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Notes on S-folds and $\mathcal{N}=3$ theories

Prarit Agarwal, Antonio Amariti

TL;DR

This work analyzes D3-brane configurations near S-folds to realize $\mathcal{N}=3$ theories and investigates non-perturbative SUSY enhancement to $\mathcal{N}=4$. It develops an explicit DOF counting from $(p,q)$ strings, derives a projection rule for trivial discrete torsion, and establishes a concrete isomorphism between the rank-2 $\mathcal{N}=3$ BPS spectrum and the ${\cal N}=4$ $SU(3)$ spectrum at $\tau=\omega$, including a correspondence for three-pronged string junctions and wall-crossing. The analysis shows how BPS central charges and string junctions encode the enhancement mechanism and consistency with flat-space results, providing a non-perturbative bridge between S-fold backgrounds and conventional ${\cal N}=4$ gauge theories. The results illuminate the spectrum structure in S-fold backgrounds, guide extensions to other rank-2 cases, and motivate exploration of higher-fold constructions such as $S_{4,1}$ and $S_{6,1}$ with potential links to $SO(5)$ and $G_2$ ${\cal N}=4$ SYM.

Abstract

We consider D3 branes in presence of an S-fold plane. The latter is a non perturbative object, arising from the combined projection of an S-duality twist and a discrete orbifold of the R-symmetry group. This construction naively gives rise to 4d $\mathcal{N}=3$ SCFTs. Nevertheless it has been observed that in some cases supersymmetry is enhanced to $\mathcal{N}=4$. In this paper we study the explicit counting of degrees of freedom arising from vector multiplets associated to strings suspended between the D3 branes probing the S-fold. We propose that, for trivial discrete torsion, there is no vector multiplet associated to $(1,0)$ strings stretched between a brane and its image. We then focus on the case of rank 2 $\mathcal{N}=3$ theory that enhances to $SU(3)$ $\mathcal{N}=4$ SYM, explicitly spelling out the isomorphism between the BPS-spectrum of the manifestly $\mathcal{N}=3$ theory and that of three D3 branes in flat spacetime. Subsequently, we consider 3-pronged strings in these setups and show how wall-crossing in the S-fold background implies wall crossing in the flat geometry. This can be considered a consistency check of the \emph{conjectured} SUSY enhancement. We also find that the above isomorphism implies that a $(1,0)$ string, suspended between a brane and its image in the S-fold, corresponds to a 3-string junction in the flat geometry. This is in agreement with our claim on the absence of a vector multiplet associated to such $(1,0)$ strings. This is because the 3-string junction in flat geometry gives rise to a $1/4$-th BPS multiplet of the $\mathcal{N}=4$ algebra. Such multiplets always include particles with spin $>1$ as opposed to a vector multiplet which is restricted by the requirement that the spins must be $\leq 1$.

Notes on S-folds and $\mathcal{N}=3$ theories

TL;DR

This work analyzes D3-brane configurations near S-folds to realize theories and investigates non-perturbative SUSY enhancement to . It develops an explicit DOF counting from strings, derives a projection rule for trivial discrete torsion, and establishes a concrete isomorphism between the rank-2 BPS spectrum and the spectrum at , including a correspondence for three-pronged string junctions and wall-crossing. The analysis shows how BPS central charges and string junctions encode the enhancement mechanism and consistency with flat-space results, providing a non-perturbative bridge between S-fold backgrounds and conventional gauge theories. The results illuminate the spectrum structure in S-fold backgrounds, guide extensions to other rank-2 cases, and motivate exploration of higher-fold constructions such as and with potential links to and SYM.

Abstract

We consider D3 branes in presence of an S-fold plane. The latter is a non perturbative object, arising from the combined projection of an S-duality twist and a discrete orbifold of the R-symmetry group. This construction naively gives rise to 4d SCFTs. Nevertheless it has been observed that in some cases supersymmetry is enhanced to . In this paper we study the explicit counting of degrees of freedom arising from vector multiplets associated to strings suspended between the D3 branes probing the S-fold. We propose that, for trivial discrete torsion, there is no vector multiplet associated to strings stretched between a brane and its image. We then focus on the case of rank 2 theory that enhances to SYM, explicitly spelling out the isomorphism between the BPS-spectrum of the manifestly theory and that of three D3 branes in flat spacetime. Subsequently, we consider 3-pronged strings in these setups and show how wall-crossing in the S-fold background implies wall crossing in the flat geometry. This can be considered a consistency check of the \emph{conjectured} SUSY enhancement. We also find that the above isomorphism implies that a string, suspended between a brane and its image in the S-fold, corresponds to a 3-string junction in the flat geometry. This is in agreement with our claim on the absence of a vector multiplet associated to such strings. This is because the 3-string junction in flat geometry gives rise to a -th BPS multiplet of the algebra. Such multiplets always include particles with spin as opposed to a vector multiplet which is restricted by the requirement that the spins must be .

Paper Structure

This paper contains 14 sections, 36 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A brief summary of developments so far
  3. An explicit counting of the degrees of freedom
  4. A lesson from the orientifold: the $Z_2$ projection
  5. Generalization to the $\mathbb{Z}_k$ case
  6. $1/2$ BPS states and SUSY enhancement
  7. Charges and masses of $(p,q)$ strings
  8. $1/2$-BPS states of rank 2 S$_{3,1}$-fold and enhancement to $SU(3)$ ${\cal N}=4$ SYM
  9. 3-pronged string in $S$-fold background
  10. Warm-up: $M_{BPS}$ of an F1 string probing the rank 2 $S_{3,1}$ geometry
  11. The case of the 3-pronged string
  12. Wall crossing
  13. 3-pronged strings in ${\cal N}=4$ $SU(3)$ theory.
  14. Further directions and open questions

Figures (4)

  • Figure 1: Strings stretched between two D3 branes on top of an $O3$-plane
  • Figure 2: (1,0) strings stretching between D3 branes in the S-fold geometry.
  • Figure 3: The two ways in which a 3-string with $(1,0),(0,1)$ and $(-1,-1)$ prongs can be made to end on the first brane, the second brane and the first image of the second brane.
  • Figure 4: Wall of marginal stability in the S-fold and flat backgrounds.