Classical Spin Chains and Exact Three-dimensional Superpotentials

We study exact effective superpotentials of four-dimensional N = 2 supersymmetric gauge theories with gauge group U(N) and various amounts of fundamental matter on R3 × S1, broken to N = 1 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 N = 2 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 SL(2,C) 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) Higgs and Coulomb branch vacua. The baryonic branch root is on the edge of some coordinate patches. 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 and obtain three and four dimensional Affleck-Dine-Seiberglike superpotentials. [email protected] [email protected]