Nilpotent orbit Coulomb branches of types AD

TL;DR

This paper introduces a new algebraic method for constructing Coulomb branch chiral rings of 3d N=4 theories, focusing on types A and D, and explicitly describes their global symmetries and deformations.

Contribution

It develops a novel approach combining operator counting and algebraic construction to explicitly build Coulomb branch rings for nilpotent orbit closures in types A and D.

Findings

Constructed Coulomb branch rings for all balanced A and D type quivers.

Explicitly identified complex mass deformations of nilpotent orbit Coulomb branches.

Provided a framework for potential generalization to non-simply laced quivers.

Abstract

We develop a new method for constructing $3d$ $\mathcal{N}=4$ Coulomb branch chiral rings in terms of gauge-invariant generators and relations while making the global symmetry manifest. Our examples generalise to all balanced quivers of type $A$ and $D$ whose Coulomb branches are closures of nilpotent orbits. This new approach is a synthesis of operator counting using Hilbert series and explicit algebraic construction introduced by Bullimore, Dimofte and Gaiotto with significant potential for further generalisation to other quivers, including non-simply laced. The method also identifies complex mass deformations of many Coulomb branches, providing an explicit construction for complex deformations of nilpotent orbits.