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

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.

💡 Research Summary

The paper presents a novel construction of the triplet vertex operator algebra (VOA) Wₚ,₍q₎ for coprime integers p, q ≥ 2, avoiding the traditional screening‑operator approach of Feigin‑Gainutdinov‑Semikhatov‑Tipunin. Building on the authors’ earlier work on the braided tensor category 𝒪_{cₚ,₍q₎} of C₁‑cofinite modules for the universal Virasoro VOA V_{cₚ,₍q₎} (central charge cₚ,₍q₎ = 1 − 6(p − q)²/(pq)), they first compute the fusion product of the simple Virasoro modules L_{2np‑1,1} (n ≥ 2) that appear in the known Virasoro decomposition of Wₚ,₍q₎: \