A tale of two cones: the Higgs Branch of Sp(n) theories with 2n flavours

TL;DR

The paper demonstrates that the Higgs branch of the 4d $\mathcal{N}=2$ theories with $Sp(n)$ gauge groups and $2n$ flavours (and more generally theories with classical gauge groups) can split into two hyperkähler cones with a nontrivial intersection, a phenomenon traced to the reducible top antisymmetric representation of $SO(4n)$. Using Hilbert series and Highest Weight Generating Functions, the authors provide explicit descriptions of the two components and their intersection, and show how this structure emerges from the underlying representation theory and nilpotent orbit pictures. They further connect this geometric splitting to 3d mirror symmetry and brane constructions, identifying a dual flavoured $D_N$ quiver whose Coulomb branch matches the Higgs branch, and illustrating the duality with concrete examples (e.g., $n=1$ giving $\mathbb{C}^2/\mathbb{Z}_2$). The work highlights a rare, highly structured splitting of Higgs branches in meson-generated sectors and clarifies how such decompositions arise in both algebraic and brane-theoretic frameworks, with implications for understanding moduli spaces of vacua in $\mathcal{N}=2$ theories.

Abstract

The purpose of this short note is to highlight a particular phenomenon which concerns the Higgs branch of a certain family of 4d N = 2 theories with SO(2N) flavour symmetry. By studying the Higgs branch as an algebraic variety through Hilbert series techniques we find that it is not a single hyperkahler cone but rather the union of two cones with intersection a hyperkahler subvariety which we specify. This remarkable phenomenon is not only interesting per se but plays a crucial role in understanding the structure of all Higgs branches that are generated by mesons.

Figures (4)

• Figure 1: Coulomb branch of $Sp(k)$ with $N$ flavours. Each black line corresponds to a half-D3 brane. Here $k=2$ and $N=8$. The one presented here is the double cover of the orientifold $O5^-$ theory.
• Figure 2: The origin of the moduli space for $Sp(k)$ with $N$ flavours: the $2k$ half-D3 branes are at the same position as the $N$ D5 branes and the $O5^-$ on the 345 direction. On the left is the double cover of the origin of the moduli space and on the right the physical space. The picture has been simplified: the green dots represent D5 branes (and their images), the cross is the orientifold plane and the blue line the NS5 brane (and its image).
• Figure 3: The Higgs branch is achieved by maximally breaking the D3 branes between the D5 branes. Near the orientifold plane, the right projection must be adopted, i.e. D3 branes cannot stretch between a D5 brane and its image. At the NS5 end of the system, caution must also be used: a supersymmetric configuration is achieved when at most one D3 brane stretches between a D5 and an NS5 brane. A D3 brane that stretches leftward from the NS5 brane towards a D5 brane can end on the latter provided it is the first to do so: otherwise it must continue onwards to the next left D5 brane. This is how the configuration sketched in (a) is achieved. There is still freedom to move the NS5 brane across the D5 branes, as this does not affect the moduli space. Each motion of the NS5 across a D5 brane results in the annihilation of a $D3$ brane. Moving the NS5 brane across $2k$ intervals results in the set-up of (b)
• Figure 4: (a) The brane set up for the Coulomb branch of the mirror dual of $Sp(k)$ with $2N$ flavours. (b) The resulting quiver gauge theory can be read off directly from the branes.

Theorems & Definitions (1)

• Theorem : Macaulay