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Coulomb branch Hilbert series and Hall-Littlewood polynomials

Stefano Cremonesi, Amihay Hanany, Noppadol Mekareeya, Alberto Zaffaroni

TL;DR

This work develops a robust framework for Coulomb branch Hilbert series in 3d ${\cal N}=4$ theories by incorporating background magnetic fluxes for flavor symmetries, enabling gluing of building blocks to construct more general theories. It proves a Hall-Littlewood formula for the Coulomb branch Hilbert series of $T_{\bm{\rho}}(G)$ (with $G$ classical) and demonstrates that these branches are complete intersections when background fluxes vanish; mirror symmetry is shown to map fluxes to baryonic charges on the Higgs branch. The results unify monopole-based and Hall-Littlewood approaches across $SU(N)$, $SO(N)$, and $USp(2N)$ families, provide explicit quiver constructions and partition classifications, and reveal analytic structures where moving a box in the partition generates new theories via residues. Collectively, these insights offer a versatile toolkit for computing and interpreting Coulomb branch moduli spaces, with implications for dualities and gluing constructions in 3d ${\cal N}=4$ theories.

Abstract

There has been a recent progress in understanding the chiral ring of 3d $\mathcal{N}=4$ superconformal gauge theories by explicitly constructing an exact generating function (Hilbert series) counting BPS operators on the Coulomb branch. In this paper we introduce Coulomb branch Hilbert series in the presence of background magnetic charges for flavor symmetries, which are useful for computing the Hilbert series of more general theories through gluing techniques. We find a simple formula of the Hilbert series with background magnetic charges for $T_ρ(G)$ theories in terms of Hall-Littlewood polynomials. Here $G$ is a classical group and $ρ$ is a certain partition related to the dual group of $G$. The Hilbert series for vanishing background magnetic charges show that Coulomb branches of $T_ρ(G)$ theories are complete intersections. We also demonstrate that mirror symmetry maps background magnetic charges to baryonic charges.

Coulomb branch Hilbert series and Hall-Littlewood polynomials

TL;DR

This work develops a robust framework for Coulomb branch Hilbert series in 3d theories by incorporating background magnetic fluxes for flavor symmetries, enabling gluing of building blocks to construct more general theories. It proves a Hall-Littlewood formula for the Coulomb branch Hilbert series of (with classical) and demonstrates that these branches are complete intersections when background fluxes vanish; mirror symmetry is shown to map fluxes to baryonic charges on the Higgs branch. The results unify monopole-based and Hall-Littlewood approaches across , , and families, provide explicit quiver constructions and partition classifications, and reveal analytic structures where moving a box in the partition generates new theories via residues. Collectively, these insights offer a versatile toolkit for computing and interpreting Coulomb branch moduli spaces, with implications for dualities and gluing constructions in 3d theories.

Abstract

There has been a recent progress in understanding the chiral ring of 3d superconformal gauge theories by explicitly constructing an exact generating function (Hilbert series) counting BPS operators on the Coulomb branch. In this paper we introduce Coulomb branch Hilbert series in the presence of background magnetic charges for flavor symmetries, which are useful for computing the Hilbert series of more general theories through gluing techniques. We find a simple formula of the Hilbert series with background magnetic charges for theories in terms of Hall-Littlewood polynomials. Here is a classical group and is a certain partition related to the dual group of . The Hilbert series for vanishing background magnetic charges show that Coulomb branches of theories are complete intersections. We also demonstrate that mirror symmetry maps background magnetic charges to baryonic charges.

Paper Structure

This paper contains 35 sections, 84 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Note added:
  3. Coulomb branch Hilbert series of a 3d $\mathcal{N}=4$ gauge theory
  4. The monopole formula for the Coulomb branch Hilbert series
  5. The Hilbert series with background magnetic fluxes
  6. Hilbert series and Hall-Littlewood polynomials
  7. The theory $T_{\bm{\rho}}(SU(N))$
  8. Monopole formula for the Coulomb branch Hilbert series
  9. Hall-Littlewood formula for the Coulomb branch Hilbert series
  10. The Coulomb branch of $T_{\bm{\rho}}(SU(N))$ is a complete intersection
  11. Coulomb branch of $T_\rho (SO(N))$ and $T_\rho (USp(2N))$
  12. $B$, $C$ and $D$ partitions
  13. Quiver diagrams
  14. $T_{\bm{\rho}}(SO(2N))$ theory.
  15. $T_{\bm{\rho}}(SO(2N+1))$ theory.
  16. ...and 20 more sections

Figures (2)

  • Figure 1: The brane configuration of $T_{\bm{\rho}}(SU(N))$. The numbers in red indicates the number of D3-branes in each intervals. The labels in blue denote the fugacities $x_i$ for NS5-branes and the background fluxes $n_j$ for D5-branes.
  • Figure 2: The quiver diagram of $T(SU(N))_{\bm{J}}$. In this quiver, the flavor group $U(N)$ of $T(SU(N))$ is replaced by $N$$U(1)$ nodes joined to the $U(N-1)$ node, with the overall $U(1)$ gauge group factored out.