Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge

TL;DR

This paper studies the structure and fusion properties of Virasoro Kac modules in logarithmic minimal models at specific central charges, revealing their rigidity, duality, and how they generate the module category.

Contribution

It establishes rigidity and self-duality of certain Virasoro Kac modules within logarithmic minimal models and characterizes their tensor product relations, advancing understanding of their categorical structure.

Findings

Kac modules $\\mathcal{K}_{r,s}$ are rigid and self-dual for $r\leq p$, $s\leq q$

Not all Kac modules are rigid when $r>p$ or $s>q$

Tensor products of Kac modules generate all modules in the category

Abstract

Let $\mathcal{O}_c$ be the category of finite-length modules for the Virasoro Lie algebra at central charge $c$ whose composition factors are irreducible quotients of reducible Verma modules. For any $c\in\mathbb{C}$, this category admits the vertex algebraic braided tensor category structure of Huang, Lepowsky, and Zhang. Here, we begin the detailed study of $\mathcal{O}_{c_{p,q}}$ where $c_{p,q} = 1-\frac{6(p-q)^2}{pq}$ for relatively prime integers $p, q \geq 2$; in conformal field theory, $\mathcal{O}_{c_{p,q}}$ corresponds to a logarithmic extension of the central charge $c_{p,q}$ Virasoro minimal model. We particularly focus on the Virasoro Kac modules $\mathcal{K}_{r,s}$, $r,s\in\mathbb{Z}_{\geq 1}$, in $\mathcal{O}_{c_{p,q}}$ defined by Morin-Duchesne, Rasmussen, and Ridout, which are finitely-generated submodules of Feigin-Fuchs modules for the Virasoro algebra. We prove that $\mathcal{K}_{r,s}$ is rigid and self-dual when $1\leq r\leq p$ and $1\leq s\leq q$, but that not all $\mathcal{K}_{r,s}$ are rigid when $r>p$ or $s>q$. That is, $\mathcal{O}_{c_{p,q}}$ is not a rigid tensor category. We also show that all Kac modules and all simple modules in $\mathcal{O}_{c_{p,q}}$ are homomorphic images of repeated tensor products of $\mathcal{K}_{1,2}$ and $\mathcal{K}_{2,1}$, and we determine completely how $\mathcal{K}_{1,2}$ and $\mathcal{K}_{2,1}$ tensor with Kac modules and simple modules in $\mathcal{O}_{c_{p,q}}$. In the process, we prove some fusion rule conjectures of Morin-Duchesne, Rasmussen, and Ridout.