On the Gaudin model of type G$_2$

TL;DR

This paper explores the Gaudin model of type G2, providing explicit solutions to Bethe ansatz equations, proving completeness in certain cases, and establishing a geometric correspondence with self-self-dual polynomial spaces.

Contribution

It introduces explicit Bethe ansatz solutions for G2, proves their completeness in specific tensor products, and links the spectrum to self-self-dual Grassmannian structures.

Findings

Explicit formulas for Bethe ansatz solutions

Completeness of Bethe ansatz in certain tensor products

Bijection between spectrum points and self-self-dual spaces

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.