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

TL;DR

The paper analyzes the Coulomb branches of 3d $N=4$ quiver theories $T_\rho(SU(N))$ associated with non-maximal nilpotent orbits, computing exact unrefined Hilbert series for $N=4,5,6$ via Hall–Littlewood formulas and corroborating with the monopole formula. Across all non-maximal partitions studied, the Coulomb branches are shown to be complete intersections, with a remarkably uniform generator/relation pattern governed by the transpose partition $\rho^T$ and with exactly $N-1$ relations irrespective of $\rho$. The proposed dimension and generator counts satisfy $\dim_{\mathbb{C}}\mathcal{C}_\rho=\sum_i(\rho_i^T)^2-N$ and $\#\text{generators}=\sum_i(\rho_i^T)^2-1$, while the relations remain fixed in number to $N-1$. Motivated by these results, the authors conjecture that the complete-intersection property and the related counting extend to arbitrary $N$, highlighting a surprising rigidity in the algebraic structure of Coulomb branches within the $T_\rho(SU(N))$ family at low rank and suggesting broader implications for symplectic/representation-theoretic aspects of 3d $N=4$ theories.

Abstract

We study the Coulomb branches of three-dimensional $\mathcal N=4$ quiver gauge theories of type $T_ρ(SU(N))$ 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 $ρ\vdash N$ 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 $ρ^T$, with exactly $N-1$ relations appearing independently of $ρ$ 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 $T_ρ(SU(N))$ family at low rank.