{"paper":{"title":"Semi-infinite cohomology and the linkage principle for $W$-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th"],"primary_cat":"math.RT","authors_text":"Gurbir Dhillon","submitted_at":"2019-05-16T00:08:33Z","abstract_excerpt":"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 Drie"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.06477","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}