The 2-categorical S-matrix of a braided fusion 1-category is a character table

TL;DR

This paper shows that the 2-categorical S-matrix of the Drinfeld center of module categories over a braided fusion category simplifies to the character table of its M"uger center, linking higher categorical invariants to classical algebraic data.

Contribution

It establishes that the 2-categorical S-matrix for the Drinfeld center reduces to a character table, connecting higher categorical invariants with classical algebraic structures.

Findings

The 2-categorical S-matrix of the Drinfeld center is equivalent to the character table of the M"uger center.

This reduction provides a new perspective on the invariant's interpretation in terms of classical algebra.

The result simplifies the understanding of the 2-categorical S-matrix in the context of braided fusion categories.

Abstract

The semisimple module categories over a braided fusion category $\mathcal{C}$ form a connected fusion 2-category $\text{Mod}(\mathcal{C})$. Its Drinfeld center $\mathcal{Z}(\text{Mod}(\mathcal{C}))$ is a braided fusion 2-category. To any braided fusion 2-category, Johnson-Freyd and Reutter arXiv:2105.15167v3 [math.QA] have associated a matrix-valued invariant, the 2-categorical $S$-matrix. In this short note we investigate this matrix of $\mathcal{Z}(\text{Mod}(\mathcal{C}))$ as an invariant for the braided fusion 1-category $\mathcal{C}$ and show that it reduces to the character table of the M\"uger center of $\mathcal{C}$.