On the triplet vertex algebra W(p)

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We study the triplet vertex operator algebra $\mathcal{W}(p)$ of central charge $1-\frac{6(p-1)^2}{p}$, $p \geq 2$.

• 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.