S-folds and $AdS_{3}$ flows from the D3-brane

We investigate supersymmetric flows in type IIB supergravity that preserve an $SO(2,2)$ space-time symmetry and asymptote to $AdS_{5} \times S^5$ at both endpoints. The flows are constructed as Janus-type $\mathbb{R} \times AdS_{3}$ BPS domain-walls in the effective four-dimensional $[SO(1,1)\times{S}O(6)]\ltimes\mathbb{R}^{12}$ gauged maximal supergravity describing the massless sector of type IIB supergravity compactified on $S^1 \times S^{5}$. The compactification includes an S-duality hyperbolic twist along the $S^1$ which, when combined with an appropriate choice of boundary conditions for the running scalars, generates special flows that develop an S-fold regime at their core, thus enhancing the space-time symmetry there to $SO(2,3)$. Via the AdS/CFT correspondence, the flows constructed here are conjectured to describe conformal interfaces in a circle compactification of $\mathcal{N}=4$ $\textrm{SYM}_{4}$.


Introduction
Via the AdS/CFT correspondence [1], Janus solutions of supergravity theories [2] provide a simple and explicit realisation of interfaces in dual field theories [3,4,5]. The original type IIB Janus solutions are characterised by two properties: i) they feature an AdS d−1 space-time geometry that asymptotes to an AdS d vacuum on each side of the Janus. ii) the type IIB dilaton field jumps across the interface living at the boundary of space-time and takes different values ±Φ 0 on each side. However, Janus solutions have been generalised to theories without a dilaton field so that they are solely characterised by the first property. Examples are the Janus solutions of [6,7] constructed in the four-dimensional SO(8) maximal supergravity that arises upon consistent truncation of eleven-dimensional supergravity on a seven-sphere [8,9]. In this work we will relax the first property and investigate regular Janus-type solutions featuring an AdS d−2 space-time geometry that asymptotes to an AdS d geometry on each side of the solution. For the type IIB scenario investigated in this work one has d = 5 whereas d = 4 is the relevant value for the M-theory scenario.
The field theory living on the interface with N = 4 & SO(4) symmetry led to the discovery of new 3d N = 4 SCFTs dubbed T [SU (N )] theories in [5]. They provide the building blocks for 3d N = 4 SCFTs. The gravity dual was found to be closely related to the N = 4 & SO(4) Janus, but to be a distinct class of solutions [20,21]. Furthermore, by gauging the U (N )×U (N ) global symmetry of 3d T [SU (N )] theories with vector multiplets, yet another new class of strongly coupled 3d SCFTs, named S-folds, was put forward in [22] (see also [23,24]). The gravity duals of the S-fold theories were realised by first compactifying one spatial direction -we denote it η -into a circle S 1 in a putative AdS 5 space-time, and then imposing a non-trivial SL(2, Z) monodromy of hyperbolic type along the circle [22,25]. The corresponding type IIB backgrounds are of the form AdS 4 × S 1 × S 5 with a characteristic dilaton profile that is linear on the coordinate η along the S 1 . As they are the gravity duals of 3d S-fold theories closely related to Janus solutions, they were named J-fold solutions in [22]. In [18,19], a systematic method was implemented to obtain J-fold solutions from all the families of Janus solutions in the N = 8 and SO(6)-gauged supergravity in five dimensions. These solutions were also uplifted to type IIB supergravity in [18,19]. Using the effective five-dimensional approach, new S-fold solutions as well as RG-flows between S-fold CFT's have been investigated in [26,27,28].
In this work we continue the above four-dimensional program and construct supersymmetric domain-wall solutions of the form R × AdS 3 in the [SO(1, 1) × SO(6)] R 12 dyonically-gauged maximal supergravity in four dimensions. When uplifted to type IIB supergravity, they describe Janus-type solutions of the form involving a non-trivial SL(2, Z) monodromy along the S 1 . The S-duality twist inducing the monodromy is codified in a parameter c that is also responsible for the dyonic nature of the gauging in four dimensions [41,25]. 2 However, and unlike for standard Janus solutions in type IIB [19] and M-theory [6,7], this time there is no maximally supersymmetric round vacuum of the form AdS 4 × S 1 × S 5 that can serve as an endpoint on each side of the Janus solution. 