Higher Groupoid Bundles, Higher Spaces, and Self-Dual Tensor Field Equations

TL;DR

This paper develops a framework for higher gauge theory using higher groupoids, enabling the description of complex gauge structures and their applications to six-dimensional superconformal field theories via a Penrose-Ward transform.

Contribution

It introduces a first-principles approach to higher gauge theory with higher groupoids, generalizing principal bundles and connections to higher categorical structures.

Findings

Unified description of gauge theories and sigma models with higher structures

Differentiation procedure linking higher groupoids to $L_ olinebreak$-algebroids

Application to constructing six-dimensional superconformal field theories

Abstract

We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general class of (higher) spaces comprising presentable differentiable stacks, as e.g. orbifolds. We start off with a self-contained review on simplicial sets as models of $(\infty,1)$-categories. We then discuss principal bundles in terms of simplicial maps and their homotopies. We explain in detail a differentiation procedure, suggested by Severa, that maps higher groupoids to $L_\infty$-algebroids. Generalising this procedure, we define connections for higher groupoid bundles. As an application, we obtain six-dimensional superconformal field theories via a Penrose-Ward transform of higher groupoid bundles over a twistor space. This construction reduces the search for non-Abelian self-dual tensor field equations in six dimensions to a search for the appropriate (higher) gauge structure. The treatment aims to be accessible to theoretical physicists.