Flat principal 2-group bundles and flat string structures

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.

42 pages. The previous version of this paper was changed significantly and split into two articles: this one, and arxiv:2510.10773