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

الأصل · EN

We consider N=1 supersymmetric U(N) gauge theories with Zₖ symmetric tree-level superpotentials W for an adjoint chiral multiplet. We show that (for integer 2N/k) this Zₖ symmetry survives in the quantum effective theory as a corresponding symmetry of the effective superpotential Wₑff(Sᵢ) under permutations of the Sᵢ. For W(x)=ʷ(h(x)) with h(x)=xᵏ, this allows us to express the prepotential F₀ and effective superpotential Wₑff on certain submanifolds of the moduli space in terms of an ᶠ₀ and ʷₑff of a different theory with tree-level superpotential ʷ. In particular, if the Zₖ symmetric polynomial W(x) is of degree 2k, then ʷ is gaussian and we obtain very explicit formulae for F₀ and Wₑff. Moreover, in this case, every vacuum of the effective Veneziano-Yankielowicz superpotential ʷₑff is shown to give rise to a vacuum of Wₑff. Somewhat surprisingly, at the level of the prepotential F₀(Sᵢ) 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.