Quartic Hamiltonians, and higher Hamiltonians at next-to-leading order, for the affine $\mathfrak{sl}_2$ Gaudin model

TL;DR

This paper constructs and verifies the commutation relations of higher-order Hamiltonians, including quartic and next-to-leading-order terms, for the affine sl2 Gaudin model, advancing understanding of its integrable structure.

Contribution

It introduces a systematic method to construct higher local Hamiltonians for the affine sl2 Gaudin model and proves their mutual commutativity, including in a semi-classical limit.

Findings

Quartic Hamiltonians commute with quadratic ones.

Higher local Hamiltonians form a commuting hierarchy.

Semi-classical limit allows explicit construction of higher Hamiltonians.

Abstract

In this work we will use a general procedure to construct higher local Hamiltonians for the affine $\mathfrak{sl}_2$ Gaudin model. We focus on the first non-trivial example, the quartic Hamiltonians. We show by direct calculation that the quartic Hamiltonians commute amongst themselves and with the quadratic Hamiltonians which define the model. We go on to introduce a certain next-to-leading-order semi-classical limit of the model. In this limit, we are able to write down the full hierarchy of higher local Hamiltonians and prove that they commute.