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

We relate the prepotentials F0 and effective superpotentials Weff of N = 1 supersymmetric U(N) gauge theories with different tree-level superpotentials W (Φ) and Ŵ (Φ) for an adjoint chiral multiplet. For W (x) = Ŵ (h(x)) we show how F0 and Weff on certain submanifolds of the moduli space can be expressed in terms of F̂0 and Ŵeff . In particular, if W (x) is a Zk symmetric polynomial of degree 2k, Ŵ is gaussian and we obtain very explicit formulae for F0 and Weff . Moreover, in this case, every vacuum of the effective Veneziano-Yankielowicz superpotential Ŵeff is shown to give rise to a vacuum of Weff . While Weff has a permutation symmetry which is the quantum counterpart of the classical Zk symmetry, for F0 this quantum symmetry is anomalous if k ≥ 3.