Mirror Symmetry in Three-Dimensional Gauge Theories, SL(2,Z) and D-Brane Moduli Spaces

TL;DR

This paper develops a comprehensive brane-based realization of mirror symmetry for three-dimensional N=4 gauge theories, tying dual pairs to SL(2,Z) transformations of Type IIB brane configurations and to Kronheimer–Nakajima quiver varieties. It provides explicit constructions of A- and B-models, derives precise mirror maps between FI parameters and masses, and analyzes instanton corrections via open string effects and M-theory on Calabi–Yau four-folds. A general framework for mixed branches, a criterion for complete Higgsing, and a robust Abelian dual-pairs toolkit are presented, with field-theory and string-theory proofs that reinforce the duality and illuminate the geometric structure of moduli spaces. The work connects 3D mirror symmetry with level-rank duality of affine Lie algebras, and offers a unified picture of phase transitions in brane moduli spaces as a bridge between distinct gauge theories, including explicit mappings for general charges and implications for instanton corrections.

Abstract

We construct intersecting D-brane configurations that encode the gauge groups and field content of dual N=4 supersymmetric gauge theories in three dimensions. The duality which exchanges the Coulomb and Higgs branches and the Fayet-Iliopoulos and mass parameters is derived from the SL(2,Z) symmetry of the type IIB string. Using the D-brane configurations we construct explicitly this mirror map between the dual theories and study the instanton corrections in the D-brane worldvolume theory via open string instantons. A general procedure to obtain mirror pairs is presented and illustrated. We encounter transitions among different field theories that correspond to smooth movements in the D-brane moduli space. We discuss the relation between the duality of the gauge theories and the level-rank duality of affine Lie algebras. Examples of other dual theories are presented and explained via T-duality and extremal transitions in type II string compactifications. Finally we discuss a second way to study instanton corrections in the gauge theory, by wrapping five-branes around six-cycles in M-theory compactified on a Calabi-Yau 4-fold.

Figures (16)

• Figure 1: The D-brane configuration of the A-model is plotted on the left. The circles consist of $k$ Dirichlet 3-branes, the $n$ dashed lines are Dirichlet 5-branes and the solid line is an NS 5-brane. The corresponding quiver diagram is plotted on the right.
• Figure 2: The D-brane configuration of the B-model is plotted on the left, and the corresponding quiver diagram is plotted on the right.
• Figure 3: $k$ D2-brane probes near $n$ D6-branes in type I'
• Figure 4: The image in space-time of open string instantons. There are no instanton corrections from figure 4a, but there are from figure 4b.
• Figure 5: An $U(k)$ gauge theory with $n$ fundamentals and no adjoint hypermultiplet is plotted in its Coulomb branch on the left and in its Higgs branch on the right.