3 This implies that there is no maximally supersymemtric AdS 4 vacuum at the origin of the scalar potential of the effective [SO(1, 1) × SO(6)] R 12 gauged supergravity. As a consequence, the flows we construct in this work are attracted towards a particular non-AdS 4 background at both endpoints of the domain-walls. Such a non-AdS 4 behaviour was recognised as the fourdimensional incarnation of a deformation of the AdS 5 ×S 5 background of type IIB supergravity. And the deformation parameter c (the same one codifying the S-duality twist) was interpreted as a source of anisotropy along the compactified S 1 direction in the dual SYM 4 theory [34]. Thus, the seemingly non-conformal asymptotics from a four-dimensional perspective actually correspond to a deformed D3-brane with a five-dimensional R × AdS 3 × S 1 space-time metric in (1.1) given by in terms of the four-dimensional gauge coupling g and two functions f 1,2 (λ) of an slicing coordinate λ ∈ [0, ∞) . The standard D3-brane solution is recovered at c = 0 upon setting f 1 (λ) = cosh 2 λ and f 2 (λ) = sinh 2 λ so that (1.2) reproduces an AdS 5 space-time metric with a compactified S 1 direction parameterised by η . As already stated, the goal of this work is to construct supersymmetric R × AdS 3 domainwall solutions in the effective [SO(1, 1) × SO(6)] R 12 gauged supergravity that arises upon compactifying type IIB supergravity on S 1 × S 5 including a non-trivial SL(2, Z) monodromy along S 1 . Employing numerical techniques, we find two classes of regular solutions: • The generic Janus-type solutions preserve the SO(2, 2) isometries of the AdS 3 factor in (1.2) all along the flow and are attracted towards the deformed D3-brane discussed above. From a 5D perspective, they correspond to flows in the bulk where the space-time geometry along the flows conforms to AdS 5 − AdS 3 − AdS 5 .
2 As argued in [41], the parameter c is a discrete (on/off) parameter that can be set to either 1 or 0 without loss of generality. 3 There is an AdS 4 × S 1 × S 5 S-fold with SO(6) symmetry featuring a round S 5 internal geometry but it is non-supersymmetric and perturbatively unstable [29].
• Upon tuning of the boundary conditions, special Janus-type solutions appear which still approach the deformed D3-brane asymptotically but feature an intermediate S-fold regime. This regime is characterised by a factorised space-time in (1.2) of the form AdS 4 × S 1 , so there is a symmetry enhancement to SO(2, 3) associated with the AdS 4 factor. From a 5D perspective, these are flows in the bulk with a space-time geometry that conforms to In addition there are also singular solutions (akin to the flows to Hades discussed in [6,7]) which diverge at some finite value of the radial direction. All together, these three classes of solutions are the analogue of the Janus solutions in the SO(8) [8] and ISO(7) [42,43,44] maximal supergravities constructed in [6,7] and [45,46], respectively. The work is organised as follows. In section 2 we review the Z 3 2 -invariant sector of the dyonically-gauged [SO(1, 1) × SO(6)] R 12 maximal supergravity in order to set up the model and derive the BPS equations. In section 3 we first identify the (undeformed, c = 0) D3-brane solution with a curved worldvolume within our four-dimensional effective model, and then semianalytically identify the (deformed, c = 0) D3-brane behaviour that controls the asymptotics at both sides of the Janus-type solutions. Then we numerically construct and characterise such Janus-type solutions paying special attention to the choice of boundary conditions. Upon tuning of the latter, we present examples of Janus-type solutions displaying an AdS 4 intermediate S-fold regime for all the S-folds discussed in the introduction. Our conclusions and a final discussion are presented in Section 4. Some results regarding the type IIB uplift of the Janus-type solutions presented here are collected in the Appendix.

The Z 3 -invariant truncation
We consider the Z 3 2 -invariant sector of the dyonically-gauged [SO(1, 1) × SO(6)] R 12 maximal supergravity investigated in [30]. This sector has previously been considered in the ISO(7) theory [47] as well as in the SO(8) theory [48]. It describes a minimal N = 1 supergravity coupled to seven chiral fields with scalar components z i and i = 1 , . . . , 7. In a canonical N = 1 formulation, the Einstein-scalar action is given by The Kähler potential describes a [SL(2)/SO(2)] 7 coset space for the scalar geometry and is given by The Kähler metric is then defined as with K z izj denoting its inverse. The holomorphic superpotential is given by where g and c are the gauge coupling constant and the electromagnetic deformation parameter, respectively. The scalar potential is obtained from (2.2)-(2.4) as where the Kähler covariant derivative is We also introduce the (complex) gravitino mass term which defines a real superpotential that enters the BPS flow equations (see next section).

