On the relation between effective supersymmetric actions in different dimensions

We make two remarks: (i) Renormalization of the effective charge in a four-dimensional (supersymmetric) gauge theory is determined by the same graphs and is rigidly connected to the renormalization of the metric on the moduli space of the classical vacua of the corresponding reduced quantum-mechanical system. Supersymmetry provides constraints for possible modifications of the metric, and this gives us a simple proof of nonrenormalization theorems for the original four-dimensional theory. (ii) We establish a nontrivial relationship between the effective (0+1)-dimensional and (1+1)-dimensional Lagrangians. (The latter represent conventional Kaehlerian σ models.)