Geometry of superspaces with Bose and Fermi coordinates and applications to graded Lie bundles and supergravity

Abstract

The geometry of superspaces with Bose- and Fermi-type coordinates is presented from a coordinate independent point of view. Various geometrical quantities of conventional manifolds are generalized so as to be applicable to superspaces. It is shown that these generalizations can be basically arrived at algebraically by replacing, in the definitions of various geometrical quantities, the Lie derivative of the conventional manifolds with a generalized graded Lie bracket. Explicit expressions for connection coefficients, Riemann curvature tensor, etc., are derived. The general formalism is then applied to graded Lie bundles the relevance of which to supergravity theories is demonstrated.