A Tale of Two Orbits: Non-Simply Laced Mirror

TL;DR

This paper constructs a 3D $ abla=4$ gauge theory whose Higgs branch matches a symplectic singularity linked to non-simply laced Lie algebras, and proposes a mirror dual magnetic quiver with matching Coulomb branch.

Contribution

It introduces a novel gauge theory and a non-simply laced magnetic quiver that realize and mirror a specific symplectic singularity, expanding understanding of mirror symmetry in non-simply laced cases.

Findings

Hilbert series computed for the Higgs branch.

Stratification of the singularity determined.

Mirror dual magnetic quiver proposed and validated.

Abstract

A three-dimensional $\mathcal{N}=4$ gauge theory is constructed whose Higgs branch realizes the affine closure of the cotangent bundle of the minimal nilpotent orbit of $\mathfrak{sl}_n$. This space is a symplectic singularity recently identified by Fu and Liu as a $\mathrm{U}(1)$ hyperk\"ahler quotient of the closure of the minimal nilpotent orbit of $\mathfrak{so}_{2n+2}$. The theory arises by gauging an $\mathrm{SO}(2)\cong\mathrm{U}(1)$ subgroup of the flavour symmetry of $\mathrm{SU}(2)$ SQCD with $n+1$ flavours. The Hilbert series is computed and the stratification is determined. A non-simply laced magnetic quiver is proposed whose Coulomb branch reproduces the same singularity. Evidence is thereby provided for a mirror pair involving a non-simply laced quiver, further tested through quiver subtraction and Hasse diagram inversion. A related $\mathbb{Z}_2$ quotient of the magnetic lattice is also analysed, and the exceptional behaviour in the case $n=2$, where $A_1 \cong C_1$, is explained. This construction provides a concrete example in which the Higgs-branch structure associated with a non-simply laced magnetic quiver can be inferred and validated through its mirror dual.