Finiteness of Ricci flat N=2 supersymmetric σ-models

We study the ultraviolet behavior of two dimensional supersymmetric non-linear σ-models with target space an arbitrary Kähler manifold M, so that the models are N=2 supersymmetric. We point out that these models have an additional fermionic axial symmetry if and only if the metric on M is Ricci flat. We show that the preservation of this symmetry in perturbation theory implies that both bare and renormalized metrics on M are Ricci flat. Combining this result with the constraint of N=2 supersymmetry requiring that all counter-terms to the metric beyond one-loop order be cohomologically trivial, we argue that N=2 models defined on Ricci flat Kähler manifolds are on-shell ultraviolet finite to all orders of perturbation theory.