A general mirror equivalence theorem for coset vertex operator algebras

We prove a general mirror duality theorem for a subalgebra $U$ of a simple conformal vertex algebra $A$ and its commutant $V=\mathrm{Com}_A(U)$. Specifically, we assume that $A\cong\bigoplus_{i\in I} U_i\otimes V_i$ as a $U\otimes V$-module, where the $U$-modules $U_i$ are simple and distinct and are objects of a semisimple braided ribbon category of $U$-modules, and the $V$-modules $V_i$ are semisimple and contained in a (not necessarily rigid) braided tensor category of $V$-modules. We also assume $U=\mathrm{Com}_A(V)$. Under these conditions, we construct a braid-reversed tensor equivalence $\tau: \mathcal{U}_A\rightarrow\mathcal{V}_A$, where $\mathcal{U}_A$ is the semisimple category of $U$-modules with simple objects $U_i$, $i\in I$, and $\mathcal{V}_A$ is the category of $V$-modules whose objects are finite direct sums of the $V_i$. In particular, the $V$-modules $V_i$ are simple and distinct, and $\mathcal{V}_A$ is a rigid tensor category. As an application, we find a rigid semisimple tensor subcategory of modules for the Virasoro algebra at central charge $13+6p+6p^{-1}$, $p\in\mathbb{Z}_{\geq 2}$, which is braided tensor equivalent to an abelian $3$-cocycle twist of the category of finite-dimensional $\mathfrak{sl}_2$-modules. Consequently, the Virasoro vertex operator algebra at central charge $13+6p+6p^{-1}$ is the $PSL_2(\mathbb{C})$-fixed-point subalgebra of a simple conformal vertex algebra $\mathcal{W}(-p)$, analogous to the realization of the Virasoro vertex operator algebra at central charge $13-6p-6p^{-1}$ as the $PSL_2(\mathbb{C})$-fixed-point subalgebra of the triplet algebra $\mathcal{W}(p)$.