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

TL;DR

This paper develops a framework for quantum invariants of higher ribbons in 4-dimensional space using combinatorial quantization of 2-Chern-Simons theory, extending previous work to construct new topological invariants.

Contribution

It introduces a categorification of Wilson surface correlations and constructs invariants of 2-ribbons in 4D from Lie 2-group data, advancing higher gauge theory.

Findings

Constructed non-Abelian Wilson surface correlations on polyhedra

Proved invariance under 3D handlebody moves

Derived new 2-ribbon invariants in 4D from Lie 2-groups

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 $\omega:\mathfrak{C}_q(\mathbb{G}^{\Gamma^2})\rightarrow\mathsf{Hilb}$, which serve as a categorification of linear *-functionals (ie. a state) on a $C^*$-algebra. These allow us to construct non-Abelian Wilson surface correlations $\widehat{\mathfrak{C}}_q(\mathbb{G}^{P})$ 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.