On twisted Verlinde formulae for modular categories

TL;DR

This paper introduces two twisted Verlinde formula analogues for modular categories, relating fusion coefficients to S-matrices and crossed S-matrices in a setting involving module categories and autoequivalences.

Contribution

It develops new twisted Verlinde formulas for modular categories involving module categories and autoequivalences, extending classical results.

Findings

Derived fusion coefficient formulas using crossed S-matrices.

Established twisted fusion algebra structure.

Connected fusion coefficients to modular autoequivalences.

Abstract

In this note, we describe two analogues of the Verlinde formula for modular categories in a twisted setting. The classical Verlinde formula for a modular category $\mathscr{C}$ describes the fusion coefficients of $\mathscr{C}$ in terms of the corresponding S-matrix $S(\mathscr{C})$. Now let us suppose that we also have an invertible $\mathscr{C}$-module category $\mathscr{M}$ equipped with a $\mathscr{C}$-module trace. This gives rise to a modular autoequivalence $F:\mathscr{C}\xrightarrow{\cong}\mathscr{C}$. In this setting, we can define a crossed S-matrix $S(\mathscr{C},\mathscr{M})$. As our first twisted analogue of the Verlinde formula, we will describe the fusion coefficients for $\mathscr{M}$ as a $\mathscr{C}$-module category in terms of the S-matrix $S(\mathscr{C})$ and the crossed S-matrix $S(\mathscr{C},\mathscr{M})$. In this twisted setting, we can also define a twisted fusion $\mathbb{Q}^{ab}$-algebra $K_{\mathbb{Q}^{ab}}(\mathscr{C},F)$. As another analogue of the Verlinde formula, we describe the fusion coefficients of the twisted fusion algebra in terms of the crossed S-matrix $S(\mathscr{C},\mathscr{M})$.