## Advanced Studies in Pure Mathematics

### Three sides of the geometric Langlands correspondence for $\mathfrak{gl}_N$ Gaudin model and Bethe vector averaging maps

#### Abstract

We consider the $\mathfrak{gl}_N$ 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. Our main technical tool is the Bethe vector averaging maps, which is a new object.

#### Article information

Dates
Revised: 14 March 2010
First available in Project Euclid: 24 November 2018

https://projecteuclid.org/ euclid.aspm/1543085019

Digital Object Identifier
doi:10.2969/aspm/06210475

Mathematical Reviews number (MathSciNet)
MR2933807

Zentralblatt MATH identifier
1260.82025

#### Citation

Mukhin, Eugene; Tarasov, Vitaly; Varchenko, Alexander. Three sides of the geometric Langlands correspondence for $\mathfrak{gl}_N$ Gaudin model and Bethe vector averaging maps. Arrangements of Hyperplanes — Sapporo 2009, 475--511, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06210475. https://projecteuclid.org/euclid.aspm/1543085019