Relating prepotentials and quantum vacua of N=1 gauge theories with   different tree-level superpotentials
  We consider N=1 supersymmetric U(N) gauge theories with Z_k symmetric tree-level superpotentials W for an adjoint chiral multiplet. We show that (for integer 2N/k) this Z_k symmetry survives in the quantum effective theory as a corresponding symmetry of the effective superpotential W_eff(S_i) under permutations of the S_i. For W(x)=^W(h(x)) with h(x)=x^k, this allows us to express the prepotential F_0 and effective superpotential W_eff on certain submanifolds of the moduli space in terms of an ^F_0 and ^W_eff of a different theory with tree-level superpotential ^W. In particular, if the Z_k symmetric polynomial W(x) is of degree 2k, then ^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. Somewhat surprisingly, at the level of the prepotential F_0(S_i) the permutation symmetry only holds for k=2, while it is anomalous for k>2 due to subtleties related to the non-compact period integrals. Some of these results are also extended to general polynomial relations h(x) between the tree-level superpotentials. 
Adel Bilal and Steffen Metzger