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 (8, 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-infinity-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.