## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Arrangements of Hyperplanes — Sapporo 2009, H. Terao and S. Yuzvinsky, eds. (Tokyo: Mathematical Society of Japan, 2012), 475 - 511

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

Eugene Mukhin, Vitaly Tarasov, and Alexander Varchenko

#### 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

**Source***Arrangements of Hyperplanes — Sapporo 2009*, H. Terao and S. Yuzvinsky, eds. (Tokyo: Mathematical Society of Japan, 2012), 475-511

**Dates**

Received: 25 October 2009

Revised: 14 March 2010

First available in Project Euclid:
24 November 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/aspm/06210475

**Mathematical Reviews number (MathSciNet)**

MR2933807

**Zentralblatt MATH identifier**

1260.82025

**Subjects**

Primary: 82B23: Exactly solvable models; Bethe ansatz

Secondary: 17B80: Applications to integrable systems 32S22: Relations with arrangements of hyperplanes [See also 52C35]

**Keywords**

Bethe algebra Bethe anzats Bethe vector averaging map master function critical points Wronsky map

#### 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