ADE subalgebras of the triplet vertex algebra W(p): A-series

TL;DR

This paper studies fixed point subalgebras of the triplet vertex algebra W(p) under A-series subgroups, proving their C2-cofiniteness, constructing modules, and analyzing their automorphism groups and modular properties.

Contribution

It introduces the A-series fixed subalgebras of W(p), proves their C2-cofiniteness, constructs modules, and analyzes their automorphism groups and modular invariance.

Findings

Proved C2-cofiniteness of A-series fixed subalgebras.

Constructed a family of modules expected to be all irreducible.

Established the automorphism group as PSL(2,C).

Abstract

Motivated by \cite{am1}, for every finite subgroup $\Gamma \subset PSL(2,\mathbb{C})$ we investigate the fixed point subalgebra $\triplet^{\Gamma}$ of the triplet vertex $\mathcal {W}(p)$, of central charge $1-\frac{6(p-1)^{2}}{p}$, $p\geq2$. This part deals with the $A$-series in the ADE classification of finite subgroups of $PSL(2,\mathbb{C})$. First, we prove the $C_2$-cofiniteness of the $A_m$-fixed subalgebra $\triplet^{A_m}$. Then we construct a family of $\am$-modules, which are expected to form a complete set of irreps. As a strong support to our conjecture, we prove modular invariance of (generalized) characters of the relevant (logarithmic) modules. Further evidence is provided by calculations in Zhu's algebra for $m=2$. We also present a rigorous proof of the fact that the full automorphism group of $\triplet$ is $PSL(2,\mathbb{C})$.