Families of AdS 4 vacua
Four multi-parametric families of supersymmetric AdS 4 vacua have been constructed within this sector of theory.
subject to the constraint The symmetry enhances to U (2) when the parameters are identified pairwise and to SU (3) when they all vanish [29].
This vacuum also belongs to Family III and is obtained from (2.12) upon setting ϕ = 1.
Recently, two axion-like flat deformations breaking supersymmetry have been identified also for this vacuum [37,40]. Higher-dimensional and holographic aspects of these axionic deformations have been further investigated in [39].
Within a given family, the radius of the corresponding AdS 4 vacua, L, is always the same and given by where W 0 and V 0 denote values at the vacuum. Note that different values of the free parameters within a given family of AdS 4 vacua do not change L. Moreover, families II, III and IV feature the same value of L and have been shown to be connected by marginal deformations in the two-dimensional conformal manifold of three-dimensional N = 2 S-fold CFT's [36].

Further truncations
In order to construct supersymmetric Janus-type solutions numerically, we will further truncate the Z 3 2 invariant sector described above to simpler subsectors containing a smaller number of scalar fields. We will do it in a minimal manner, namely, for each of the N = 1 & SU (3), N = 2 & U (2) and N = 4 & SO(4) AdS 4 vacua, we will consider the simplest supersymmetric model accommodating that specific AdS 4 vacuum. This will improve the efficiency of the numerical integration method as there are no other supersymmetric AdS 4 vacua towards which the flow is attracted.
The set of minimal models we will consider in this work are: i) The SU (3)  Needless to say, working with a truncated model necessarily entails a loss of generality in the structure of solutions we will present. We leave a more thorough characterisation/classification of numerical solutions for the future.

Supersymmetric Janus-type solutions
Janus solutions correspond to AdS 3 -sliced domain-wall configurations for which the fourdimensional metric takes the form in terms of the AdS 3 line element ds 2 AdS 3 and a metric function A(r) that only depends on the radial coordinate r ∈ (−∞, ∞) transverse to the domain-wall. The metric (3.1) conforms to AdS 4 when where L and l are the radii of AdS 4 and AdS 3 , respectively.

BPS flow equations
Let us now solve the supersymmetry variations of fermionic fields on the curved background, as it was done on the Lorentzian AdS 3 -sliced domain-walls in [6,45]. The supersymmetric BPS equations for the complex scalar fields read whereas for the metric function one finds in terms of the AdS 3 radius l and κ = ±1. The choice κ = ±1 is not related to the ± sign in (3.4). Reversing the sign of κ merely generates a solution reflected along the coordinate r. It will prove convenient to introduce the upper-half plane parameterisation for the Z 3 2invariant scalars in the theory, namely, These scalars span an [SL(2)/SO(2)] 7 coset geometry and, in terms of their real and imaginary components, the supersymmetric BPS equations take the form

Second-order equations of motion
In our numerical analysis we will look at the second-order equations of motion. The ones for the scalar fields are given by whereas the Einstein equations yield two additional equations

Numerical methodology
In order to construct supersymmetric Janus solutions numerically, we closely follow the method employed in [6,45]. It also proves efficient to solve the differential equations starting from a given AdS 4 vacuum and perturbing it to trigger the flow. Moreover we set l = 1, g = 1, c = 1, and κ = +1 for all numerical solutions in this work.
As the BPS equation for the warp factor (3.4) involves a square root, it involves a branch cut. In order to obtain regular Janus solutions in the domain r ∈ (−∞, ∞), we must choose the positive sign for r > 0 and the negative sign for r < 0 [13,14,6,49]. Then, in order to avoid dealing with branch cuts, we solve the second-order equations of motion (3.7)-(3.8) instead of the first-order BPS equations (3.3)-(3.4). To numerically solve the second-order equations of motion, we should specify the initial conditions for the scalar fields (ϕ i , χ i ), the warp factor A, and their derivatives. We impose a smoothness condition at the interface As a consequence of the smoothness condition in (3.9), and once the initial conditions for the scalar fields {ϕ i (0), χ i (0)} are fixed, the initial condition for the warp factor A(0) is automatically determined through its BPS equation in (3.4). Similarly, {ϕ i (0), χ i (0)} are also determined through the BPS equations for the scalar fields in (3.6). Summarising, we have a set of fourteen free parameters specifying the boundary conditions and, once they are fixed, the rest of the parameters are also fixed by the smoothness condition (3.9) and the BPS equations. Finally, once we have obtained a numerical solution of the secondorder equations of motion, we numerically verify that it satisfies the BPS equations.

