Combinatorial quantization of 4d 2-Chern-Simons theory I: the Hopf category of higher-graph states

Abstract

2-Chern-Simons theory, or more commonly known as 4d BF-BB theory with gauged shift symmetry, is a natural generalization of Chern-Simons theory to 4-dimensional manifolds. It is part of the bestiary of higher-homotopy Maurer-Cartan theories. In this article, we present a framework towards the combinatorial quantization of 2-Chern-Simons theory on the lattice, taking inspiration from the work of Aleskeev-Grosse-Schomerus three decades ago. The central geometric input is a "2-graph" $Γ^2$ embedded in a 3d Cauchy slice $Σ$, which has equipped the structure of a discrete 2-groupoid. Upon such 2-graphs, we model the extended Wilson surface operators in 2-Chern-Simons holonomies as Crane-Yetter's {\it measureable fields}. We show that the 2-Chern-Simons action endows these 2-graph operators -- as well as their quantum 2-gauge symmetries -- the structure of a Hopf category, and that their associated higher $R$-matrix gives it a categorical quasitriangularity structure, which we call the {\it cobraiding}. This is an explicit realization of the categorical ladder proposal of Baez-Dolan, in the context of Lie group 2-gauge theories on the lattice. Moreover, we will also analyze the lattice 2-algebra on the graph $Γ$, and extract the observables from it.