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arXiv · 2602.01028

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

Ayush Kumar
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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.

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Ayush Kumar. 2026-02-01. Hilbert Series and Complete-Intersection Structure of Coulomb Branches for Non-Maximal Nilpotent Orbits of $SL(N)$. https://arxiv.org/abs/2602.01028

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