Mathematical Aspects of Superspace

ISBN-13: 9789027718051 / Angielski / Twarda / 1984 / 214 str.

H. J. Seifert; C. J. S. Clarke; A. Rosenblum

Opis

Over the past five years, through a continually increasing wave of activity in the physics community, supergravity has come to be regarded as one of the most promising ways of unifying gravity with other particle interaction as a finite gauge theory to explain the spectrum of elementary particles. Concurrently im portant mathematical works on the arena of supergravity has taken place, starting with Kostant's theory of graded manifolds and continuing with Batchelor's work linking this with the superspace formalism. There remains, however, a gap between the mathematical and physical approaches expressed by such unanswered questions as, does there exist a superspace having all the properties that physicists require of it? Does it make sense to perform path integral in such a space? It is hoped that these proceedings will begin a dialogue between mathematicians and physicists on such questions as the plan of renormalisation in supergravity. The contributors to the proceedings consist both of mathe maticians and relativists who bring their experience in differen tial geometry, classical gravitation and algebra and also quantum field theorists specialized in supersymmetry and supergravity. One of the most important problems associated with super symmetry is its relationship to the elementary particle spectrum."

Spis treści

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 Schrödinger 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.