Moduli space singularities for $3d$ $\mathcal{N} = 4$ circular quiver gauge theories

TL;DR

This paper investigates the singularity structures of Coulomb and Higgs branches in 3d N=4 circular quiver gauge theories, using Kraft--Procesi transitions to classify and understand their complex geometric features.

Contribution

It introduces a comprehensive classification of singularities in circular quiver gauge theories using higher-level Hasse diagrams and extends known results from linear quivers to the circular case.

Findings

Full determination of singularity structures for a class of CQGTs.

Introduction of higher-level Hasse diagrams for compact representation.

Generalization of linear quiver results to circular quivers.

Abstract

The singularity structure of the Coulomb and Higgs branches of good $3d$ $\mathcal{N}=4$ circular quiver gauge theories (CQGTs) with unitary gauge groups is studied. The central method employed is the Kraft--Procesi transition. CQGTs are described as a generalisation of a class of linear quivers. This class degenerates into the familiar class $T_{\rho}^{\sigma}(SU(N))$ in the linear case, however the circular case does not have the degeneracy and so the class of CQGTs contains many more theories and much more structure. We describe a collection of good, unitary, CQGTs from which the entire class can be found using Kraft--Procesi transitions. The singularity structure of a general member of this collection is fully determined, encompassing the singularity structure of a generic CQGT. Higher-level Hasse diagrams are introduced in order to write the results compactly. In higher-level Hasse diagrams, single nodes represent lattices of nilpotent orbit Hasse diagrams and edges represent traversing structure between lattices. The results generalise the case of linear quiver moduli spaces which are known to be nilpotent varieties of $\mathfrak{sl}_n$.