Instanton Effects in Three-Dimensional Supersymmetric Gauge Theories with Matter

TL;DR

This work analyzes instanton effects in three-dimensional SU(2) gauge theories with N=2 and N=4 supersymmetry, computing perturbative and nonperturbative contributions to the Coulomb branch. Using one-loop renormalization and a careful instanton calculus, the authors verify Seiberg–Witten proposals: the N=4 Coulomb branch with one massless flavor is the double cover of the Atiyah–Hitchin monopole moduli space, while a hypermultiplet mass deforms this space in line with Dancer’s construction. For N=2 theories, they determine the one-loop metric and complex structure and derive the one-instanton superpotential, showing holomorphy in the one-loop-corrected complex coordinate Z and confirming expected RG flows via mass decoupling. Collectively, the results provide explicit semiclassical checks of symmetry- and holomorphy-based predictions and illuminate the structure of the 3D Coulomb branch across different amounts of supersymmetry and matter content.

Abstract

Using standard field theory techniques we compute perturbative and instanton contributions to the Coulomb branch of three-dimensional supersymmetric QCD with N=2 and N=4 supersymmetry and gauge group SU(2). For the N=4 theory with one massless flavor, we confirm the proposal of Seiberg and Witten that the Coulomb branch is the double-cover of the centered moduli space of two BPS monopoles constructed by Atiyah and Hitchin. Introducing a hypermultiplet mass term, we show that the asymptotic metric on the Coulomb branch coincides with the metric on Dancer's deformation of the monopole moduli space. For the N=2 theory with $N_f$ flavors, we compute the one-loop corrections to the metric and complex structure on the Coulomb branch. We then determine the superpotential including one-loop effects around the instanton background. These calculations provide an explicit check of several results previously obtained by symmetry and holomorphy arguments.