Abstract
Let g be a simple Lie algebra, and let Wκ be the affine W-algebra associated to a principal nilpotent element of g and level κ. We explain a duality between the categories of smooth W modules at levels κ+κc and −κ+κc, where κc 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.
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