On the triplet vertex algebra W(p)

Abstract

We study the triplet vertex operator algebra $\mathcal{W}(p)$ of central charge $1-\frac{6(p-1{)}^2}{p}$, $p\ge 2$. We show that $\trip$ is $C_2$-cofinite but irrational since it admits indecomposable and logarithmic modules. Furthermore, we prove that $\trip$ is of finite-representation type and we provide an explicit construction and classification of all irreducible $\mathcal{W}(p)$-modules and describe block decomposition of the category of ordinary $\trip$-modules. All this is done through an extensive use of Zhu's associative algebra together with explicit methods based on vertex operators and the theory of automorphic forms. Moreover, we obtain an upper bound for $\dim (A(\mathcal{W}(p)))$. Finally, for $p$ prime, we completely describe the structure of $A(\trip)$. The methods of this paper are easily extendable to other $\mathcal{W}$-algebras and superalgebras.

32 pages; v2: a few minor changes, to appear in Advances in Mathematics

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• Logarithmic W-algebras and Argyres-Douglas theories at higher rank
• Modular Transformations and Verlinde Formulae for Logarithmic $(p_{+},p_{-})$-Models
• Bosonic ghostbusting -- The bosonic ghost vertex algebra admits a logarithmic module category with rigid fusion
• The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector
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• Critical dense polymers with Robin boundary conditions, half-integer Kac labels and ${\mathbb{Z}}_4$ fermions
• The logarithmic Cardy case: Boundary states and annuli
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• W-extended Kac representations and integrable boundary conditions in the logarithmic minimal models WLM(1,p)
• The second bosonization of the CKP hierarchy
• C-Graded Vertex Algebras and Conformal Flow
• Cofiniteness conditions, projective covers and the logarithmic tensor product theory