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

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.

arXiv comment56 pages, final version incorporating referee's comments

References in corpus (22)

• Vertex Algebras and Algebraic Curves
• Modular invariance of trace functions in orbifold theory
• Indecomposable Fusion Products
• Logarithmic Minimal Models
• Associative-algebraic approach to logarithmic conformal field theories
• Tensor categories for vertex operator superalgebra extensions
• Schur-Weyl Duality for Heisenberg Cosets
• Nonmeromorphic operator product expansion and C_2-cofiniteness for a family of W-algebras
• Logarithmic tensor category theory, II: Logarithmic formal calculus and properties of logarithmic intertwining operators
• Fusion Algebras of Logarithmic Minimal Models
• Logarithmic tensor category theory, VI: Expansion condition, associativity of logarithmic intertwining operators, and the associativity isomorphisms
• Logarithmic tensor category theory, VIII: Braided tensor category structure on categories of generalized modules for a conformal vertex algebra
• Logarithmic tensor category theory, V: Convergence condition for intertwining maps and the corresponding compatibility condition
• Logarithmic tensor category theory, VII: Convergence and extension properties and applications to expansion for intertwining maps
• Logarithmic tensor category theory, III: Intertwining maps and tensor product bifunctors
• Logarithmic tensor category theory, IV: Constructions of tensor product bifunctors and the compatibility conditions
• Boundary algebras and Kac modules for logarithmic minimal models
• On the tensor structure of modules for compact orbifold vertex operator algebras
• Kazhdan-Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models
• Classification of Kac representations in the logarithmic minimal models LM(1,p)
• Twisted modules and $G$-equivariantization in logarithmic conformal field theory
• C_1-cofiniteness and Fusion Products of Vertex Operator Algebras

Cited by in corpus (6)

• Quantum SL(2) and logarithmic vertex operator algebras at (p,1)-central charge
• Characterizing braided tensor categories associated to logarithmic vertex operator algebras
• 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
• On rationality for $C_2$-cofinite vertex operator algebras
• Tensor Categories arising from the Virasoro Algebra