Complete Graphs, Hilbert Series, and the Higgs branch of the 4d N=2 $(A_n,A_m)$ SCFT's

TL;DR

The paper identifies a nontrivial Higgs branch for the 4d N=2 (A_n,A_m) SCFTs when h(A_n) divides h(A_m), using 4d/2d correspondence to reveal flavor symmetries and employing 3d N=4 mirror symmetry combined with CHZ Hilbert-series techniques. It introduces the A_{s,p} subclass and derives a universal framework for the refined Coulomb-branch Hilbert series of their 3d mirrors, which equals the refined Higgs-branch Hilbert series of the 4d theories. The authors provide explicit calculations for s=1,2,3, showing SU(2) and SU(3) flavor enhancements at p=1 that break to U(1)^2 for p>1, and they reveal the Higgs-branch algebraic structures (notably non-complete-intersection cases) via plethystic analysis and, in select cases, brane constructions. These results connect non-Lagrangian 4d dynamics to tractable 3d Coulomb-branch data and sharpen understanding of Higgs-branch geometry in this class.

Abstract

The strongly interacting 4d N=2 SCFT's of type $(A_n,A_m)$ are the simplest examples of models in the $(G,G^\prime)$ class introduced by Cecotti, Neitzke, and Vafa in arXiv:1006.3435. These systems have a known 3d N=4 mirror only if $h(A_n)$ divides $h(A_m)$, where $h$ is the Coxeter number. By 4d/2d correspondence, we show that in this case these systems have a nontrivial global flavor symmetry group and, therefore, a non-trivial Higgs branch. As an application of the methods of arXiv:1309.2657, we then compute the refined Hilbert series of the Coulomb branch of the 3d mirror for the simplest models in the series. This equals the refined Hilbert series of the Higgs branch of the $(A_n,A_m)$ SCFT, providing interesting information about the Higgs branch of these non-lagrangian theories.

Figures (3)

• Figure 1: Some of the quivers for the 3d ${\cal N}=4$ mirrors of the models of type ${\cal A}_{s,p}$: $s$ is the number indicated under the quivers, and each edge has multeplicity $p$. Notice that these graphs are all $\mathfrak{S}_{s+1}$ symmetric under permutation of the nodes, and that removing one node maps the 3d ${\cal A}_{s,p}$ quiver into the ${\cal A}_{s-1,p}$ one.
• Figure 2: Examples of graphs of the class ${\cal A}^{k}_{s,p}$ for $s = 1,2,3$.
• Figure 3: The graph ${\cal A}^1_{3,1}$ and the corresponding brane network.