Parallel Transport on Principal Bundles over Stacks

TL;DR

This paper extends the concept of parallel transport for principal bundles with connections from manifolds to differentiable stacks, establishing a correspondence that generalizes previous results and simplifies existing definitions.

Contribution

It introduces a notion of parallel transport over stacks and proves that principal bundles with connections can be reconstructed from their transport, generalizing prior manifold-based results.

Findings

Principal bundles over stacks can be recovered from their parallel transport.

Simplified the definition of transport functors for principal bundles.

Extended the correspondence between bundles with connections and transport functors to stacks.

Abstract

In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby extending the results of Barrett, Caetano and Picken, and Schreiber and Waldof from manifolds to stacks. In the process of proving our main result we simplify Schreiber and Waldorf's definition of a transport functor for principal bundles with connections over manifolds and provide a more direct proof of the correspondence between principal bundles with connections and transport functors.