D3-brane
Here we describe the generic non-AdS 4 background towards which the regular BPS flows are attracted when reaching the boundary at r → ±∞. As argued in [34], this non-AdS 4 configuration uplifts to a deformation of the AdS 5 × S 5 geometry arising as the near-horizon geometry of a D3-brane in type IIB supergravity. The deformation is induced by the four-dimensional electromagnetic parameter c in (2.4).
and then perform a change of radial coordinate In this manner we obtain an analytic solution with (continuous) U (3) symmetry of the form and in terms of two arbitrary parameters (g , Φ 0 ) and the scaling parameter σ . 4 Note that demanding Imz 1,2,3 ≥ 0 , as imposed by (3.5), and e 2A ≥ 0 requires λ ∈ [0, ∞) for the choice σ > 0 and λ ∈ (−∞, 0] for the choice σ < 0 . Moreover, taking σ → 0 renders the solution in (3.13) pathological as z 1,2,3 → 0 in this limit. Using the (finite) unit-disk parameterisation of the complex scalars, namely, various plots of thez 1,2,3 scalar in (3.16) are presented in Figure 1 for different values of σ . Note that the scalarz 4,5,6,7 in (3.16) covers the real interval [−1, 1] in Figure 1 as a function of the constant parameter Φ 0 setting the asymptotic value of the type IIB dilaton field. This four-dimensional solution uplifts to a D3-brane in ten-dimensional type IIB supergravity (see Appendix A for more details on the geometry of the uplift). More concretely the metric reads  4). This is given by 15) and the metric function in (3.14), so it allows for an additional parameterσ. Settingσ = 0 implies Rez 1,2,3 = 0 and the (continuous) symmetry group of the solution is enhanced from U (3) to SO(6). by From (3.17) one observes that the parameter σ can be reabsorbed in a redefinition of the coordinate η thus rendering the limit σ → 0 pathological this time from a higher-dimensional perspective. In addition, having Rez 1,2,3 = − 1 3 g −1 σ = 0 formally induces a monodromy (see (A.7)) on the S 1 in S 5 = CP 2 S 1 when moving along the compactified η direction. This results in the global breaking of the S 5 isometries from SO(6) to U (3) as discussed in [34]. Unlike for the curved-sliced D3-brane discussed here, the flat-sliced D3-brane in [34] corresponds to a four-dimensional analytic flow solution with Rez 1,2,3 = 0 and, therefore, features a larger SO(6) symmetry.

D3-brane at c = 0
Let us study a perturbation of the analytic solution at c = 0 in (3.13)-(3.14) with U (3) symmetry. We will redefine the radial coordinate as ρ = sinh λ , (3.19) so that ρ ∈ [0, ∞) and the solution in (3.13)-(3.14) at c = 0 takes the form Using the new radial coordinate ρ it is possible to analytically solve the BPS equations (3.3)-(3.4) order by order as a power series in the deformation parameter c (which can be set to any desired numerical value without loss of generality). At linear order in c , and further expanding around the region at ρ → ∞, the scalar fields conform to the power series

SU (3)-invariant flows
The (SU (3) ∩ Z 3 2 )-invariant sector of the theory is recovered upon the identifications in (3.11), namely, This yields an effective two-field model and the holomorphic superpotential in (2.4) reduces to The identifications in (3.23) are compatible with the Family I of AdS 4 vacua presented in (2.9). However, the condition (2.10) requires Rez 1 = 0 at the corresponding AdS 4 vacuum. Still, as we will see (e.g. red flow in Figure 3), a generic Janus-type solution involving such an AdS 4 vacuum in the intermediate regime will arrive at the asymptotic D3-brane behaviour at r → ±∞ with a non-zero value of the axion Rez 1 .

Numerical study
Following the numerical methodology introduced before, a numerical Janus-type flow is constructed upon suitable choice of boundary conditions {z 1 (0) , z 4 (0)} . It proves covenient to switch again to the unit-disk parameterisation for the complex scalars The second natural choice of boundary conditions is to distribute the turning points of the numerical flows along the Imz 4 imaginary axis while keepingz 1 fixed at its value in the AdS 4 vacuum. This isz However, we find no regular flows for this choice of boundary conditions. Actually, we cannot find regular flows with boundary conditions different from (3.26).

