Remarks on the analytic structure of supersymmetric effective actions

TL;DR

This paper explores the algebraic curve structure of the effective superpotential in N=1 supersymmetric gauge theories, linking the degree of the curve to the number of semiclassical branches, and verifies this in specific models.

Contribution

It proposes a novel relation between the algebraic curve degree and gauge theory branches, verified in models with multiple adjoints and superpotentials.

Findings

The curve degree matches the number of semiclassical branches.

Different sheets of the curve correspond to distinct classical vacua.

Continuous interpolation between branches is possible via coupling variations.

Abstract

We study the effective superpotential of N=1 supersymmetric gauge theories with a mass gap, whose analytic properties are encoded in an algebraic curve. We propose that the degree of the curve equals the number of semiclassical branches of the gauge theory. This is true for supersymmetric QCD with one adjoint and polynomial superpotential, where the two sheets of its hyperelliptic curve correspond to the gauge theory pseudoconfining and higgs branches. We verify this proposal in the new case of supersymmetric QCD with two adjoints and superpotential V(X)+XY^2, which is the confining phase deformation of the D_{n+2} SCFT. This theory has three kinds of classical vacua and its curve is cubic. Each of the three sheets of the curve corresponds to one of the three semiclassical branches of the gauge theory. We show that one can continuously interpolate between these branches by varying the couplings along the moduli space.