Mathematical Aspects of Superspace

Non-linear Realization of Supersymmetry.- 1. Introduction.- 2. The Akulov-Volkov field.- 3. Superfields.- 4. Standard fields.- 5. N > 1/N = 1.- 6. N = 1 supergravity.- References.- Fields, Fibre Bundles and Gauge Groups.- 1. Manifolds.- 2. Fibre bundles.- 2.1 Fields.- 2.2 Coordinate bundles.- 2.3 Fibre bundles.- 2.4 Examples.- 2.5 Fields and geometry.- 2.6 Principal bundles.- 2.7 Cross-sections.- 2.8 Bundles with structure: sheaves.- 2.9 Associated bundles.- 2.10 Connections.- 2.11 Examples.- 3. Gauge Groups.- 3.1 Proposition: Gauge transformations.- 3.2 Gauge action on associate bundles.- 3.3 Quasi-gauge groups.- 3.4 Gauge algebras.- 3.5 Gauge-invariance.- 3.6 Gauge theory.- 4. Space-Time.- 4.1 Spinors.- 4.2 Soldering forms.- 4.3 Achtbeine.- 4.4 Example: Lie derivatives.- 4.5 Supersymmetries.- Path Integration on Manifolds.- 1. Introduction.- 2. Gaussian measures, cylinder set measures, and the Feynman-Kac formula.- 2.1 Basic difficulties; terminology.- 2.2 Gaussian measures.- 2.3 Cylinder set measures.- 2.4 Radonification.- 2.5 Feynman-Kac formula.- 2.6 Time slicing.- 3. Feynman path integrals.- 3.1 Oscillatory integrals and Fresnal integrals.- 3.2 Feynman maps.- 3.3 Feynman path integrals and the Schrodinger equation.- 4. Path integration on Riemannian manifolds.- 4.1 Wiener measure and rolling without slipping.- 4.2 The Pauli-Van-Vleck-De Witt propagator.- 5. Gauge invariant equations; diffusion and differential forms.- 5.1 Quantum particle in a classical magnetic field.- 5.2 Heat equation for differential forms.- Acknowledgements, References.- Graded Manifolds and Supermanifolds.- Preface and cautionary note.- 0. Standard notation.- 1. The category GM.- 1.1 Definitions and examples of graded manifolds.- 1.2 Bundles in GM.- 2. The geometric approach.- 2.1 The general idea.- 2.2 The graded commutative algebra B and supereuclidan space.- 2.3 Smooth maps on Er,s.- 2.4 Examples of supermanifolds.- 2.5 Bundles over supermanifolds.- 3. Comparisons.- 3.1 Comparing GM and SSM.- 3.2 Comparison of geometric manifolds.- 3.3 A direct method of comparing GM and G?.- 4. Lie supergroups.- 4.1 Lie supergroups in the geometric categories.- 4.2 Graded Lie groups.- Table: "All I know about supermanifolds".- References.- Aspects of the Geometrical Approach to Supermanifolds.- 1. Abstract.- 2. Building superspace over an arbitrary spacetime.- 3. Super Lie groups.- 4. Compact supermanifolds with non-Abelian fundamental group.- 5. Developments and applications.- References.- Integration on Supermanifolds.- 1. Introduction.- 2. Standard integration theory.- 3. Integration over odd variables.- 4. Superforms.- 5. Integration on Er,s.- 6. Integration on supermanifolds.- References.- Remarks on Batchelor's Theorem.- Classical Supergravity.- 1. Definition of classical supergravity.- 2. Dynamical analysis of classical field theories.- 3. Formal dynamical analysis of classical supergravity.- 4. The exterior algebra formulation of classical supergravity.- 5. Does classical supergravity make sense?.- Appendix: Notations and conventions.- References.- List of participants.