# Three sides of the geometric Langlands correspondence for gl_N Gaudin model and Bethe vector averaging maps

## TL;DR

This paper explores a new aspect of the geometric Langlands correspondence for the gl_N Gaudin model by introducing an algebra related to the critical points of a master function and establishing isomorphisms with existing structures.

## Contribution

It introduces a third algebra related to the critical set of a master function and constructs isomorphisms with the Bethe algebra and differential operators.

## Key findings

- Established isomorphisms between the three algebras
- Introduced Bethe vector averaging maps
- Extended the geometric Langlands correspondence

## Abstract

We consider the gl_N Gaudin model of a tensor power of the standard vector representation. The geometric Langlands correspondence in the Gaudin model relates the Bethe algebra of the commuting Gaudin Hamiltonians and the algebra of functions on a suitable space of N-th order differential operators. In this paper we introduce a third side of the correspondence: the algebra of functions on the critical set of a master function. We construct isomorphisms of the third algebra and the first two. A new object is the Bethe vector averaging maps.