Chern-Simons classes of flat connections on supermanifolds

TL;DR

This paper introduces Chern-Simons classes for superconnections on supervector bundles, extending classical theories to supermanifolds and providing new invariants for morphisms between flat bundles.

Contribution

It defines Chern-Simons classes for superconnections on supervector bundles and extends Reznikov's theorem to these classes on certain complex manifolds.

Findings

Defined Chern-Simons classes for superconnections on supervector bundles.

Extended Reznikov's theorem to supermanifolds and non-flat morphisms.

Proved triviality of these classes in certain geometric contexts.

Abstract

In this note we define Chern-Simons classes of a superconnection $D+L$ on a complex supervector bundle $E$ such that $D$ is flat and preserves the grading, and $L$ is an odd endomorphism of $E$ on a supermanifold. As an application we obtain a definition of Chern-Simons classes of a (not necessarily flat) morphism between flat vector bundles on a smooth manifold. We extend Reznikov's theorem on triviality of these classes when the manifold is a compact K\"ahler manifold or a smooth complex quasi--projective variety, in degrees > 1.