Exact Results for SupersymmetricσModels

We show that the metric and Berry's curvature for the ground states of $N=2\mathrm{}$ supersymmetric $\sigma$ models can be computed exactly as one varies the Kahler structure. For the case of ${\mathrm{CP}}^n$ these are related to special solutions of affine Toda equations. This allows us to extract exact results. We find that the ground-state metric is nonsingular as the size of the manifold shrinks to zero, suggesting that 2D quantum field theory makes sense even beyond zero radius. Thus it seems that manifolds with zero size are nonsingular as target spaces for string theory (even when they are not conformal).