A representation formula for maps on supermanifolds

TL;DR

This paper develops a representation formula for morphisms between function algebras on manifolds and supermanifolds, enabling better understanding and computation of supermanifold maps.

Contribution

It establishes a general representation formula for morphisms from ordinary manifolds to supermanifolds, facilitating analysis and computation of such maps.

Findings

Derived a representation formula for supermanifold morphisms.

Showed how to integrate morphism data to obtain maps on ordinary spaces.

Provided a simple method to compute pull-backs of functions on manifolds.

Abstract

In this paper we analyze the notion of morphisms of rings of superfunctions which is the basic concept underlying the definition of supermanifolds as ringed spaces (i.e. following Berezin, Leites, Manin, etc.). We establish a representation formula for all morphisms from the algebra of functions on an ordinary manifolds to the superalgebra of functions on an open subset of R^{p|q}. We then derive two consequences of this result. The first one is that we can integrate the data associated with a morphism in order to get a (non unique) map defined on an ordinary space (and uniqueness can achieved by restriction to a scheme). The second one is a simple and intuitive recipe to compute pull-back images of a function on a manifold by a map defined on a superspace.