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

Abstract

Motivated by [On the triplet vertex algebra W(p), Adv. Math. 217 (2008) 2664-2699, for every finite subgroup Γ ⊂ PSL(2, C) we investigate the fixed point subalgebra W(p) Γ of the triplet vertex W(p), of central charge 1 − 6(p−1) 2 p , p ≥ 2. This part deals with the A-series in the ADE classification of finite subgroups of PSL(2, C). First, we prove the C 2 -cofiniteness of the Am-fixed subalgebra W(p) Am . Then we construct a family of W(p) Am -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 W(p) is PSL(2, C).

Vertex Algebras W(p)Amand W(p)Dmand Constant Term Identities

“…In this section we shall slightly extend results from [3] so we shall introduce two new combinatorial conjectures which will imply the classification of irreducible modules.…”

“…We begin with a short review of the triplet algebra W(p), its orbifold subalgebras W(p) Γ , and status of classifications of their modules; more details can be found in [2,3,5].…”

“…Note also that Ψ is also an automorphism of M (1) such that Ψ(H) = −H and its fixed point subalgebra M (1) + is a subalgebra of W(p) Dm (for details see [2,3]). …”

“…Results from [3] show that Zhu's algebra A(W(p) Dm ) is a commutative, finite-dimensional algebra such that dim A(W(p) Dm ) ≥ (m 2 + 8)p − 1. In order to prove that modules constructed in [3] provide a complete list of irreducible W(p) Dm -modules, it suffices to prove inequality…”

“…In [3], we initiated the study of representation theory of the vertex operator algebra W(p) Dm , m ≥ 2. We proved C 2 -cofiniteness of these vertex algebras and showed that the associated Zhu's algebra is a finite-dimensional commutative algebra.…”