Global properties of supermanifolds and their bodies

Abstract

AbstractIn supersymmetric field theories, the use of Berezin-like integrals for curved superspace presupposes the existence of a smooth manifold structure on the underlying curved space (the body). The aim of this paper is to address the following question: when does a Rogers supermanifold have a body manifold?Elementary techniques from the theory of foliations are used to derive some necessary and sufficient conditions. The main result is the necessity of regularity of the soul foliation in the sense of Catenacci, Reina and Teofilatto. Regularity is not sufficient (for example, punctured superspace is regular with non-Hausdorff body). A sufficient condition (albeit highly sensitive to the choice of charts) is the transitivity of the body relation ˜, a property enjoyed by De Witt supermanifolds.We show that the body of a quotient supermanifold S/Γ is the quotient of the body (under the induced action). Several examples in the text illustrate that this is extremely useful.New characterizations ρ-supermanifolds and their saturations are given and the saturation construction is extended to all bodied supermanifolds.