Relating prepotentials and quantum vacua of N=1 gauge theories with different tree-level superpotentials

We relate the prepotentials F_0 and effective superpotentials W_eff of N=1 supersymmetric U(N) gauge theories with different tree-level superpotentials W and ^W for an adjoint chiral multiplet. For W(x) = ^W(h(x)) we show how F_0 and W_eff on certain submanifolds of the moduli space can be expressed in terms of ^F_0 and ^W_eff. In particular, if W is a Z_k symmetric polynomial of degree 2k, ^W is gaussian and we obtain very explicit formulae for F_0 and W_eff. Moreover, in this case, every vacuum of the effective Veneziano-Yankielowicz superpotential ^W_eff is shown to give rise to a vacuum of W_eff. While W_eff has a permutation symmetry which is the quantum counterpart of the classical Z_k symmetry, for F_0 this quantum symmetry is anomalous if k>2