Integrable Models, SUSY Gauge Theories, and String Theory

TL;DR

This paper explores the deep connections between dualities in N=2 supersymmetric gauge theories, integrable models, and string theory, revealing how various models relate to the quantum moduli space and hyperelliptic curves.

Contribution

It establishes explicit links between integrable models and SUSY gauge theories, and derives the stringy origin of integrability via exact nonperturbative limits of string compactifications.

Findings

Relation between integrable models and quantum moduli space of SUSY theories

Identification of hyperelliptic curves with integrable systems like Toda and Calogero models

Derivation of hyperelliptic curves from string compactification limits

Abstract

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 N_f = 0,1,2, where a system with seven dimensional phase space has the relevant hyperelliptic curve appear in the Painlev\'e 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 N_f=1 hypermultiplet.