The center of monoidal 2-categories in 3+1D Dijkgraaf-Witten Theory

TL;DR

This work computes the center of the ω-twisted G-graded monoidal 2-category 2Vec_G^ω, revealing a braided monoidal 2-category with trivial sylleptic center and providing a concrete 3+1D Dijkgraaf-Witten interpretation of topological defects. The center decomposes as a direct sum over conjugacy classes: $\mathcal{Z}(\operatorname{2Vec}_G^{\omega}) \simeq \boxplus_{[h]} \operatorname{2Rep}(C_G(h),\tau_h(\omega))$, where $\tau_h(\omega)$ is the transgression of the 4-cocycle to $C_G(h)$. The computation proceeds by analyzing the center componentwise, restricting to a single grading, and establishing an equivalence with $2$-representations of centralizers twisted by $\tau_h(\omega)$. The unit component is identified with $\operatorname{2Rep}(G)$, and the sylleptic center is shown to be trivial for the center and to yield a minimal modular extension of $\operatorname{2Rep}(G)$. These results advance the program of defining and classifying modular tensor 2-categories and illuminate the structure of 3+1D DW theory via categorical centers.

Abstract

In this work, for a finite group $G$ and a 4-cocycle $ω\in Z^4(G, \mathbf{k}^\times)$, we compute explicitly the center of the monoidal 2-category $\operatorname{2Vec}_G^ω$ of $ω$-twisted $G$-graded 1-categories of finite dimensional $\mathbf{k}$-vector spaces. This center gives a precise mathematical description of the topological defects in the associated 3+1D Dijkgraaf-Witten TQFT. We prove that this center is a braided monoidal 2-category with a trivial sylleptic center.

Theorems & Definitions (25)

• Theorem 1.1
• Conjecture 1.2
• Theorem 1.3
• Remark 2.1
• Theorem 2.2
• proof
• Remark 3.1
• Lemma 3.2
• proof
• Theorem 3.3