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

Abstract

Motivated by \cite{am1}, for every finite subgroup $\Gamma \subset PSL(2,\mathbb{C})$ we investigate the fixed point subalgebra $\triplet^Γ$ of the triplet vertex $\mathcal{W}(p)$, of central charge $1-\frac{6(p-1{)}^2}{p}$, $p\ge 2$. 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 $PSL(2,\mathbb{C})$.

24 pages. To appear in Communications of Contemporary Mathematics

References in corpus (4)

• Logarithmic Conformal Field Theory: Beyond an Introduction
• The Triplet Vertex Operator Algebra W(p) and the Restricted Quantum Group at Root of Unity
• The N=1 triplet vertex operator superalgebras
• C_2-cofinite W-algebras and their logarithmic representations

Cited by in corpus (2)

• Vertex Algebras $\mathcal{W}(p{)}^{A_m}$ and $\mathcal{W}(p{)}^{D_m}$ and Constant Term Identities
• A new proof of the ${\mathfrak{sl}}_2$ action on the triplet vertex algebra