Flat principal 2-group bundles and flat string structures

TL;DR

The paper develops a concrete framework for flat principal 2-group bundles on differentiable stacks, modeled as a localization of functor bicategories to yield Bun_𝒢. It proves that a flat 𝒢-bundle on a stack X is equivalent to a G-bundle with a trivialization of the associated 2-gerbe λ_{P,α}, and it specializes this to flat string structures via trivializations of the flat Chern–Simons 2-gerbe CS_V. The construction provides explicit Čech-type data for 2-group bundles and clarifies how nontrivial associators lead to 2-gerbe geometry, in contrast to strict 2-groups whose data reduces to ordinary gerbes. This framework connects higher-categorical descriptions of string structures to differential-geometric objects, offering a pathway toward a differential model for TMF orientations within the Stolz–Teichner program and supporting explicit computations on differentiable stacks.

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.

Theorems & Definitions (139)

• Definition 1.1: Definition \ref{['defn:stringstruc']}
• Theorem 1.2: Theorem \ref{['thm:flatstringdata']}
• Remark 1.3
• Theorem 1.4: Theorem \ref{['thm:2gerbetriv']}
• Remark 1.5
• Conjecture 1.6
• Definition 2.1
• Proposition 2.2
• Definition 2.3: BL04
• Definition 2.4: BL04