Monopole Operators and Mirror Symmetry in Three Dimensions

TL;DR

The paper develops a controlled framework to study monopole operators in the IR of 3d $N=2$ and $N=4$ SQED via radial quantization on ${ m S}^2 imes{ m R}$ and a large-$N_f$ expansion. It shows that these monopole operators organize into short superconformal multiplets with quantum numbers predicting and confirming 3d mirror symmetry, including explicit identifications $V_+=q_1 o q_{N_f}$ and $V_-= ilde q_1 o ilde q_{N_f}$ and a chiral-ring relation $V_+V_- o ext{Phi}^{N_f}$ in the $N=4$ case; in the special case $N_f=1$, the monopole sector is argued to be free, providing a concrete realization of the basic duality. The work further proves non-renormalization of vortex-related charges via index theory and outlines a path to derive the basic $N=4$ mirror symmetry beyond large $N_f$, while acknowledging open problems such as extending to non-Abelian theories and constructing monopoles directly in the Hamiltonian formalism. Together, these results strengthen the evidence for 3d mirror symmetry and offer a framework toward a more rigorous, general understanding of dualities in low-dimensional gauge theories.

Abstract

We study vortex-creating, or monopole, operators in 3d CFTs which are the infrared limit of N=2 and N=4 supersymmetric QEDs in three dimensions. Using large-Nf expansion, we construct monopole operators which are primaries of short representations of the superconformal algebra. Mirror symmetry in three dimensions makes a number of predictions about such operators, and our results confirm these predictions. Furthermore, we argue that some of our large-Nf results are exact. This implies, in particular, that certain monopole operators in N=4 d=3 SQED with Nf=1 are free fields. This amounts to a proof of 3d mirror symmetry in a special case.