Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras

TL;DR

This paper generalizes the link between differential equations and Bethe ansatz equations to models based on classical Lie algebras, introducing new pseudo-differential equations and exploring their spectral properties.

Contribution

It proposes new families of pseudo-differential equations related to classical Lie algebras and establishes their connection to Bethe ansatz solutions in integrable models.

Findings

New pseudo-differential equations for classical Lie algebras

Link between eigenvalue problems and Bethe ansatz

Observation of dualities in boundary conditions and dimensions

Abstract

The correspondence between ordinary differential equations and Bethe ansatz equations for integrable lattice models in their continuum limits is generalised to vertex models related to classical simple Lie algebras. New families of pseudo-differential equations are proposed, and a link between specific generalised eigenvalue problems for these equations and the Bethe ansatz is deduced. The pseudo-differential operators resemble in form the Miura-transformed Lax operators studied in work on generalised KdV equations, classical W-algebras and, more recently, in the context of the geometric Langlands correspondence. Negative-dimension and boundary-condition dualities are also observed.