Adjusted Connections I: Differential Cocycles for Principal Groupoid Bundles with Connection

TL;DR

This paper introduces a novel approach to principal bundles with connection using differential graded Lie groupoids, extending to principal groupoid bundles with an adjustment akin to a Cartan connection, and relates to curved Yang-Mills-Higgs theories.

Contribution

It develops a new perspective on principal bundles with connection via dg-Lie groupoids and extends it to principal groupoid bundles with a Cartan connection adjustment.

Findings

Provides a global formulation of curved Yang-Mills-Higgs theories

Identifies the adjustment data as a Cartan connection for Lie groupoids

Extends the theory to principal groupoid bundles

Abstract

We develop a new perspective on principal bundles with connection as morphisms from the tangent bundle of the underlying manifold to a classifying dg-Lie groupoid. This groupoid can be identified with a lift of the inner homomorphisms groupoid arising in \v{S}evera's differentiation procedure of Lie quasi-groupoids. Our new perspective readily extends to principal groupoid bundles, but requires an adjustment, an additional datum familiar from higher gauge theory. We show that for Lie groupoids, the additional adjustment data amounts to a Cartan connection. The resulting adjusted connections naturally provide a global formulation of the kinematical data of curved Yang-Mills-Higgs theories as described by Kotov-Strobl (arXiv:1510.07654) and Fischer (arXiv:2104.02175).