On the Quantum Moduli Space of Vacua of $N=2$ Supersymmetric $SU(N_c)$ Gauge Theories

TL;DR

The paper develops a systematic method to construct exact quantum moduli spaces for four-dimensional $\mathcal{N}=2$ $SU(N_c)$ gauge theories with $N_f$ fundamental flavors by associating them with families of hyper-elliptic curves constrained by $R$-symmetry, classical singularities, and instanton corrections. It derives explicit curve forms for the regimes $N_f<N_c$, $N_f=N_c$, and $N_f> N_c$, and supplements them with weak-coupling monodromy checks and detailed $N_c=2$ and $N_c=3$ examples, yielding the exact metrics and BPS spectra. For $N_f< N_c$ the moduli space is fully determined; for $N_f\ge N_c$ a two-parameter (or more generally constrained) family is obtained, with global symmetry guiding the remaining undetermined constants. A novel symmetry exchanging a moduli parameter and a bare mass is revealed, and the methods are shown to extend to $N=1$ theories in the Coulomb phase, hinting at deep connections with integrable systems.

Abstract

We construct families of hyper-elliptic curves which describe the quantum moduli spaces of vacua of $N=2$ supersymmetric $SU(N_c)$ gauge theories coupled to $N_f$ flavors of quarks in the fundamental representation. The quantum moduli spaces for $N_f < N_c$ are determined completely by imposing $R$-symmetry, instanton corrections and the proper classical singularity structure. These curves are verified by residue and weak coupling monodromy calculations. The quantum moduli spaces for $N_f\geq N_c$ theories are parameterized and their general structure is worked out using residue calculations. Global symmetry considerations suggest a complete description of them. The results are supported by weak coupling monodromy calculations. The exact metrics on the quantum moduli spaces as well as the exact spectrum of stable massive states are derived. We find an example of a novel symmetry of a quantum moduli space: Invariance under the exchange of a moduli parameter and the bare mass. We apply our method for the construction of the quantum moduli spaces of vacua of $N=1$ supersymmetric theories in the coulomb phase.