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

TL;DR

The paper analyzes the Virasoro module category $\\mathcal{O}_c$ at central charges $c_{p,1}$, proving rigidity and constructing projective covers within the tensor subcategory $\\mathcal{O}_c^0$. It computes all tensor products among irreducibles and their projective covers, and shows a semisimplification of $\\mathcal{O}_c$ as the Deligne product of Kazhdan–Lusztig categories for $\\widehat{\\mathfrak{sl}}_2$ at levels $-2+1/p$ and $-2+p$, linking to logarithmic minimal models. The results recover and extend known fusion and rigidity phenomena for triplet algebras $W(p)$, and, via VOA-extension theory, yield a rapid derivation of their module structure and projective covers. A key achievement is proving Negron’s conjecture that $\\O_c^0$ is braided tensor equivalent to the $PSL(2,\\mathbb{C})$-equivariantization of $\\mathcal{C}_{W(p)}$, connecting the Virasoro and triplet theories in a precise categorical framework.

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.

Theorems & Definitions (95)

• Theorem 1.1
• Definition 2.1
• Lemma 2.2
• proof
• Corollary 2.3
• Definition 2.4
• Proposition 2.5
• proof
• Theorem 2.6
• Remark 2.7