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Название: Gauge Theories of Gravity and Supergravity
Автор: Schaer C.H.
Differential geometry and its applications to general relativity and linear gauge theories are first reviewed. A fiber bundle formalism for gravity is developed using spacetime as the base space and the de-Sitter group as the fiber. The de-Sitter group is broken to the Lorentz group, and the translation generators are interlocked with the covariant derivatives. This leads to nonlinear gauge transformations which are equivalent to general coordinate transformations on spacetime. It is found that Einstein's Lagrangian is invariant under these transformations up to a total divergence. Conformal gravity and gravity directly coupled to a Yang-Mills theory are then examined. This formalism is extended to supergravity by extending spacetime to a dimensional superspace and using the orthosymplectic groups OSp(4;N) for the fiber. The geometry of superspace is first discussed. Nonlinear gauge transformations are then derived making use of an interlocking condition, and Lagrangians invariant under these transformation laws up to a total divergence are constructed. Transformation laws for conformal supergravity are also derived. Finally, an alternative to superspace in which the line element of general relativity is maintained is developed. It is argued that the metric on such a space contains just enough fields to possibly be able to include the gravitational, weak, and electromagnetic interactions.