Flat principal 2-group bundles and flat string structures

TL;DR

This paper develops a geometric framework for flat 2-group bundles over differentiable stacks, linking them to ordinary bundles and 2-gerbes, and applies this to characterize flat string structures.

Contribution

It introduces a bicategory of flat 2-group bundles over stacks and provides explicit geometric descriptions, including a novel characterization of flat string structures.

Findings

Flat 2-group bundles are described via ordinary G-bundles and 2-gerbe trivializations.

A bicategory of flat 2-group bundles over differentiable stacks is constructed.

Characterization of flat string structures on vector bundles over stacks is achieved.

Abstract

For a weak 2-group, we construct a bicategory of flat 2-group bundles over differentiable stacks as a localization of a functor bicategory. This description is amenable to explicit geometric constructions. For example, we show that flat 2-group bundles can be described in terms of ordinary $G$-bundles together with a trivialization of a certain 2-gerbe. This specializes to a characterization of flat string structures on vector bundles over differentiable stacks.