-dimensional spaces
14. Bosonic String Theory
14.1 Differential geometry on Riemann surfaces
14.2 Quantum theory of bosonic strings
14.3 One-loop amplitudes