2025/01/11 by Chen, Hank · 1 citation
#16T05 #18N25 (Secondary) #81T25 (Primary) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2501.06486
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 the 2-truncation Γ2 of the ∞-groupoid of simplices formed by the underlying lattice Γ. On such a "2-graph", we model states of 2-Chern-Simons holonomies as Crane-Yetter's measureable fields. We show that the 2-Chern-Simons action endows the 2-graph states -- as well as their quantum 2-gauge symmetries -- the structure of a Hopf category, and that their associated higher R-matriex gives it a comonoidal \it cobraiding structure. This is an explicit realization of the categorical ladder proposal of Baez-Dolan, in the context of Lie group lattice 2-gauge theories. Moreover, we will also analyze the lattice 2-algebra on the graph Γ, and extract the observables of discrete 2-Chern-Simons theory from it.