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

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 $p,q\ge 2$; 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 ${\mathcal{K}}_{r,s}$, $r,s\in {\mathbb{Z}}_{\ge 1}$, 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 $1\le r\le p$ and $1\le s\le q$, 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.

55 pages, final version incorporating referee comments

References in corpus (22)

• Logarithmic extensions of minimal models: characters and modular transformations
• Associative-algebraic approach to logarithmic conformal field theories
• Braided tensor categories and extensions of vertex operator algebras
• From Percolation to Logarithmic Conformal Field Theory
• Kazhdan--Lusztig-dual quantum group for logarithmic extensions of Virasoro minimal models
• Virasoro representations and fusion for general augmented minimal models
• From boundary to bulk in logarithmic CFT
• Fusion Algebras of Logarithmic Minimal Models
• Logarithmic CFTs connected with simple Lie algebras
• Logarithmic M(2,p) Minimal Models, their Logarithmic Couplings, and Duality
• W-Extended Logarithmic Minimal Models
• On the Percolation BCFT and the Crossing Probability of Watts
• W-Extended Fusion Algebra of Critical Percolation
• Unitary and non-unitary $N=2$ minimal models
• Ribbon tensor structure on the full representation categories of the singlet vertex algebras
• On W-algebras associated to (2,p) minimal models and their representations
• Quantum SL(2) and logarithmic vertex operator algebras at (p,1)-central charge
• Characterizing braided tensor categories associated to logarithmic vertex operator algebras
• An ${\mathfrak{sl}}_2$-type tensor category for the Virasoro algebra at central charge $25$ and applications
• C_2-cofinite W-algebras and their logarithmic representations
• Staggered modules of $N=2$ superconformal minimal models
• Polynomial Fusion Rings of Logarithmic Minimal Models

Cited by in corpus (4)

• $N=1$ super Virasoro tensor categories
• Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras
• 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