Frobenius-like structure in Gaudin model

TL;DR

This paper introduces a Frobenius-like structure for the $rak{sl}_2$ Gaudin model, linking potential functions to key algebraic forms and Hamiltonian actions, advancing the mathematical understanding of integrable systems.

Contribution

It presents a novel Frobenius-like structure for the Gaudin model, defining potential functions and relating them to the Shapovalov form and Hamiltonians.

Findings

Expressed the Shapovalov form via derivatives of the potential of the first kind.

Described Gaudin Hamiltonians in terms of derivatives of the potential of the second kind.

Established a Frobenius-like structure connecting algebraic forms and potentials.

Abstract

We introduce a Frobenius-like structure for the $\frak{sl}_2$ Gaudin model. Namely, we introduce potential functions of the first and second kind. We describe the Shapovalov form in terms of derivatives of the potential of the first kind and the action of Gaudin Hamiltonians in terms of derivatives of the potential of the second kind.