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On the Gaudin model of type G$_2$

Kang Lu, E. Mukhin

TL;DR

This work extends Gaudin-model analysis to the exceptional Lie type $G_2$ by constructing the Bethe algebra $\\mathcal{B}$ and a universal differential operator $\\mathcal{D}^{\\mathcal{B}}$, linking eigenvectors to scalar differential operators whose kernels are self-self-dual spaces. A central achievement is proving completeness of the Bethe ansatz for tensor products of the form $V_{\\lambda} \otimes V_{\\omega_2}^{\otimes k}$ with generic parameters, and deriving explicit two-point Bethe solutions that generate all eigenvectors via reproduction procedures. The authors then build a rich geometric picture by identifying Gaudin eigenvalues with self-self-dual spaces, organizing them into the self-self-dual Grassmannian $\\mathbb{S}\mathrm{Gr}_d$ and its $\\mathfrak{g}$-stratification, with strata labeled by unordered weight data and nonnegative integers. This stratification is connected to Schubert calculus, the $^t\mathfrak{g}$-opers (via Miura transformations), and the Wronski map, yielding a precise description of closures and degrees of the covering maps that relate strata to spectral data. Overall, the paper provides a deep synthesis of representation theory, algebraic geometry, and differential-operator methods to describe the Gaudin spectrum for $G_2$ and its geometric stratification.

Abstract

We derive a number of results related to the Gaudin model associated to the simple Lie algebra of type G$_2$. We compute explicit formulas for solutions of the Bethe ansatz equations associated to the tensor product of an arbitrary finite-dimensional irreducible module and the vector representation. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We show that the points of the spectrum of the Gaudin model in type G$_2$ are in a bijective correspondence with self-self-dual spaces of polynomials. We study the set of all self-self-dual spaces - the self-self-dual Grassmannian. We establish a stratification of the self-self-dual Grassmannian with the strata labeled by unordered sets of dominant integral weights and unordered sets of nonnegative integers, satisfying certain explicit conditions. We describe closures of the strata in terms of representation theory.

On the Gaudin model of type G$_2$

TL;DR

This work extends Gaudin-model analysis to the exceptional Lie type by constructing the Bethe algebra and a universal differential operator , linking eigenvectors to scalar differential operators whose kernels are self-self-dual spaces. A central achievement is proving completeness of the Bethe ansatz for tensor products of the form with generic parameters, and deriving explicit two-point Bethe solutions that generate all eigenvectors via reproduction procedures. The authors then build a rich geometric picture by identifying Gaudin eigenvalues with self-self-dual spaces, organizing them into the self-self-dual Grassmannian and its -stratification, with strata labeled by unordered weight data and nonnegative integers. This stratification is connected to Schubert calculus, the -opers (via Miura transformations), and the Wronski map, yielding a precise description of closures and degrees of the covering maps that relate strata to spectral data. Overall, the paper provides a deep synthesis of representation theory, algebraic geometry, and differential-operator methods to describe the Gaudin spectrum for and its geometric stratification.

Abstract

We derive a number of results related to the Gaudin model associated to the simple Lie algebra of type G. We compute explicit formulas for solutions of the Bethe ansatz equations associated to the tensor product of an arbitrary finite-dimensional irreducible module and the vector representation. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We show that the points of the spectrum of the Gaudin model in type G are in a bijective correspondence with self-self-dual spaces of polynomials. We study the set of all self-self-dual spaces - the self-self-dual Grassmannian. We establish a stratification of the self-self-dual Grassmannian with the strata labeled by unordered sets of dominant integral weights and unordered sets of nonnegative integers, satisfying certain explicit conditions. We describe closures of the strata in terms of representation theory.

Paper Structure

This paper contains 33 sections, 32 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. The Lie algebra of type G$_2$
  3. Simple Lie algebra of type G$_2$
  4. The matrix $G$
  5. Current algebra $\mathfrak{g}[t]$
  6. Gaudin Model
  7. Bethe algebra
  8. Universal differential operator
  9. Quadratic Gaudin Hamiltonians
  10. Completeness of Bethe ansatz of $V_{\lambda}\otimes V_{\omega_2}^{\otimes k}$
  11. Bethe ansatz
  12. Reproduction procedure
  13. Differential operator associated with a Bethe vector
  14. Parameterization of solutions
  15. The explicit $2$-point solutions of the Bethe ansatz equation
  16. ...and 18 more sections

Key Result

Lemma 3.1

We have an isomorphism of vector spaces: given by the projection to a lowest weight vector in $V_{\lambda}$. The map $\pi$ is an isomorphism of $\mathcal{B}$-modules $(M'\otimes V_{\lambda}(\infty))^{\mathfrak{g}}\to (M')^{\mathrm{sing}}_{\lambda}$. ∎

Theorems & Definitions (55)

  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.3
  • Theorem 4.1: RV
  • Theorem 4.2: FFRe
  • Theorem 4.3: V
  • Theorem 4.4: MV4
  • Proposition 4.5
  • proof
  • Lemma 4.6
  • ...and 45 more