An $\mathfrak{sl}_2$-type tensor category for the Virasoro algebra at central charge $25$ and applications

TL;DR

This paper constructs and analyzes a braided tensor category related to the Virasoro algebra at central charge 25, revealing an $rak{sl}_2$-type structure and applications to conformal vertex algebras and centralizer algebras.

Contribution

It establishes the rigidity and $rak{sl}_2$-type structure of the category at c=25 and explores its connections to other Virasoro categories and vertex algebras.

Findings

The category $oxtimes_{25}$ is rigid and generated by simple objects.

It is braided tensor equivalent to an abelian 3-cocycle twist of $rak{sl}_2$-modules.

The category at c=25 is braid-reversed equivalent to the one at c=1.

Abstract

Let $\mathcal{O}_{25}$ be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge $25$ whose composition factors are the irreducible quotients of reducible Verma modules. We show that $\mathcal{O}_{25}$ is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian $3$-cocycle twist of the category of finite-dimensional $\mathfrak{sl}_2$-modules. We also show that this $\mathfrak{sl}_2$-type subcategory is braid-reversed tensor equivalent to a similar category for the Virasoro algebra at central charge $1$. As an application, we construct a simple conformal vertex algebra which contains the Virasoro vertex operator algebra of central charge $25$ as a $PSL_2(\mathbb{C})$-orbifold. We also use our results to study Arakawa's chiral universal centralizer algebra of $SL_2$ at level $-1$, showing that it has a symmetric tensor category of representations equivalent to $\mathrm{Rep}\,PSL_2(\mathbb{C})$. This algebra is an extension of the tensor product of Virasoro vertex operator algebras of central charges $1$ and $25$, analogous to the modified regular representations of the Virasoro algebra constructed earlier for generic central charges by I. Frenkel-Styrkas and I. Frenkel-M. Zhu.