Atiyah sequence and Gauge transformations of a principal $2$-bundle over a Lie groupoid

TL;DR

This paper introduces a new framework for principal 2-bundles over Lie groupoids, establishing an Atiyah sequence, different connection notions, and analyzing gauge transformations and symmetries within this higher geometric context.

Contribution

It develops the theory of principal 2-bundles over Lie groupoids, including the Atiyah sequence, connection structures, and gauge transformation actions, advancing higher gauge theory.

Findings

Constructed a short exact sequence of VB-groupoids (Atiyah sequence) for principal 2-bundles.

Defined strict and semi-strict connections and their relation to the Atiyah sequence.

Analyzed the action of gauge transformations and identified extended symmetries in the connection category.

Abstract

In this paper, a notion of a principal $2$-bundle over a Lie groupoid has been introduced. For such principal $2$-bundles, we produced a short exact sequence of VB-groupoids, namely, the Atiyah sequence. Two notions of connection structures viz. strict connections and semi-strict connections on a principal $2$-bundle arising respectively, from a retraction of the Atiyah sequence and a retraction up to a natural isomorphism have been introduced. We constructed a class of principal $\mathbb{G}=[G_1\rightrightarrows G_0]$-bundles and connections from a given principal $G_0$-bundle $E_0\rightarrow X_0$ over $[X_1\rightrightarrows X_0]$ with connection. An existence criterion for the connections on a principal $2$-bundle over a proper, \'etale Lie groupoid is proposed. The action of the $2$-group of gauge transformations on the category of strict and semi-strict connections has been studied. Finally we noted an extended symmetry of the category of semi-strict connections.