ODE/IM correspondence and supersymmetric affine Toda field equations

TL;DR

This paper explores the ODE/IM correspondence for supersymmetric affine Toda field equations linked to affine Lie superalgebras, revealing how linear problems reduce to ODEs and connecting to supersymmetric models.

Contribution

It extends the ODE/IM correspondence to affine Lie superalgebras with purely odd roots, including explicit ODEs for specific superalgebras like $osp(2,2)^{(2)}$.

Findings

Derived ODEs for supersymmetric affine Toda equations.

Connected the $osp(2,2)^{(2)}$ case to ${ m N}=1$ supersymmetric minimal models.

Generalized ODEs for classical affine Lie superalgebras.

Abstract

We study the linear differential system associated with the supersymmetric affine Toda field equations for affine Lie superalgebras, which has a purely odd simple root system. For an affine Lie algebra, the linear problem modified by conformal transformation leads to an ordinary differential equation (ODE) that provides the functional relations in the integrable models. This is known as the ODE/IM correspondence. For the affine Lie superalgebras, the linear equations modified by a superconformal transformation are shown to reduce to a couple of ODEs for each bosonic subalgebra. In particular, for $osp(2,2)^{(2)}$, the corresponding ODE becomes the second-order ODE with squared potential, which is related to the ${\cal N}=1$ supersymmetric minimal model via the ODE/IM correspondence. We also find ODEs for classical affine Lie superalgebras with purely odd simple root systems.