A Tensor Category Construction of the $W_{p,q}$ Triplet Vertex Operator Algebra and Applications

TL;DR

The paper presents a tensor-category construction of the triplet VOA $W_{p,q}$ for coprime $p,q\ge 2$, linking the Virasoro module category at central charge $c_{p,q}$ with the representation category of $PSL_2(\mathbb{C})$. It identifies a PSL$_2$-type fusion structure among the Virasoro summands in $W_{p,q}$ and builds a rigid symmetric subcategory equivalent to $\operatorname{Rep} PSL_2$, enabling a commutative algebra construction in a Deligne product that yields $W_{p,q}$ as a non-simple extension with automorphism group $PSL_2(\mathbb{C})$. A key development is the definition of a Virasoro-induced category $\mathcal{O}_{c_{p,q}}^0$ and its embedding into the $PSL_2$-equivariantization of $\operatorname{Rep}(W_{p,q})$, together with conjectures about projective objects and suitability for logarithmic minimal models. The work also clarifies the relationship between Virasoro and triplet representation theories via induction, and outlines future directions for extending these methods to broader Feigin–Tipunin algebras and related superalgebras, with potential impact on logarithmic CFT and low-dimensional topology.

Abstract

For coprime $p,q\in\mathbb{Z}_{\geq 2}$, the triplet vertex operator algebra $W_{p,q}$ is a non-simple extension of the universal Virasoro vertex operator algebra of central charge $c_{p,q}=1-\frac{6(p-q)^2}{pq}$, and it is a basic example of a vertex operator algebra appearing in logarithmic conformal field theory. Here, we give a new construction of $W_{p,q}$ different from the original screening operator definition of Feigin-Gainutdinov-Semikhatov-Tipunin. Using our earlier work on the tensor category structure of modules for the Virasoro algebra at central charge $c_{p,q}$, we show that the simple modules appearing in the decomposition of $W_{p,q}$ as a module for the Virasoro algebra have $\mathrm{PSL}_2$-fusion rules and generate a symmetric tensor category equivalent to $\operatorname{Rep}\mathrm{PSL}_2$. Then we use the theory of commutative algebras in braided tensor categories to construct $W_{p,q}$ as an appropriate non-simple modification of the canonical algebra in the Deligne tensor product of $\operatorname{Rep}\mathrm{PSL}_2$ with this Virasoro subcategory. As a consequence, we show that the automorphism group of $W_{p,q}$ is $\mathrm{PSL}_2(\mathbb{C})$. We also define a braided tensor category $\mathcal{O}_{c_{p,q}}^0$ consisting of modules for the Virasoro algebra at central charge $c_{p,q}$ that induce to untwisted modules of $W_{p,q}$. We show that $\mathcal{O}_{c_{p,q}}^0$ tensor embeds into the $\mathrm{PSL}_2(\mathbb{C})$-equivariantization of the category of $W_{p,q}$-modules and is closed under contragredient modules. We conjecture that $\mathcal{O}_{c_{p,q}}^0$ has enough projective objects and is the correct category of Virasoro modules for constructing logarithmic minimal models in conformal field theory.

Theorems & Definitions (26)

• Theorem 2.1
• Lemma 3.1
• Proposition 3.2
• Theorem 3.3
• Theorem 3.4
• Remark 3.5
• Remark 3.6
• Remark 3.7
• Theorem 3.8
• Remark 3.9