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Gaudin models with irregular singularities

B. Feigin, E. Frenkel, V. Toledano-Laredo

TL;DR

The work extends Gaudin models by embedding them in the center of the affine Kac–Moody algebra at the critical level, allowing non-highest weight representations and irregular singularities. It identifies generalized Gaudin algebras with algebras of functions on spaces of opers, and proves that their spectra on modules correspond to opers with specified singular behavior, including monodromy constraints. A Bethe Ansatz framework via Wakimoto modules constructs eigenvectors and connects eigenvalues to Miura-transformed opers, tying the quantum models to ramified geometric Langlands data. The results unify Hitchin-type integrable systems, irregular singularities, and Langlands duality, and give a concrete program for diagonalizing irregular Gaudin models and understanding their global geometric significance.

Abstract

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.

Gaudin models with irregular singularities

TL;DR

The work extends Gaudin models by embedding them in the center of the affine Kac–Moody algebra at the critical level, allowing non-highest weight representations and irregular singularities. It identifies generalized Gaudin algebras with algebras of functions on spaces of opers, and proves that their spectra on modules correspond to opers with specified singular behavior, including monodromy constraints. A Bethe Ansatz framework via Wakimoto modules constructs eigenvectors and connects eigenvalues to Miura-transformed opers, tying the quantum models to ramified geometric Langlands data. The results unify Hitchin-type integrable systems, irregular singularities, and Langlands duality, and give a concrete program for diagonalizing irregular Gaudin models and understanding their global geometric significance.

Abstract

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.

Paper Structure

This paper contains 41 sections, 44 theorems, 340 equations.

Table of Contents

  1. Introduction
  2. Non-highest weight representations and irregular singularities
  3. The shift of argument subalgebra and the DMT Hamiltonians
  4. Diagonalizing quantum Hamiltonians
  5. Connection to the geometric Langlands correspondence
  6. Plan of the paper
  7. Construction of generalized Gaudin algebras
  8. Affine Kac--Moody algebra and its modules
  9. Spaces of coinvariants
  10. The algebra of endomorphisms of $\mathbb{V}_{0,\kappa}$
  11. Action of ${\mathfrak z}_\kappa(\widehat{{\mathfrak g}})$ on coinvariants
  12. The universal Gaudin algebra and its quotients
  13. Example: the Gaudin model
  14. Another example: the algebra ${\mathcal{A}}_\chi$
  15. Multi-point generalization
  16. ...and 26 more sections

Key Result

Lemma 2.1

The composition of the above embedding and the projection gives rise to an isomorphism

Theorems & Definitions (55)

  • Lemma 2.1: FFR, Lemma 1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Lemma 2.9: FFR, Prop. 1
  • Theorem 3.1
  • ...and 45 more