On the Quantum Moduli Space of N=2 Supersymmetric Gauge Theories

TL;DR

The paper constructs quantum moduli spaces for $N=2$ supersymmetric $SO(N_c)$ gauge theories with $N_f$ vector flavors by formulating families of hyperelliptic curves constrained by $R$-symmetry, instanton corrections, and classical singularities. It derives exact moduli-space metrics and the BPS spectrum, validating the curves with residue calculations and weak-coupling monodromies, and demonstrates a case split: for $N_f\mu<h-l$ the curves are fully determined, while for $N_f\ge N_c-l-2$ they admit mass- and moduli-dependent terms constrained by residues and symmetry. The results suggest an underlying integrable structure and offer a path to generalizing the curves to other gauge groups in line with Martinec–Warner’s spectral-curve program. These findings provide exact, nonperturbative insights into the Coulomb branch dynamics of four-dimensional ${N}=2$ gauge theories and establish a framework for connecting Seiberg–Witten theory to broader groups and representations.

Abstract

Families of hyper-elliptic curves which describe the quantum moduli spaces of vacua of $N=2$ supersymmetric $SO(N_c)$ gauge theories coupled to $N_f$ flavors of quarks in the vector representation are constructed. The quantum moduli spaces for $2N_f < N_c-1$ are determined completely by imposing $R$-symmetry, instanton corrections and the proper classical singularity structure. These curves are verified by residue calculations. The quantum moduli spaces for $2N_f\geq N_c-1$ theories are parameterized and their general structure is worked out using residue calculations. The exact metrics on the quantum moduli spaces as well as the exact spectrum of stable massive states are derived. The results presented here together with recent results of Martinec and Warner provide a natural conjecture for the form of the curves for the other gauge groups.