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

Understand

Let $\mathfrak{g}$ be a simple Lie algebra, and let $W_\kappa$ be the affine ${W}$-algebra associated to a principal nilpotent element of $\mathfrak{g}$ 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.