Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p > 1$

TL;DR

This paper determines the structure of the Virasoro tensor category at specific central charges, showing rigidity, computing tensor products, and relating it to known categories like Kazhdan-Lusztig and triplet VOAs.

Contribution

It explicitly describes the tensor category structure of Virasoro modules at certain central charges, including rigidity, projective covers, and category equivalences.

Findings

Proves rigidity of the category at specified central charges

Constructs projective covers of irreducible modules

Shows the category's semisimplification as a Deligne product

Abstract

Let $\mathcal{O}_c$ be the category of finite-length central-charge-$c$ modules for the Virasoro Lie algebra whose composition factors are irreducible quotients of reducible Verma modules. Recently, it has been shown that $\mathcal{O}_c$ admits vertex algebraic tensor category structure for any $c\in\mathbb{C}$. Here, we determine the structure of this tensor category when $c=13-6p-6p^{-1}$ for an integer $p>1$. For such $c$, we prove that $\mathcal{O}_{c}$ is rigid, and we construct projective covers of irreducible modules in a natural tensor subcategory $\mathcal{O}_{c}^0$. We then compute all tensor products involving irreducible modules and their projective covers. Using these tensor product formulas, we show that $\mathcal{O}_c$ has a semisimplification which, as an abelian category, is the Deligne product of two tensor subcategories that are tensor equivalent to the Kazhdan-Lusztig categories for affine $\mathfrak{sl}_2$ at levels $-2+p^{\pm 1}$. Next, as a straightforward consequence of the braided tensor category structure on $\mathcal{O}_c$ together with the theory of vertex operator algebra extensions, we rederive known results for triplet vertex operator algebras $\mathcal{W}(p)$, including rigidity, fusion rules, and construction of projective covers. Finally, we prove a recent conjecture of Negron that $\mathcal{O}_c^0$ is braided tensor equivalent to the $PSL(2,\mathbb{C})$-equivariantization of the category of $\mathcal{W}(p)$-modules.