Affine Algebras, Langlands Duality and Bethe Ansatz

TL;DR

This paper explores the deep connections between affine algebra representation theory at the critical level, the geometric Langlands correspondence, and the Bethe ansatz in Gaudin models, providing new insights into their interrelations.

Contribution

It offers a novel interpretation of the Bethe ansatz and Sklyanin's separation of variables through the lens of the geometric Langlands correspondence in genus zero.

Findings

D-modules on moduli spaces relate to Gaudin model diagonalization

New interpretation of Bethe ansatz via Langlands correspondence

Connections established between affine algebras, Langlands duality, and integrable systems

Abstract

We review various aspects of representation theory of affine algebras at the critical level, geometric Langlands correspondence, and Bethe ansatz in the Gaudin models. Geometric Langlands correspondence relates D-modules on the moduli space of G-bundles on a complex curve X and flat G^L-bundles on X. Beilinson and Drinfeld construct it by applying a localization functor to representations of affine algebras of critical level. We show that in genus zero the corresponding D-modules are closely related to the diagonalization problem in the Gaudin model associated to G. This allows us to give a new interpretation of the Bethe ansatz and Sklyanin's separation of variables in the Gaudin model in terms of Langlands correspondence.