1|1 Parallel Transport and Connections

TL;DR

This paper demonstrates that every parallel transport along superpaths on a vector bundle over a manifold arises from a connection, establishing a fundamental link between parallel transport and connections in supergeometry.

Contribution

It proves that all parallel transports along superpaths originate from a vector bundle with a connection over a manifold, extending classical results to supergeometry.

Findings

Every parallel transport along superpaths is induced by a vector bundle with connection.

The result holds when the base supermanifold is a classical manifold.

Establishes a one-to-one correspondence between parallel transport and connections in this setting.

Abstract

A vector bundle with connection over a supermanifold leads naturally to a notion of parallel transport along superpaths. In this note we show that {\it every} such parallel transport along superpaths comes form a vector bundle with connection, at least when the base supermanifold is a manifold.