Vertex algebras with big centre and a Kazhdan-Lusztig correspondence

Abstract

We study the semiclassical limit $\kappa \to \infty$ of the generalized quantum Langlands kernel associated to a Lie algebra $g$ and an integer level p. This vertex algebra acquires a big centre, containing the ring of functions over the space of $g$-connections. We conjecture that the fibre over the zero connection is the Feigin-Tipunin vertex algebra, whose category of representations should be equivalent to the small quantum group, and that the other fibres are precisely its twisted modules, and that the entire category of representations is related to the quantum group with a big centre. In this sense we present a generalized Kazhdan-Lusztig conjecture, involving deformations by any $g$-connection. We prove our conjectures in small cases $(g,1)$ and $({\mathrm{sl}}_2,2)$ by explicitly computing all vertex algebras and categories involved.