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Gaudin model and opers

Edward Frenkel

TL;DR

This survey blends representation theory, integrable systems, and the geometric Langlands program to connect Gaudin spectra with ^L G-opers on the projective line. The authors describe the Gaudin algebra as a center-quotient at the critical level, whose spectrum is realized as spaces of opers with prescribed singularities, and develop a Wakimoto-module framework to construct eigenvectors via the Bethe Ansatz. Central to the narrative are Miura opers and Cartan connections, which provide a concrete parametrization (via flag varieties) of eigenvalues and Bethe vectors, and extend to finite-dimensional g-modules where monodromy restrictions yield a precise oper-theoretic description. The work also analyzes degeneracies and degenerate opers, outlining geometric approaches toward completeness and potential Langlands-analytic proofs of the Bethe Ansatz correspondence for general g.

Abstract

This is a review of our previous works (some of them joint with B. Feigin and N. Reshetikhin) on the Gaudin model and opers. We define a commutative subalgebra in the tensor power of the universal enveloping algebra of a simple Lie algebra g. This algebra includes the hamiltonians of the Gaudin model, hence we call it the Gaudin algebra. It is constructed as a quotient of the center of the completed enveloping algebra of the affine Kac-Moody algebra g^ at the critical level. We identify the spectrum of the Gaudin algebra with the space of opers associated to the Langlands dual Lie algebra of g on the projective line with regular singularities at the marked points. Next, we recall the construction of the eigenvectors of the Gaudin algebra using the Wakimoto modules over g^ of critical level. The Wakimoto modules are naturally parameterized by Miura opers (or, equivalently, Cartan connections), and the action of the center on them is given by the Miura transformation. This allows us to relate solutions of the Bethe Ansatz equations to Miura opers and ultimately to the flag varieties associated to the Langlands dual Lie algebra of g.

Gaudin model and opers

TL;DR

This survey blends representation theory, integrable systems, and the geometric Langlands program to connect Gaudin spectra with ^L G-opers on the projective line. The authors describe the Gaudin algebra as a center-quotient at the critical level, whose spectrum is realized as spaces of opers with prescribed singularities, and develop a Wakimoto-module framework to construct eigenvectors via the Bethe Ansatz. Central to the narrative are Miura opers and Cartan connections, which provide a concrete parametrization (via flag varieties) of eigenvalues and Bethe vectors, and extend to finite-dimensional g-modules where monodromy restrictions yield a precise oper-theoretic description. The work also analyzes degeneracies and degenerate opers, outlining geometric approaches toward completeness and potential Langlands-analytic proofs of the Bethe Ansatz correspondence for general g.

Abstract

This is a review of our previous works (some of them joint with B. Feigin and N. Reshetikhin) on the Gaudin model and opers. We define a commutative subalgebra in the tensor power of the universal enveloping algebra of a simple Lie algebra g. This algebra includes the hamiltonians of the Gaudin model, hence we call it the Gaudin algebra. It is constructed as a quotient of the center of the completed enveloping algebra of the affine Kac-Moody algebra g^ at the critical level. We identify the spectrum of the Gaudin algebra with the space of opers associated to the Langlands dual Lie algebra of g on the projective line with regular singularities at the marked points. Next, we recall the construction of the eigenvectors of the Gaudin algebra using the Wakimoto modules over g^ of critical level. The Wakimoto modules are naturally parameterized by Miura opers (or, equivalently, Cartan connections), and the action of the center on them is given by the Miura transformation. This allows us to relate solutions of the Bethe Ansatz equations to Miura opers and ultimately to the flag varieties associated to the Langlands dual Lie algebra of g.

Paper Structure

This paper contains 25 sections, 35 theorems, 242 equations.

Table of Contents

  1. Opers
  2. Definition of opers
  3. $PGL_2$-opers
  4. Opers with regular singularities
  5. The Gaudin model
  6. The definition of the Gaudin model
  7. Gaudin model and coinvariants
  8. The center of ${\mathbb V}_0$ and $^L G$-opers
  9. Example
  10. The Gaudin algebra
  11. Miura opers and Cartan connections
  12. Miura opers and flag manifolds
  13. Cartan connections
  14. Singularities of Cartan connections
  15. Wakimoto modules and Bethe Ansatz
  16. ...and 10 more sections

Key Result

Lemma 1.1

The gauge action of $N(R)$ on $\widetilde{\operatorname{Op}}_G(\operatorname{Spec} R)$ is free, and each gauge equivalence class contains a unique operator of the form $\nabla = \partial_t + p_{-1} + {\mathbf v}(t)$, where ${\mathbf v}(t) \in V_{\operatorname{can}}(R)$, so that we can write

Theorems & Definitions (41)

  • Lemma 1.1
  • Lemma 1.2: F:opers, Lemma 2.4
  • Lemma 2.1
  • Proposition 2.2: FFR, Prop. 1
  • Proposition 2.3: FFR, Prop. 2
  • Theorem 2.4: FF:gdF:wak
  • Theorem 2.5: F:wak
  • Theorem 2.6
  • Theorem 2.7
  • proof
  • ...and 31 more