The three-point Gaudin model and branched coverings of the Riemann sphere

TL;DR

This paper investigates the geometric structure of the three-point quantum -Gaudin model by analyzing algebraic curves arising from the joint spectrum of Gaudin Hamiltonians as branched coverings of the Riemann sphere.

Contribution

It provides a novel algebraic and geometric analysis of Gaudin model spectra as branched coverings, with explicit determinant equations for the spectral curves.

Findings

Spectral curves are described as determinants of tridiagonal matrices.

The compactification of the parameter space is the Riemann sphere.

Results reveal geometric properties of Gaudin coverings.

Abstract

We study the three-point quantum $\mathfrak{sl_2}$-Gaudin model. In this case the compactification of the parameter space is $\overline{M_{0,4}(\mathbb{C})}$, which is the Riemann sphere. We analyze sphere coverings by the joint spectrum of the Gaudin Hamiltonians treating them as algebraic curves. We write equations of these curves as determinants of tridiagonal matrices and deduce some consequences regarding the geometric structure of the Gaudin coverings.