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

TL;DR

This paper constructs and verifies hyper-elliptic curves describing the quantum moduli spaces of vacua in $N=2$ supersymmetric $SU(N_c)$ gauge theories with various flavors, revealing new symmetries and exact spectra.

Contribution

It provides a complete construction and analysis of the quantum moduli spaces for $N=2$ supersymmetric $SU(N_c)$ theories with fundamental flavors, including new symmetry insights.

Findings

Explicit hyper-elliptic curves for $N_f < N_c$ and $N_f \\geq N_c$ cases.

Verification of curves via residue and monodromy calculations.

Discovery of a novel symmetry involving exchange of a moduli parameter and the bare mass.

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.