Cubic hypergeometric integrals of motion in affine Gaudin models

TL;DR

This paper constructs and analyzes cubic Hamiltonians in affine Gaudin models, demonstrating their commutation properties and expressing their eigenvalues via hypergeometric functions on affine opers.

Contribution

It introduces new cubic Hamiltonians for affine Gaudin models using hypergeometric integrals and proves their commutation and eigenvalue properties.

Findings

Cubic Hamiltonians commute with quadratic ones.

Eigenvalues are expressed as hypergeometric functions on affine opers.

Constructs a new class of integrals of motion for affine Gaudin models.

Abstract

We construct cubic Hamiltonians for quantum Gaudin models of affine types $\hat{\mathfrak{sl}}_M$. They are given by hypergeometric integrals of a form we recently conjectured in arXiv:1804.01480. We prove that they commute amongst themselves and with the quadratic Hamiltonians. We prove that their vacuum eigenvalues, and their eigenvalues for one Bethe root, are given by certain hypergeometric functions on a space of affine opers.