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

TL;DR

This paper extends the ODE/IM correspondence to non-simply laced Lie algebras by analyzing meromorphic connections, constructing the $ ext{ extPsi}$-system, and deriving solutions to the Bethe Ansatz equations for the quantum $ ext{ extg}$-KdV model.

Contribution

It introduces a new approach for non-simply laced Lie algebras using meromorphic connections and constructs explicit solutions to the Bethe Ansatz equations.

Findings

Proved that spectral determinants satisfy Bethe Ansatz equations.

Constructed the $ ext{ extPsi}$-system for non-simply laced cases.

Developed explicit solutions for generalized Airy functions.

Abstract

We assess the ODE/IM correspondence for the quantum $\mathfrak{g}$-KdV model, for a non-simply laced Lie algebra $\mathfrak{g}$. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra ${\mathfrak{g}}^{(1)}$, and constructing the relevant $\Psi$-system among subdominant solutions. We then use the $\Psi$-system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum $\mathfrak{g}$-KdV model. We also consider generalized Airy functions for twisted Kac--Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.