Quillen superconnections and connections on supermanifolds

TL;DR

This paper explores the relationship between Quillen superconnections on supervector bundles and graded connections on supermanifolds, providing methods to compute Chern classes for these structures.

Contribution

It establishes a parametrization of Quillen superconnections via graded connections on supermanifolds and relates their curvatures and Chern classes.

Findings

Parametrization of Quillen superconnections by graded connections.

Relation between curvatures of both connections types.

Method to compute Chern classes for graded vector bundles.

Abstract

Given a supervector bundle $E = E_0\oplus E_1 \to M$, we exhibit a parametrization of Quillen superconnections on $E$ by graded connections on the Cartan-Koszul supermanifold $(M;\Omega (M))$. The relation between the curvatures of both kind of connections, and their associated Chern classes, is discussed in detail. In particular, we find that Chern classes for graded vector bundles on split supermanifolds can be computed through the associated Quillen superconnections.