SU (2)-invariant flows
The (SU (2) ∩ Z 3 2 )-invariant sector of the theory is obtained upon identifying the complex scalars as so the resulting effective model involves four chiral fields. The holomorphic superpotential in (2.4) simplifies to The identifications in (3.28) are compatible with the Family II of AdS 4 vacua in (2.11) provided For generic choices of boundary conditions, the algebraic condition (3.30) will not be preserved along the flows we will construct. However, as we will see, a specific choice of boundary conditions will preserve the condition (3.30).

Numerical study
The SU (2) invariant sector we are considering involves four complex scalars and, therefore, it is hard to perform an exhaustive study of numerical flows. Still the location of the N = 2 & U (2) AdS 4 vacuum and the D3-brane attractor point (see Figure 4) suggest again various natural choices of boundary conditions. The first choice of boundary conditions consists in distributing the turning points of the numerical flows (pink points in Figure 4) along the Rez 1 and Rez 2 axes, while keepingz 4 and   Figure 5. We find again two limiting/bounding flows associated with the boundary values of the parameter ≈ 0 and ≈ crit , which respectively correspond to the solid red and orange lines in Figure 4 and Figure 5. The flow with ≈ 0 (red solid lines) develops an AdS 4 intermediate behaviour with constant scalars around the turning point at r = 0. This AdS 4 regime uplifts to the N = 2 & U (2) type IIB S-fold of [30]. In addition, we observe the presence of a crossing point with Imz 2 = Rez 2 = 0 at finite r = ±r 0 in all the flows having 0 < < * with * ≈ 0.23428 . An example of this is the flow with = 0.15 < * (blue dotted line) depicted in Figure 4 and Figure 5. 5 On the contrary, note that, for example, the crossing point is not present in the flow with = 0.30 > * (blue dashed line) in the same figures. Lastly, the flow at the special value ≈ * has Imz 2 = Rez 2 = 0 at r 0 → ±∞, namely, the crossing point has been pushed to the endpoints of the flow.
One could also think of relaxing the z 1,3 (0) = z 2 (0) symmetry of the boundary conditions in (3.31). For instance, we can distribute the turning points of the numerical flows (pink points in Figure 6) only along the Rez 1 real axis while keepingz 2 ,z 4 andz 5 fixed at their values in the AdS 4 vacuum. This is This choice of boundary conditions produces regular flows only within the range 0 < < crit with crit ≈ 0.44262 . Furthermore, as it can be seen from Figure 6, the boundary conditions (3.32) localise the flows in the real axis both forz 1 andz 4 . In our numerical scanning, we could not find regular flows with boundary conditions different from (3.31) and (3.32).

SO(3)-invariant flows
The (SO(3) ∩ Z 3 2 )-invariant sector of the theory is a three field model that is obtained upon the identifications z 1 = z 2 = z 3 , z 4 = z 5 = z 6 , z 7 . The above choice of boundary conditions produces regular flows only within the range 0 < < crit with crit ≈ 0.03472 . Some examples are depicted in Figure 9. Performing an exhaustive study of the entire six-dimensional parameter space goes beyond the scope of this paper.

