S-fold magnetic quivers

TL;DR

This work constructs and cross-validates magnetic quivers for the Higgs branches of 4d $\mathcal{N}=2$ SCFTs arising from $\mathbb{Z}_{\ell}$ S-folds. It develops three independent derivations—quiver-based inference from symmetry and dimension, FI-deformed reductions of 6d theories with folding, and T-dual brane-web constructions—tying the 4d quivers to their 6d UV origins. A key insight is that ungauging choices on long vs short nodes yield discrete gaugings, expanding the landscape to include discretely gauged variants $\mathring{\mathcal{S}}_{G,\ell}^{(r)}$ and $\mathring{\mathcal{T}}_{G,\ell}^{(r)}$, with moduli spaces matching instanton moduli on orbifolds in many cases. The authors provide explicit magnetic quivers for several families (S/T theories with $(G,\ell)=(E_6,2),(D_4,2),(A_2,2),(D_4,3),(A_1,3),(A_2,4)$) and compute Hilbert series and refined plethystic data to corroborate global symmetry enhancements. They also map 6d twisted compactifications and brane-web realizations to 4d quivers, offering a cohesive framework to study Hasse diagrams and the structure of Higgs branches across a broader class of SCFTs. The results illuminate the interplay between higher-dimensional origins, discrete gauging, and the rich moduli space geometry of 4d $\mathcal{N}=2$ theories, with practical impact on classifying SCFTs and computing their Higgs-branch data. All mathematical expressions are consistently presented in $...$ math delimiters.

Abstract

Magnetic quivers and Hasse diagrams for Higgs branches of rank $r$ 4d $\mathcal{N}=2$ SCFTs arising from $\mathbb{Z}_{\ell}$ $\mathcal{S}$-fold constructions are discussed. The magnetic quivers are derived using three different methods: 1) Using clues like dimension, global symmetry, and the folding parameter $\ell$ to guess the magnetic quiver. 2) From 6d $\mathcal{N}=(1,0)$ SCFTs as UV completions of 5d marginal theories, and specific FI deformations on their magnetic quiver, which is further folded by $\mathbb{Z}_{\ell}$. 3) From T-duality of Type IIA brane systems of 6d $\mathcal{N}=(1,0)$ SCFTs and explicit mass deformation of the resulting brane web followed by $\mathbb{Z}_{\ell}$ folding. A choice of the ungauging scheme, either on a long node or on a short node, yields two different moduli spaces related by an orbifold action, thus suggesting a larger set of SCFTs in four dimensions than previously expected.

Figures (12)

• Figure 1: Summary of the derivations of the 4d magnetic quivers discussed in Sections \ref{['SectionFI']} and \ref{['sectionBW']}.
• Figure 2: (\ref{['fig:Kac']}) The Kac labels $\{m_i,m^\prime_j\}$, which are non-negative integers, need to satisfy $\sum_{i=1}^6 a_i m_i + \sum_{j=2}^4 a^\prime_j m^\prime_j = \ell$ to define an embedding $\mathbb{Z}_\ell \hookrightarrow E_8$. The $\{a_i,a^\prime_j\}$ are the Dynkin labels of affine $E_8$, which are the numbers displayed inside the nodes. (\ref{['fig:cases']}) The relevant embeddings for the $\mathcal{S}$-fold theories considered here.
• Figure 3: The FI deformation of the 6d magnetic quiver \ref{['sukMQ']}. We turn on FI parameters at the nodes in red, resulting in the second quiver.
• Figure 4: The FI deformations leading from the magnetic quiver of the 5d SCFT (the second quiver in Figure \ref{['DeftoUSpUV']}) to the 3d mirror of the $\mathrm{USp}(2r)^2$ gauge theory \ref{['lagrir']}.
• Figure 5: The FI deformations of the last quiver in Figure \ref{['DeftoUSpUV']} leading to the 3d mirror of the $\mathrm{USp}(2r)^2$ gauge theory \ref{['eq:2_USp2r_USp2r_2']} with 2 flavors on each side.