On twisted Verlinde formulae for modular categories

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 $S(\mathscr{C},\mathscr{M})$.