Classical spin chains and exact three-dimensional superpotentials Original Research Article

We study exact effective superpotentials of four-dimensional image supersymmetric gauge theories with gauge group image and various amounts of fundamental matter on image, broken to image by turning on a classical superpotential for the adjoint scalar. On general grounds these superpotentials can easily be constructed once we identify a suitable set of coordinates on the moduli space of the gauge theory. These coordinates have been conjectured to be the phase space variables of the classical integrable system which underlies the image gauge theory. The sought low energy effective superpotential can then be constructed from the conserved quantities in the integrable system. For the gauge theory under study these integrable systems are degenerations of the classical, inhomogeneous, periodic image spin chain. Ambiguities in the degeneration provide multiple coordinate patches on the gauge theory moduli space. By studying the vacua of these superpotentials in several examples we find that the spin chain provides coordinate patches that parametrize holomorphically the part of the gauge theory moduli space which is connected to the electric (as opposed to magnetic or baryonic) Higgs and Coulomb branch vacua. The baryonic branch root is on the edge of some coordinate patches. As a product of our analysis all maximally confining (non-baryonic) Seiberg–Witten curve factorizations for image are obtained, explicit up to one constraint for equal mass flavors and up to two constraints for unequal mass flavors. Gauge theory addition and multiplication maps are shown to have a natural counterpart in this construction. Furthermore it is shown how to integrate in the meson fields in this formulation in order to obtain three- and four-dimensional Affleck–Dine–Seiberg-like superpotentials.