S-folds, String Junctions, and 4D $\mathcal{N} = 2$ SCFTs

TL;DR

This work develops a general framework for extracting the flavor symmetry seen by D3-branes probing 7-branes in the presence of S-folds, by extending orientifold projections to non-perturbative string junctions. It shows that discrete torsion crucially alters the resulting flavor algebras, and provides explicit Z_k quotients (k = 2,3,4) without torsion and with torsion, linking them to rank-one 4D \\mathcal{N}=2 SCFTs via Seiberg–Witten geometry and, when torsion is present, proposing an operational F-theory definition through those curves. The paper catalogs the allowed flavor algebras and admissible representations across all S-fold cases, and verifies anomaly structures (a, c, κ_G, κ_SU(2)) in the large-N limit, aligning with known rank-one classifications. It also extends the discussion to F-theory with discrete torsion, providing rank-one SW curves and outlining how to interpret the geometry in torsion backgrounds. Overall, the results illuminate a deep correspondence between S-fold quotients of 7-brane configurations and the spectrum of rank-one 4D \\mathcal{N}=2 SCFTs, with implications for defining F-theory in nonperturbative settings and for future 4D \\mathcal{N}=1 explorations.

Abstract

S-folds are a non-perturbative generalization of orientifold 3-planes which figure prominently in the construction of 4D $\mathcal{N} = 3$ SCFTs and have also recently been used to realize examples of 4D $\mathcal{N} = 2$ SCFTs. In this paper we develop a general procedure for reading off the flavor symmetry experienced by D3-branes probing 7-branes in the presence of an S-fold. We develop an S-fold generalization of orientifold projection which applies to non-perturbative string junctions. This procedure leads to a different 4D flavor symmetry algebra depending on whether the S-fold supports discrete torsion. We also show that this same procedure allows us to read off admissible representations of the flavor symmetry in the associated 4D $\mathcal{N} = 2$ SCFTs. Furthermore this provides a prescription for how to define F-theory in the presence of S-folds with discrete torsion.

Figures (9)

• Figure 1: Realization of the different rank one 4D $\mathcal{N} = 2$ SCFTs starting from the $E_8$ Minahan--Nemeschansky theory, written as $[II^{\ast} , E_8]$. We can perform mass deformations (as indicated by downward blue arrows), or we can act by a discrete twist by an outer automorphism of an algebra, possibly composed with an inner automorphism. All of the different choices can be realized by a suitable choice of S-fold projection with (diagonal red arrows and $\widehat{\mathbb{Z}}_{k}$) or without (diagonal green arrows and $\mathbb{Z}_{k}$) discrete torsion. Here, we use the conventions of references Argyres:2015ffaArgyres:2015ghaArgyres:2016xuaArgyres:2016xmcArgyres:2016yzzMartone:2020nsyArgyres:2020wmq, which labels a given theory by its Kodaira fiber type, as well as the associated flavor symmetry algebra. We note that while this notation does not necessarily uniquely specify a particular 4D SCFT, it does so for the theories listed here. The notation $\chi_a$ refers to the fact that the theory has a chiral deformation parameter which has scaling dimension $a$. The theories connected to the $[II^*, E_8]$ theory by blue arrows will be referred to as "parent" theories, and the theories determined via the red/green arrows from a given parent will be referred to as the "descendants" of that parent.
• Figure 2: Illustration of orientifold projection acting on perturbative open strings. We denote the orientifold image branes by open shapes, and image strings by dashed blue lines.
• Figure 3: Projection rules for S-fold planes acting on string junctions. We denote the orientifold image branes by open shapes, and image strings by dashed blue lines.
• Figure 4: $\mathbb Z_2$ symmetric configuration for $E_6$ theory.
• Figure 5: String junctions for the S-fold projection of $E_6$ to $F_4$. We denote the orientifold image branes by open shapes, and image strings by dashed blue lines.