The Factorizable Feigin-Frenkel center

TL;DR

This paper proves a factorizable version of the Feigin-Frenkel theorem, establishing an isomorphism between the center of a sheaf of completed enveloping algebras at the critical level and functions on Opers, in the context of affine Kac-Moody algebras.

Contribution

It introduces a factorizable structure to the Feigin-Frenkel center and establishes a canonical isomorphism with functions on Opers for the Langlands dual Lie algebra.

Findings

The center forms a factorization algebra.

Established a canonical isomorphism with Opers.

Extended the theorem to sheaves over smooth curves.

Abstract

We prove a factorizable version of the Feigin-Frenkel theorem on the center of the completed enveloping algebra of the affine Kac-Moody algebra attached to a simple Lie algebra at the critical level. On any smooth curve C we consider a sheaf of complete topological Lie algebras whose fiber at any point is the usual affine algebra at the critical level and consider its sheaf of completed enveloping algebras. We show that the center of this sheaf is a factorization algebra and establish that it is canonically isomorphic, in a factorizable manner, with the factorization algebra of functions on Opers on the pointed disk for the Langlands dual Lie algebra.