Integrable structure in supersymmetric gauge theories with massive hypermultiplets

TL;DR

This paper explores the integrable structures underlying the quantum moduli space of vacua in N=2 supersymmetric SU(N_c) gauge theories with massive hypermultiplets, revealing connections to spectral curves and integrable systems.

Contribution

It identifies the spectral curve of a specific integrable system with the moduli space of certain supersymmetric gauge theories, extending known results to massive quark cases.

Findings

Spectral curve matches the moduli space for N_c=3, N_f=0,1,2.

Integrable systems are constructed for theories with massive quarks.

Potential generalizations to other N_c and N_f values are discussed.

Abstract

We study the quantum moduli space of vacua of $N=2$ supersymmetric $SU(N_c)$ gauge theories coupled to $N_f$ flavors of quarks in the fundamental representation. We identify the moduli space of the $N_c = 3$ and $N_f=2$ massless case with the full spectral curve obtained from the Lax representation of the Goryachev-Chaplygin top. For the case with {\it massive} quarks, we present an integrable system where the corresponding hyperelliptic curve parametrizing the Laurent solution coincides with that of the moduli space of $N_{c}=3$ with $N_{f}=0, 1, 2$. We discuss possible generalizations of the integrable systems relevant to gauge theories with $N_c \neq 3 $ and general $N_f$.