Combinatorial quantization of 4d 2-Chern-Simons theory II: Quantum invariants of higher ribbons in $D^4$

Abstract

This is a continuation of the first paper (arXiv:2501.06486) of this series, where the framework for the combinatorial quantization of the 4d 2-Chern-Simons theory with an underlying compact structure Lie 2-group $\mathbb{G}$ was laid out. In this paper, we continue our quest and characterize additive module *-functors $\ddot{I}‰:{\mathfrak{C}}_q({\mathbb{G}}^{\hat{I}{“}^2})\to \mathsf{Hilb}$, which serve as a categorification of linear *-functionals (ie. a state) on a $C^{\ast }$-algebra. These allow us to construct non-Abelian Wilson surface correlations ${\hat{\mathfrak{C}}}_q({\mathbb{G}}^P)$ on the discrete 2d simple polyhedra $P$ partitioning 3-manifolds. By proving its stable equivalence under 3d handlebody moves, these Wilson surface states extend to decorated 3-dimensional marked bordisms in a 4-disc $D^4$. This provides invariants of framed oriented 2-ribbonsin $D^4$ from the data of the given compact Lie 2-group $\mathbb{G}$. We find that these 2-Chern-Simons-type 2-ribbon invariants are given by bigraded $\mathbb{Z}$-modules, similar to the lasagna skein modules of Manolescu-Walker-Wedrich.

92 pages; 19 figures (v2: 87 pages, corrected the appearance of equivariantized cohomology theory in section 6.3.3)