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

Abstract

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. Along the way, we determine the linkage principle for Category $\mathcal{O}$ of the $W$-algebra.

41 pages, comments welcome!

References in corpus (2)

• On the structure of the category O for W-algebras
• W-algebras and Whittaker categories

Cited by in corpus (2)

• Homological methods in semi-infinite contexts
• On parafermion vertex algebras of $\mathfrak{sl}(2{)}_{-3/2}$ and $\mathfrak{sl}(3{)}_{-3/2}$