Semi-infinite cohomology and the linkage principle for W-algebras

Let g be a simple Lie algebra, and let W-kappa be the affine W-algebra associated to a principal nilpotent element of g and level kappa. We explain a duality between the categories of smooth W modules at levels kappa+kappa(c), and -kappa+kappa(c), where kappa(c) is the critical level. Their pairing amounts to a construction of semi-infinite cohomology for the W-algebra. As an application, we determine all homomorphisms between the Verma modules for W, verifying a conjecture from the conformal field theory literature of de Vos-van Driel. Along the way, we determine the linkage principle for Category O of the W-algebra. (C) 2021 Elsevier Inc. All rights reserved.