Final remarks
In this work we have investigated a class of Janus-type solutions in type IIB supergravity, some of them accommodating an S-fold intermediate regime at their core. Instead of addressing the problem directly in ten dimensions, we have made use of the effective four-dimensional [SO(1, 1) × SO(6)] R 12 gauged supergravity that arises upon a twisted compactification of type IIB supergravity on S 1 × S 5 [25]. The implementation of an S-duality twist in the reduction along the S 1 is a necessary ingredient for the intermediate S-fold region to exist. This twist totally codifies the simple and analytic dependence of the Janus-type solution on the coordinate η along the S 1 (see Appendix A). In order to obtain the flow solutions along the radial coordinate r, we have resorted to numerical methods and solved the BPS equations in the four-dimensional [SO(1, 1) × SO(6)] R 12 gauged supergravity imposing an AdS 3 -sliced domain-wall ansatz. When uplifted back to ten dimensions, the flow solutions are interpreted as type IIB Janustype solutions that interpolate between a supersymmetric deformation of AdS 5 × S 5 on each side of the Janus (r → ±∞) and generically display an R × AdS 3 × S 1 × S 5 geometry along the flow. However, upon tuning of the boundary conditions for the running scalars when solving the four-dimensional BPS equations, it is possible to obtain a special class of Janus-type solutions for which an S-fold geometry of the form AdS 4 × S 1 × S 5 emerges at the core of the solution (r = 0). We have constructed such special classes of Janus-type solutions for the three largest symmetric S-folds preserving N = 1 & SU (3) [29], N = 2 & U (2) [30] and N = 4 & SO(4) [25] symmetry. A natural extension is to consider Janus-type solutions involving less symmetric S-folds obtained, for example, by turning on the so-called flat deformations dual to marginal deformations in the dual S-fold CFT's [37,40].
One may wonder whether the solutions presented in this work could have been found in the much more studied Einstein-scalar models describing consistent subsectors of the SO(6) gauged supergravity in five dimensions [15,16,17]. However, this is not possible because the axions (pseudo-scalars) Imz 1,2,3 in four dimensions originate from Wilson lines of five-dimensional vector fields along the S 1 [34]. And even if these axions are set to zero at the intermediate Sfold regime of the Janus-type solutions, they are generically activated away from the core of the solutions. Nonetheless, there are direct flows in which Imz 1,2,3 = 0 all along the Janus-type solutions. These particular flows therefore stand a chance of being found analytically in four, five and ten dimensions. It would be very interesting to investigate these direct flows in more detail.
On the other hand, sticking to the common characteristic of usual Janus solutions -AdS d−1 solutions with AdS d asymptotics on each side of the Janus -, it would be interesting to construct flows that depart from an AdS 4 S-fold and arrive at either the same or a different AdS 4 S-fold while preserving the SO(2, 2) isometries of AdS 3 along the flow. The latter would be the curved counterparts of the holographic RG-flows (with a three-dimensional flat slicing) connecting S-fold CFT's in [34], and would holographically describe conformal interfaces in three-dimensional S-fold CFT's. In the M-theory context, examples of supersymmetric Janus solutions dual to conformal interfaces in ABJM theory [50] (and mass deformed versions thereof) were put forward in [6].
Let us conclude with some remarks on the Janus-type solutions with an S-fold at the core for which the space-time geometry conforms to AdS 5 − AdS 3 − AdS 4 − AdS 3 − AdS 5 . Each side of these Janus-type solutions can be viewed as a D3-brane to S-fold flow with a geometry of the form AdS 5 − AdS 3 − AdS 4 , and holography anticipates an interplay between field theories of different dimensionality. Despite the presence of nested AdS factors, the geometry in our solutions is different from the Janus within Janus solutions considered in [51] describing interfaces of higher codimensions in N = 4 SYM 4 . In our case there is no dependence of the ten-dimensional geometry on the fifth coordinate η on S 1 since the metric is a singlet under S-duality and therefore is not affected by the twisted reduction (see eq.(A.11)). The setup here also differs from the Janus on the brane construction of [52] realising interfaces on surface defects in N = 4 SYM 4 . This construction considers an AdS 2 × R slicing on the AdS 3 ⊂ AdS 3 × S 1 parameterising the worldvolume of the D3-brane. It would therefore be interesting to make progress in understanding the holography of the Janus-type solutions presented here in light of the AdS/CFT correspondence. We leave this and other related questions for the future.
where we have defined the metric functions As a non-trivial monodromy on the internal manifold is induced by the set of non-zero constant axion Rez 1,2,3 the real one-form is given by where the one-form potential on CP 2 is Note that, at the core of the SU (3)-invariant flow with ≈ 0 (red solid line) in Figure 3, the scalars undergo a constant behaviour so that f 1 (λ) ∝ e 2A ≈ L l 2 cosh 2 λ L and f 2 (λ) ≈ cst . As a result, the five-dimensional space-time metric conforms to AdS 4 × S 1 as required by the S-fold regime. Note also that, at finite c , the function f 2 in (A.12) vanishes whenever Im[z 1,2,3 ] = 0 (F in (A.4) is bounded both above and bellow). Having Im[z 1,2,3 ] = 0 implies |z 1,2,3 | = 1. Therefore, when this occurs,z 1,2,3 hits the boundary of the unit-disk and the solution becomes singular: the S 1 in (A.11) parameterised by η collapses. Flows of this type can be constructed for which the singularity occurs at a finite radial distance, although we are not investigating them in this work. As mentioned in the introduction, these are the counterparts of the flows to Hades investigated in [6,7].

Axion-dilaton
The axion-dilaton matrix including the ten-dimensional dilaton Φ and the Ramond-Ramond (RR) axion C 0 is given by The SL(2) hyperbolic twist matrix induced by the electromagnetic parameter (we are setting c = 1) reads Importantly, the twist matrix trivialises to A α β = δ α β when having a purely electric gauging with c = 0. This is the case studied in Section 3.1.1.

Five-form flux
The self-dual five-form field strength is given by