ODE/IM correspondence and supersymmetric affine Toda field equations

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.