Local opers with two singularities: the case of $\mathfrak{sl}(2)$

TL;DR

This paper investigates local opers with two singularities for sl(2), establishing analogues of key theorems related to the centre at critical level and endomorphism rings of Weyl modules, extending the affine Lie algebra framework.

Contribution

It introduces a novel study of local opers with two singularities for sl(2) and proves analogues of fundamental theorems connecting opers, the centre, and Weyl modules.

Findings

Proved an analogue of the Feigin-Frenkel theorem for two singularities.

Characterized endomorphism rings of Weyl modules via functions on opers.

Extended the affine Lie algebra framework to include two singularities.

Abstract

We study local opers with two singularities for the case of the Lie algebra sl(2), and discuss their connection with a two-variables extension of the affine Lie algebra. We prove an analogue of the Feigin-Frenkel theorem describing the centre at the critical level, and an analogue of a result by Frenkel and Gaitsgory that characterises the endomorphism rings of Weyl modules in terms of functions on the space of opers.