On the triplet vertex algebra W(p)

TL;DR

This paper thoroughly investigates the triplet vertex algebra W(p), establishing its properties such as $C_2$-cofiniteness, irrationality, and finite-representation type, while classifying all irreducible modules and analyzing its algebraic structure.

Contribution

It provides a complete classification of irreducible modules, block decomposition, and explicit algebraic structures for W(p), extending methods to other W-algebras and superalgebras.

Findings

W(p) is $C_2$-cofinite but irrational.

All irreducible W(p)-modules are classified.

Explicit structure of the Zhu algebra A(W(p)) is described.

Abstract

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. Moreover, we obtain an upper bound for ${\rm dim}(A(\mathcal{W}(p)))$. 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.