Superconnections and Parallel Transport

TL;DR

This paper develops a framework for defining parallel transport along superpaths and supermanifolds, extending classical concepts to the supergeometric setting with potential applications in mathematical physics.

Contribution

It introduces a novel construction of parallel transport along superpaths and supermanifolds, generalizing classical notions to supergeometry.

Findings

Constructed parallel transport along superpaths from superconnections.

Extended classical parallel transport to supermanifolds.

Provided a foundation for supergeometric gauge theories.

Abstract

This note addresses the construction of a notion of parallel transport along superpaths arising from the concept of a superconnection on a vector bundle over a manifold $M$. A superpath in $M$ is, loosely speaking, a path in $M$ together with an odd vector field in $M$ along the path. We also develop a notion of parallel transport associated with a connection (a.k.a. covariant derivative) on a vector bundle over a \emph{supermanifold} which is a direct generalization of the classical notion of parallel transport for connections over manifolds.