Global properties of supermanifolds

Abstract

We construct new examples of supermanifolds, and determine the vector bundle structure of the supermanifolds commonly used in physics. We show that any supermanifold admits a foliation whose leaves are locally tangent to the soul directions in the coordinate charts, and which is one of a nested sequence of foliations. We point out that the existence of these foliations implies restrictions on the possible topologies of supermanifolds. For example, a compact supermanifold with a single even dimension must have vanishing Euler characteristic. We also show that a globally defined superfield on a “nice” compact supermanifold must be constant along the leaves of the foliations. By this mechanism, the global topology of a supermanifold can be used to impose physically interesting constraints on superfields. As an example, we exhibit a supermanifold which has the local geometry of flat superspace but is such that all globally defined superfields are chiral.