Advanced Studies in Pure Mathematics

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

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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


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