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

Motivated by \cite{am1}, for every finite subgroup $\Gamma \subset PSL(2,$\mathbb{C}$)$weinvestigatethefixedpointsubalgebra$\triplet^{\Gamma}$ofthetripletvertex$\mathcal {W}(p)$,ofcentralcharge$1-\frac{6(p-1)^{2}}{p}$,$p\geq2$\mathrm{.}Thispartdealswiththe$A$-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 ${\text{triplet}}^{A_m}$. Then we construct a family of $\text{am}$-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 $\text{triplet}$ is $PSL(2,\mathbb{C})$.