ADE SUBALGEBRAS OF THE TRIPLET VERTEX ALGEBRA $\mathcal{W}(p)$: A-SERIES

Abstract

Motivated by [On the triplet vertex algebra [Formula: see text], Adv. Math.217 (2008) 2664–2699], for every finite subgroup Γ ⊂ PSL(2, ℂ) we investigate the fixed point subalgebra [Formula: see text] of the triplet vertex [Formula: see text], of central charge [Formula: see text], p ≥ 2. This part deals with the A-series in the ADE classification of finite subgroups of PSL(2, ℂ). First, we prove the C2-cofiniteness of the Am-fixed subalgebra [Formula: see text]. Then we construct a family of [Formula: see text]-modules, which are expected to form a complete set of irreducible representations. 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 [Formula: see text] is PSL(2, ℂ).