Hilbert Series and Complete-Intersection Structure of Coulomb Branches for Non-Maximal Nilpotent Orbits of $SL(N)$

TL;DR

This paper analyzes the algebraic structure of Coulomb branches for certain 3D supersymmetric gauge theories related to non-maximal nilpotent orbits of SL(N), revealing a uniform complete intersection pattern.

Contribution

It provides explicit Hilbert series computations for low N, uncovers a consistent pattern in generators and relations, and formulates conjectures for general N.

Findings

Coulomb branches are complete intersections in all examined cases.

Number of generators and relations follows a pattern dictated by the transpose partition.

Exactly N-1 relations appear, independent of the partition.

Abstract

We study the Coulomb branches of three-dimensional $\mathcal N=4$ quiver gauge theories of type $T_\rho(SU(N))$ associated with non-maximal nilpotent orbits of $SL(N)$. Using the Hall--Littlewood closed form for Coulomb-branch Hilbert series, together with independent checks from the monopole formula, we compute exact unrefined Hilbert series for all non-maximal partitions $\rho\vdash N$ with $N=4$, and extend the analysis to $N=5,6$. By analyzing the plethystic logarithms of the resulting Hilbert series, we find that in all cases examined the Coulomb branch is a complete intersection. The number of generators and relations follows a uniform pattern governed by the transpose partition $\rho^T$, with exactly $N-1$ relations appearing independently of $\rho$ in these examples. We summarize the results in explicit classification tables and formulate conjectures extending these patterns to arbitrary $N$. Our findings provide strong evidence for a remarkable uniformity in the algebraic structure of Coulomb branches within the $T_\rho(SU(N))$ family at low rank.