Integrable Models, Susy Gauge Theories and String Theory

We consider the close relation between duality in N = 2 SUSY gauge theories and integrable models. Various integrable models ranging from Toda lattices, Calogero models, spinning tops, and spin chains are related to the quantum moduli space of vacua of N = 2 SUSY gauge theories. In particular, SU(3) gauge theories with two flavors of massless quarks in the fundamental representation can be related to the spectral curve of the Goryachev-Chaplygin top, which is a Nahm’s equation in disguise. This can be generalized to the cases with massive quarks, and Nf = 0, 1, 2, where a system with seven dimensional phase space has the relevant hyperelliptic curve appear in the Painlevé test. To understand the stringy origin of the integrability of these theories we obtain exact nonperturbative point particle limit of type II string compactified on a Calabi-Yau manifold, which gives the hyperelliptic curve of SU(2) QCD with Nf = 1 hypermultiplet.