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

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 $PS{L}_2(\mathbb{C})$-orbifold. We also use our results to study Arakawa's chiral universal centralizer algebra of $S{L}_2$ at level $-1$, showing that it has a symmetric tensor category of representations equivalent to $\mathrm{Rep}PS{L}_2(\mathbb{C})$. 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.

39 pages, in this version, reference added for Arakawa's earlier construction of chiral universal centralizer algebra, also further results added on uniqueness and representation theory of vertex algebra extensions of the c = 25 Virasoro algebra

References in corpus (2)

• Braided tensor categories and extensions of vertex operator algebras
• Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$

Cited by in corpus (5)

• A general mirror equivalence theorem for coset vertex operator algebras
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge
• $N=1$ super Virasoro tensor categories
• The non-semisimple Kazhdan-Lusztig category for affine ${\mathfrak{sl}}_2$ at admissible levels
• Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras