Remarks on the analytic structure of supersymmetric effective actions

We study the effective superpotential of N=1 supersymmetric gauge theorieswith a mass gap, whose analytic properties are encoded in an algebraic curve.We propose that the degree of the curve equals the number of semiclassicalbranches of the gauge theory. This is true for supersymmetric QCD with oneadjoint and polynomial superpotential, where the two sheets of itshyperelliptic curve correspond to the gauge theory pseudoconfining and higgsbranches. We verify this proposal in the new case of supersymmetric QCD withtwo adjoints and superpotential V(X)+XY^2, which is the confining phasedeformation of the D_{n+2} SCFT. This theory has three kinds of classical vacuaand its curve is cubic. Each of the three sheets of the curve corresponds toone of the three semiclassical branches of the gauge theory. We show that onecan continuously interpolate between these branches by varying the couplingsalong the moduli space.Comment: 49 pages, 3 figures, harvmac; typos correcte