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 $\leq 1$.
Highlights
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 Sfold
We 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
We develop a dictionary by mapping the masses and the charges associated to the (p, q) strings connecting the D3 branes in the S-fold geometry and the corresponding states in SU(3) N = 4 theory
Summary
We start our analysis from the well known case of n D3 branes probing an O3 plane This corresponds to the Z2 projection in the language of the S-fold. 3.1 A lesson from the orientifold: the Z2 projection We consider a stack of n D3 branes on top of the O3− plane This gives rise to an N = 4, O(n) gauge theory. The spectrum of BPS-states in the theory is obtained by (p, q)strings stretched between the branes along the homotopically distinct paths that connect them. Each such state can be assigned a 4-vector of electromagnetic charges with respect to the U(1) gauge symmetry generated by the system of two branes. In the case of O3+-planes, such vector multiplets are there and we get an Sp(n) gauge group when all the branes are made coincident with the orientifold
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