Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 012, 34 pages      arXiv:2508.18895      https://doi.org/10.3842/SIGMA.2026.012
Contribution to the Special Issue on Recent Advances in Vertex Operator Algebras in honor of James Lepowsky

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

Robert McRae a and Valerii Sopin b
a) Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, P.R. China
b) Shanghai Institute for Mathematics and Interdisciplinary Sciences, Shanghai 200433, P.R. China

Received August 27, 2025, in final form January 19, 2026; Published online February 09, 2026

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.

Key words: triplet vertex operator algebras; $\mathrm{PSL}_2$ automorphism group; braided tensor categories; logarithmic conformal field theory.

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