W-algebras for Argyres-Douglas theories

Abstract

The Schur-index of the $(A_1, X_n)$-Argyres-Douglas theory is conjecturally a character of a vertex operator algebra. Here such vertex algebras are found for the $A_{\text{odd}}$ and $D_{\text{even}}$-type Argyres-Douglas theories. The vertex operator algebra corresponding to $A_{2p-3}$-Argyres-Douglas theory is the logarithmic $\mathcal B_p$-algebra of [1], while the one corresponding to $D_{2p}$, denoted by $\mathcal W_p$, is realized as a non-regular Quantum Hamiltonian reduction of $L_{k}(\mathfrak{sl}_{p+1})$ at level $k=-(p^2-1)/p$. For all $n$ one observes that the quantum Hamiltonian reduction of the vertex operator algebra of $D_n$ Argyres-Douglas theory is the vertex operator algebra of $A_{n-3}$ Argyres-Douglas theory. As corollary, one realizes the singlet and triplet algebras (the vertex algebras associated to the best understood logarithmic conformal field theories) as Quantum Hamiltonian reductions as well. Finally, characters of certain modules of these vertex operator algebras and the modular properties of their meromorphic continuations are given.