Higher Geometric Structures on Manifolds and the Gauge Theory of Deligne Cohomology

TL;DR

This paper develops a comprehensive framework for higher geometric structures and gauge theories on manifolds, including symmetry groups, moduli stacks, and higher connections, with applications to supergravity and string theory.

Contribution

It introduces a universal classification of higher symmetries, constructs moduli stacks of higher geometric data, and provides new models for higher gauge theories and supergravity solutions.

Findings

Constructed smooth higher symmetry groups for geometric structures.

Developed moduli stacks as $irc$-categorical quotients and analyzed their homotopy types.

Presented a new String group model and corrected previous moduli of higher gauge solutions.

Abstract

We study smooth higher symmetry groups and moduli $\infty$-stacks of generic higher geometric structures on manifolds. Symmetries are automorphisms which cover non-trivial diffeomorphisms of the base manifold. We construct the smooth higher symmetry group of any geometric structure on $M$ and show that this completely classifies, via a universal property, equivariant structures on the higher geometry. We construct moduli stacks of higher geometric data as $\infty$-categorical quotients by the action of the higher symmetries, extract information about the homotopy types of these moduli $\infty$-stacks, and prove a helpful sufficient criterion for when two such higher moduli stacks are equivalent. In the second part of the paper we study higher $\mathrm{U}(1)$-connections. First, we observe that higher connections come organised into higher groupoids, which further carry affine actions by Baez-Crans-type higher vector spaces. We compute a presentation of the higher gauge actions for $n$-gerbes with $k$-connection, comment on the relation to higher-form symmetries, and present a new String group model. We construct smooth moduli $\infty$-stacks of higher Maxwell and Einstein-Maxwell solutions, correcting previous such considerations in the literature, and compute the homotopy groups of several moduli $\infty$-stacks of higher $\mathrm{U}(1)$- connections. Finally, we show that a discrepancy between two approaches to the differential geometry of NSNS supergravity (via generalised and higher geometry, respectively) vanishes at the level of moduli $\infty$-stacks of NSNS supergravity solutions.