Branes and Mirror Symmetry in N=2 Supersymmetric Gauge Theories in Three Dimensions

TL;DR

This work uses type IIB brane configurations and $SL(2,\mathbb{Z})$ duality to construct and test mirror symmetry between 3D $N=2$ gauge theories, mapping Higgs and Coulomb branches as well as FI and mass parameters across abelian and non-abelian cases. The authors show that, in Abelian theories, the one-loop Coulomb-branch metric of the A-model matches the classical Higgs branch of the B-model (and vice versa), up to parameter identifications like $\zeta^{\mathbf{r}}=m_q^{\mathbf{r}}$; they also analyze how the two R-symmetry actions are exchanged under mirror symmetry. In non-Abelian setups, a richer quiver structure emerges on the B-model side, with the mirror map extending to differences of mass parameters and FI terms, though a complete non-Abelian analysis (including nonperturbative superpotentials) remains incomplete. The paper further discusses breaking $N=4$ to $N=2$ by superpotentials and explores how open D-string instantons generate nonperturbative superpotentials, providing a string-theoretic mechanism for holomorphic corrections. Finally, it comments on the limits of three-dimensional dualities descended from four-dimensional ones, highlighting the role of monopole instantons and the need for brane-based instanton analyses to understand $N=2$ dynamics in three dimensions.

Abstract

We use brane configurations and SL(2,Z) symmetry of the type IIB string to construct mirror N=2 supersymmetric gauge theories in three dimensions. The mirror map exchanges Higgs and Coulomb branches, Fayet-Iliopoulos and mass parameters and U(1)R symmetries. Some quantities that are determined at the quantum level in one theory are determined at the classical level of the mirror. One such example is the complex structure of the Coulomb branch of one theory which is determined quantum mechanically. It is mapped to the complex structure of the Higgs branch of the mirror theory, which is determined classically. We study the generation of N=2 superpotentials by open D-string instantons in the brane configurations.

Figures (10)

• Figure 1: Brane configuration for an $N=2$$U(N_c)$ gauge theory with $N_f$ flavors in three dimensions.
• Figure 2: Brane configuration for an $N=2$$U(N_c)$ gauge theory with $N_f$ flavors and three adjoints in three dimensions.
• Figure 3: $SL(2,Z)$ of the brane configuration of $N=2$$U(1)$ gauge theory with $N_f$ flavours.
• Figure 4: Quiver like diagram for the B-model. The dashed circle correspond to the meson.
• Figure 5: The Higgs branch of the B-model.