Abstract
$S$-folds are a nonperturbative generalization of orientifold 3-planes which figure prominently in the construction of four-dimensional (4D) $\mathcal{N}=3$ superconformal field theories (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 nonperturbative 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.
Highlights
One of the important ingredients in many string theory realizations of quantum field theories is the use of singular geometries in the presence of various configurations of branes
When a discrete torsion is present on the S-fold, we find that the resulting flavor symmetry of a probe D3-brane is different
Using this procedure, we show how to match each possible S-fold quotient of 7-branes to a corresponding theory appearing in the list of rank-1 4D N 1⁄4 2 superconformal field theories (SCFTs) appearing in Refs. [28,29,30,31,32,33,34], where the rank-1 theories are classified by the associated Kodaira fiber type obtained from the Seiberg-Witten curve
Summary
One of the important ingredients in many string theory realizations of quantum field theories is the use of singular geometries in the presence of various configurations of branes. We show that the presence of discrete torsion, in tandem with the geometric Zk action on the local geometry, leads to a well-defined set of rules which act on the end points of the string junction states This in turn leads to a general quotienting procedure for the resulting flavor symmetry algebras. When a discrete torsion is present on the S-fold, we find that the resulting flavor symmetry of a probe D3-brane is different In these cases, the standard F-theory geometry is not valid, but we can instead deduce its structure from the corresponding Seiberg-Witten curve of the 4D N 1⁄4 2 SCFT. For each possible discrete quotient of an F-theory Kodaira fiber as associated with a probe D3-brane in the presence of a 7-brane and an S-fold with or without discrete torsion, we find a corresponding interacting rank-1 theory as given in Table 1 of Ref. Some additional details on brane motions in the presence of S-folds are presented in the Appendix A, and an explicit example of string junction projections is worked out in Appendix B
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