Gaudin models with irregular singularities

Details Gaudin models with irregular singularities Gaudin models with irregular singularities

2006-12-27

arxiv.org/abs/math/0612798v3

Adv.Math.223:873-948,2010

Mathematics

Quantum Algebra

Latex, 72 pages. Final version to appear in Advances in Mathematics

Scientific paper

10.1016/j.aim.2009.09.007

We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac-Moody algebra at the critical level, extending the construction of higher Gaudin Hamiltonians from hep-th/9402022 to the case of non-highest weight representations of affine algebras. We show that these algebras are isomorphic to algebras of functions on the spaces of opers on P^1 with regular as well as irregular singularities at finitely many points. We construct eigenvectors of these Hamiltonians, using Wakimoto modules of critical level, and show that their spectra on finite-dimensional representations are given by opers with trivial monodromy. We also comment on the connection between the generalized Gaudin models and the geometric Langlands correspondence with ramification.

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