A general mirror equivalence theorem for coset vertex operator algebras

Abstract

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 $A\cong {⨁}_{i\in I}{U}_i\otimes V_i$ 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 $\tau :{\mathcal{U}}_A\to {\mathcal{V}}_A$, 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 $13+6p+6p^{-1}$, $p\in {\mathbb{Z}}_{\ge 2}$, 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 $PS{L}_2(\mathbb{C})$-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 $PS{L}_2(\mathbb{C})$-fixed-point subalgebra of the triplet algebra $\mathcal{W}(p)$.

54 pages; final version, to appear in Science China Mathematics

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Cited by in corpus (3)

• On rationality for $C_2$-cofinite vertex operator algebras
• $N=1$ super Virasoro tensor categories
• Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras