Affine Algebras, Langlands Duality and Bethe Ansatz

1995/06/05 by Edward Frenkel, Frenkel, Edward
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Quantum Algebra (math.QA) #alg-geom #hep-th #math.AG #math.QA #q-alg

paper · pdf · doi:10.48550/arxiv.q-alg/9506003

34 pages, Latex

arxiv created 1999/09/23 · arxiv updated 2009/11/30

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.

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