Finiteness of nonlinear sigma-models on Ricci-Flat manifolds

It is shown that nonlinear sigma-models can be made finite in 2-2 is an element of dimensions if the background bare metric is chosen to be gij(φ)=Σ gij(k) is an element ofk, where gij(0) is Ricci-flat. This means that despite the appearance of non-zero β-functions in the standard approach, we can construct conformally invariant sigma-models with Ricci-flat metric. Thus the superstring compactification on Calabi-Yau manifolds becomes possible