Higher groupoid bundles, higher spaces, and self-dual tensor field equations

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 (∞, 1)‐categories. We then discuss principal bundles in terms of simplicial maps and their homotopies. We explain in detail a differentiation procedure, suggested by Ševera, that maps higher groupoids to L∞‐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. The authors 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. The article starts off with a self‐contained review on simplicial sets as models of (∞,1) ‐categories. After discussing principal bundles in terms of simplicial maps and their homotopies, a differentiation procedure mapping higher groupoids to L∞‐algebroids is explained in detail. Generalising this procedure, connections for higher groupoid bundles are defined. As an application, one obtains 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.