Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes
Jan de Boer, Kentaro Hori, Hirosi Ooguri, Yaron Oz
TL;DR
This work develops and tests mirror symmetry for three-dimensional $N=4$ gauge theories, mapping Coulomb and Higgs branches and exchanging mass and FI parameters across multiple quiver-based dual pairs. By embedding the theories into Kronheimer–Nakajima quiver varieties and connecting to D-brane string theory, the authors derive explicit mirror maps and demonstrate consistency through dimension counting, one-loop metrics, and Higgs-branch geometry, including Hilbert schemes and ALE/D-type spaces. They establish three main dual families—$U(k)$, $Sp(k)$, and $U(k)^n$—with detailed analyses of moduli spaces, quantum corrections, and brane pictures, and they discuss the moduli space of vacua without matter. The results illuminate nonperturbative phenomena in string compactifications, monopole corrections in three dimensions, and the geometry of small instantons, offering a comprehensive framework for future explorations of dual quiver theories and their holographic/stringy origins.
Abstract
We construct and analyze dual N=4 supersymmetric gauge theories in three dimensions with unitary and symplectic gauge groups. The gauge groups and the field content of the theories are encoded in quiver diagrams. The duality exchanges the Coulomb and Higgs branches and the Fayet-Iliopoulos and mass parameters. We analyze the classical and the quantum moduli spaces of the theories and construct an explicit mirror map between the mass parameters and the the Fayet-Iliopoulos parameters of the dual. The results generalize the relation between ALE spaces and moduli spaces of SU(n) and SO(2n) instantons. We interpret some of these results from the string theory viewpoint, for SU(n) by analyzing T-duality and extremal transitions in type II string compactifications, for SO(2n) by using D-branes as probes. Finally, we make a proposal for the moduli space of vacua of these theories in the absence of matter.