T^σ_ρ(G) Theories and Their Hilbert Series

TL;DR

This work derives explicit Hilbert series for the Higgs and Coulomb branches of the 3d N=4 theories $T^{\bm{\sigma}}_{\bm{\rho}}(G)$, realized via D3 branes ending on NS5/D5 branes (with possible O3 planes). The authors develop a generalized Hall-Littlewood formalism, linking Coulomb-branch monopole data and Higgs-branch localization to HL polynomials, with a precise mirror-symmetry map between parameters. They provide complete quiver constructions for classical groups, derive and validate HL formulas for $SU(N)$ (and discuss extensions to $SO$, $USp$), and demonstrate multiple explicit examples (including $SU(4)$ and $SU(6)$) where HL, monopole, and Molien-Weyl results coincide. The results illuminate the geometric structure of moduli spaces as intersections of nilpotent orbits with Slodowy slices, and they reveal intricate relations between different quivers related by $O/SO$ gauging and parity, with implications for dualities and moduli-space stratifications in theories built from brane configurations.

Abstract

We give an explicit formula for the Higgs and Coulomb branch Hilbert series for the class of 3d N=4 superconformal gauge theories T^σ_ρ(G) corresponding to a set of D3 branes ending on NS5 and D5-branes, with or without O3 planes. Here G is a classical group, σis a partition of G and ρa partition of the dual group G^\vee. In deriving such a formula we make use of the recently discovered formula for the Hilbert series of the quantum Coulomb branch of N=4 superconformal theories. The result can be expressed in terms of a generalization of a class of symmetric functions, the Hall-Littlewood polynomials, and can be interpreted in mathematical language in terms of localization. We mainly consider the case G=SU(N) but some interesting results are also given for orthogonal and symplectic groups.

Figures (6)

• Figure 1: A brane construction for $T^{(3,2,2)}_{(2,2,2,1)}(SU(7))$. The partition $\bm{\sigma} = (3,2,2)$ gives the net number of D3 branes ending on each D5-brane from the interior to the exterior. The partition $\bm{\rho} = (2,2,2,1)$ gives the net number of D3 branes ending on each NS5-brane from the interior to the exterior. Here $x_i$ are the fugacities associated with each NS5 brane and $n_{j}$ are the background monopole charges associated with each D5-brane.
• Figure 2: Left: brane construction for $T^{(3,2,2)}_{(2,2,2,1)}(SU(7))$ after the D5-branes are moved inside the NS5-brane intervals. Right: the linear quiver is read off from the brane configuration. We adopt the convention that the $i$-th gauge group corresponds to the D3-brane interval between $x_i$ and $x_{i+1}$: hence $U(1)$, $U(2)$, $U(1)$ from left to right are regarded as the first, second and third gauge groups respectively, and similarly $U(2)$ and $U(1)$ are regarded as the second and third flavor groups respectively.
• Figure 3: Top: brane construction for $T^{(2,2,2,1)}_{(3,2,2)}(SU(7))$, obtained by exchanging D5-branes and NS5-branes in Figure \ref{['fig:BraneTs322r2221a']}. Bottom left: the D5-branes are moved inside the NS5-brane intervals. Bottom right: the quiver diagram read off from the bottom left brane configuration. We adopt the convention that the $i$-th gauge group corresponds to the D3-brane interval between $n_i$ and $n_{i+1}$: hence $U(1)$ and $U(2)$ are regarded as the first and the second gauge groups respectively, and similarly $U(1)$ and $U(3)$ are regarded as the first and the second flavor groups respectively.
• Figure 4: Brane construction for $T^{(3,2,1)}_{(2,2,1,1)}(SU(6))$. The corresponding quiver diagram is depicted in Figure \ref{['fig:BraneTs321r2211b']}.
• Figure 5: Left: brane construction for $T^{(3,2,1)}_{(2,2,1,1)}(SU(6))$ after the D5-branes are moved into the NS-brane intervals. Right: the linear quiver read off from the brane configuration. We adopt the convention that the $i$-th gauge group corresponds to the D3-brane interval between $x_i$ and $x_{i+1}$.