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

TL;DR

The work advances the combinatorial quantization of 4d 2-Chern-Simons theory by constructing a categorified state space built from Crane–Yetter measureable categories and by defining non-Abelian Wilson surface invariants for 2-ribbons in the 4-disk D^4. Central to the method are additive measureable *-functors ω: 𝔠_q(𝔊^{Γ^2}) → Hilb, a Yoneda-type embedding, and cointegrals for Hopf cocategories that realize invariant states; deformation quantization via the 2-Fock–Rosly bracket yields a quantum 2-gauge theory with a Hopf category structure. The paper constructs higher-ribbon invariants Ω as monoidal functors from PLRib'^{𝔊;q}_{(1+1)+ε}(D^4) to the Wilson surface state category, with invariance under 2d Pachner moves and a parameterization by G-equivariant cohomology classes, showing connections to lasagna skein modules. The results establish a robust framework for 4d higher-gauge invariants, providing a path toward a categorified moduli space of flat 2-connections and deep links to extended TQFT and higher categorical quantum symmetry. The formalism lays groundwork for potential applications to topological phases and higher-dimensional quantum topology, where non-Abelian Wilson surfaces encode rich topological data beyond traditional 3d knot invariants.

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 $ω:\mathfrak{C}_q(\mathbb{G}^{Γ^2})\rightarrow\mathsf{Hilb}$, 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.

Figures (18)

• Figure 1: The two ways in which a local 2-graph $(e,f)$ can be decomposed into two 2-graphs, depending on how $(e,f)$ intersects an embedded 2-cell $C$ in the 3d manifold $\Sigma$. The left denotes $\cup_h,$ in which the normal vectors of $C$ are locally tangent to the source edge $e$ of $f$ around $v=s(e)$, while the right denotes $\cup_v$, where the normal vectors are perpendicular to $e$.
• Figure 2: The graphical representation of the derivation property \ref{['derivation']}, which implements the geometric consitency between the product $\otimes$ and the 2-gauge transformation action $\Lambda$.
• Figure 3: A typical complex of 2-simplices with different choice of source paths, coloured in red. The left is a unbroken configuration, and the right is broken.
• Figure 4: The left illustrates the geometric configuration of 2-simplices upon which the interchanger $\beta$ is defined. This geometry is precisely the vertex in a trisected singular graph matveev2007algorithmic as displayed on the right; see also fig. 2 (c) of Sakata2022-il.
• Figure 5: The left illustrates the geometric configuration of 2-simplices upon which the $U(1)$-gerbe $\sigma$ is defined. This geometry is precisely the triple point in a singular graph matveev2007algorithmic as displayed on the right; see also fig. 2 (b) of Sakata2022-il.

Theorems & Definitions (200)

• Definition 1.1
• Theorem 1.2
• Remark 1.1
• Theorem 1.3
• Remark 1.2
• Theorem 1.4
• Proposition 1.5
• Definition 2.1
• Remark 2.1
• Proposition 2.2