Representation theory of $W$-algebras, II

Abstract

We study the (Ramond twisted) representations of the affine $W$-algebra $\mathcal{W}^k (\bar{\mathfrak{g}}, f)$ in the case that $f$ admits a good even grading. We establish the vanishing and the almost irreducibility of the corresponding BRST cohomology. This confirms some of the recent conjectures of Kac and Wakimoto [KW08]. In type $A$, our results give the characters of all irreducible ordinary (Ramond twisted) representations of $\mathcal{W}^k (\mathfrak{sl}_n, f)$ for all nilpotent elements $f$ and all non-critical $k$, and prove the existence of modular invariant representations conjectured in [KW08].