Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections I. The simply-laced case

TL;DR

This paper explores the connection between differential equations and integrable models for simply-laced Lie algebras, establishing that spectral determinants obey Bethe Ansatz equations through the analysis of subdominant solutions.

Contribution

It proves that subdominant solutions satisfy the $ ext{ extPsi}$-system and that spectral determinants follow Bethe Ansatz equations in the simply-laced case.

Findings

Subdominant solutions satisfy the $ ext{ extPsi}$-system.

Spectral determinants obey Bethe Ansatz equations.

Results extend the ODE/IM correspondence to simply-laced Lie algebras.

Abstract

We study the ODE/IM correspondence for ODE associated to $\hat{\mathfrak g}$-valued connections, for a simply-laced Lie algebra $\mathfrak g$. We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called $\Psi$-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.