Graded covering of a supermanifold I. The case of a Lie supergroup

TL;DR

This paper extends the Donagi-Witten construction to supermanifolds, providing a new way to embed and realize non-split supermanifolds using graded and vector bundle categories, especially for Lie supergroups.

Contribution

It generalizes the Donagi-Witten obstruction theory and introduces embeddings of supermanifolds into graded and vector bundle categories, with applications to Lie supergroups.

Findings

Realization of non-split supermanifolds via vector bundles and morphisms

Embeddings of supermanifolds into graded and vector bundle categories

Universal property of graded coverings for Lie supergroups

Abstract

We generalize the Donagi and Witten construction of a first obstruction class for splitting of a supermanifold via differential operators using the theory of $n$-fold vector bundles and graded manifolds. Applying the generalized Donagi--Witten construction we obtain a family of embeddings of the category of supermanifolds into the category of $n$-fold vector bundles and into the category of graded manifolds. This leads to a realization of any non-split supermanifold in terms of a collection of vector bundles and some morphism between them. Further we study the images of these embeddings into the category of graded manifolds in the case of a Lie supergroup and a Lie superalgebra. We show that these images satisfy universal property of a graded covering or a graded semicovering.