A Bound on 3d Mirror Pairs

TL;DR

This paper explores the structure of 3d mirror symmetry in gauge theories, expanding known dual pairs and proposing that many Lagrangian theories lack Lagrangian mirrors, especially those with exceptional affine Dynkin subquivers.

Contribution

It introduces methods to generate new 3d mirror pairs by replacing unitary groups with special unitary groups and conjectures limitations on Lagrangian mirrors involving affine Dynkin diagrams.

Findings

Expanded the family of known 3d mirror pairs using brane lockings and magnetic quivers.

Most Lagrangian 3d mirror pairs are unlikely to have Lagrangian duals, especially with certain Dynkin subquivers.

Abstract

A distinctive duality present in 3d $\mathcal{N}=4$ theories is the 3d mirror symmetry. Under this duality, the Coulomb (Higgs) branch of one theory corresponds to the Higgs (Coulomb) branch of its mirror dual. This paper is divided into two parts. In the first part, we examine quiver gauge theories constructed from unitary gauge groups arranged in the shape of ABCD-type Dynkin diagrams. This is arguably the largest family of quivers in the literature with known 3d mirror pairs. Using brane lockings and magnetic quivers, we show how this family can be vastly expanded by replacing any number of the unitary gauge groups with special unitary gauge groups and finding the mirror pairs. In the second part, we argue that in the landscape of 3d mirror pairs, most Lagrangian (quiver gauge theories) will not have 3d mirror duals that are also Lagrangian (quiver gauge theories). For unitary quiver gauge theories, we conjecture that any quiver with an exceptional affine Dynkin diagram as a subquiver cannot have a Lagrangian (quiver gauge theory) mirror.