Standard Verlinde conjecture for Bershadsky–Polyakov minimal models

Let $\mathsf{k}$ be admissible-nondegenerate. For $\ell,\ell'\in\mathbb{Z}$, $[j],[j']\in\mathbb{R}/\mathbb{Z}$, and $[\lambda],[\lambda']\in\Gamma_{\mathsf{u},\mathsf{v}}/\mathbb{Z}_3$, let $\widetilde{\mathcal{R}}_{[j],[\lambda]}^\ell$ denote standard $\mathsf{BP}\brac{\mathsf{u},\mathsf{v}}$-modules and let $\mathsf{S}$ be the modular S-matrix. Standard Verlinde conjecture. The Grothendieck fusion rules are given by

$$[\widetilde{\mathcal{R}}_{[j],[\lambda]}^\ell]\mathbin{\boxtimes}[\widetilde{\mathcal{R}}_{[j'],[\lambda']}^{\ell'}]=\sum_{\ell”\in\mathbb{Z}}\int_{\mathbb{R}/\mathbb{Z}}\sum_{[\lambda”]\in\Gamma_{\mathsf{u},\mathsf{v}}/\mathbb{Z}_3}\genfrac{(}{)}{0pt}{0}{\ell”, [j”], [\lambda”]}{\ell, [j], [\lambda]\quad\ell', [j'], [\lambda']}[\widetilde{\mathcal{R}}_{[j”],[\lambda”]}^{\ell”}]\,\mathrm{d}[j”],$$

where

$$\genfrac{(}{)}{0pt}{0}{\ell”, [j”], [\lambda”]}{\ell, [j], [\lambda]\quad\ell', [j'], [\lambda']}=\sum_{m\in\mathbb{Z}}\int_{\mathbb{R}/\mathbb{Z}}\sum_{[\mu]\in\Gamma_{\mathsf{u},\mathsf{v}}/\mathbb{Z}_3}\frac{\mathsf{S}_{\ell,[j],[\lambda]}^{m,[k],[\mu]}\mathsf{S}_{\ell',[j'],[\lambda']}^{m,[k],[\mu]}\left(\mathsf{S}_{\ell”,[j”],[\lambda”]}^{m,[k],[\mu]}\right)^*}{\mathsf{S}_{\textup{vac.}}^{m,[k],[\mu]}}\,\mathrm{d}[k].$$

Here, the asterisk denotes complex conjugation. This is the conjectural extension of the Verlinde formula to these nonrational theories; it concerns Grothendieck fusion coefficients because characters cannot distinguish modules from sums of composition factors. Its validity requires fusion with standard modules to define an exact functor, and the authors note that rigidity of the standard-module category would imply the needed consistency.