Springer Heidelberg, 2009

1 Geometry

1.1 Riemannian and Lorentzian Manifolds

1.1.1 Differential Geometry
1.1.2 Complex Manifolds
1.1.3 Riemannian and Lorentzian Metrics
1.1.4 Geodesics
1.1.5 Curvature
1.1.6 Principles of General Relativity

1.2 Bundles and Connections

1.2.1 Vector and Principal Bundles
1.2.2 Covariant Derivatives
1.2.3 Reduction of the Structure Group. The Yang–Mills Functional
1.2.4 The Kaluza–Klein Construction

1.3 Tensors and Spinors

1.3.1 Tensors
1.3.2 Clifford Algebras and Spinors
1.3.3 The Dirac Operator
1.3.4 The Lorentz Case
1.3.5 Left- and Right-handed Spinors

1.4 Riemann Surfaces and Moduli Spaces

1.4.1 The General Idea of Moduli Spaces
1.4.2 Riemann Surfaces and Their Moduli Spaces
1.4.3 Compactifications of Moduli Spaces

1.5 Supermanifolds

1.5.1 The Functorial Approach
1.5.2 Supermanifolds
1.5.3 Super Riemann Surfaces
1.5.4 Super Minkowski Space

2 Physics

2.1 Classical and Quantum Physics

2.1.1 Introduction
2.1.2 Gaussian Integrals and Formal Computations
2.1.3 Operators and Functional Integrals
2.1.4 Quasiclassical Limits

2.2 Lagrangians

2.2.1 Lagrangian Densities for Scalars, Spinors and Vectors
2.2.2 Scaling
2.2.3 Elementary Particle Physics and the Standard Model
2.2.4 The Higgs Mechanism
2.2.5 Supersymmetric Point Particles

2.3 Variational Aspects

2.3.1 The Euler–Lagrange Equations
2.3.2 Symmetries and Invariances: Noether’s Theorem

2.4 The Sigma Model

2.4.1 The Linear Sigma Model
2.4.2 The Nonlinear Sigma Model
2.4.3 The Supersymmetric Sigma Model
2.4.4 Boundary Conditions
2.4.5 Supersymmetry Breaking
2.4.6 The Supersymmetric Nonlinear Sigma Model and Morse Theory
2.4.7 The Gravitino

2.5 Functional Integrals

2.5.1 Normal Ordering and Operator Product Expansions
2.5.2 Noether’s Theorem and Ward Identities
2.5.3 Two-dimensional Field Theory

2.6 Conformal Field Theory

2.6.1 Axioms and the Energy–Momentum Tensor
2.6.2 Operator Product Expansions and the Virasoro Algebra
2.6.3 Superfields

2.7 String Theory