$\mathcal{N}=2$ supersymmetric S-folds

Multi-parametric families of AdS$_{4}$ vacua with various amounts of supersymmetry and residual gauge symmetry are found in the $\,[\textrm{SO}(1,1) \times \textrm{SO}(6)] \ltimes \mathbb{R}^{12}\,$ maximal supergravity that arises from the reduction of type IIB supergravity on $\,\mathbb{R} \times \textrm{S}^5\,$. These provide natural candidates to holographically describe new strongly coupled three-dimensional CFT's which are localised on interfaces of $\,\mathcal{N}=4\,$ super-Yang--Mills theory. One such AdS$_{4}$ vacua features a symmetry enhancement to $\,\textrm{SU}(2) \times \textrm{U}(1)\,$ while preserving $\,\mathcal{N}=2\,$ supersymmetry. Fetching techniques from the $\,\textrm{E}_{7(7)}\,$ exceptional field theory, its uplift to a class of $\,\mathcal{N}=2\,$ S-folds of type IIB supergravity of the form $\,\textrm{AdS}_{4} \times \textrm{S}^{1} \times \textrm{S}^5\,$ involving S-duality twists of hyperbolic type along $\,\textrm{S}^{1}\,$ is presented.


Introduction
Electromagnetic duality in four-dimensional maximal supergravity has provided a very rich phenomenology as far as the existence of new gaugings and vacuum solutions are concerned.The prototypical example is the dyonically-gauged SO (8) supergravity where the action of electromagnetic duality on the gauging generates a one-parameter family of inequivalent theories parameterised by a continuous parameter c ∈ 0 , √ 2 − 1 [1].Setting the parameter to c = 0 then the standard (electric) SO(8) supergravity of de Wit and Nicolai [2] is recovered which is known to arise upon dimensional reduction of eleven-dimensional supergravity on a seven-sphere S 7 .The various AdS 4 vacua of the c = 0 theory [3] (see also [4] for an updated encyclopedic reference) get generalised to one-parameter families of vacua when turning on c and, more importantly, new and genuinely dyonic AdS 4 vacua also appear which do not have a well defined (electric) c → 0 limit [1,5,6,7].Other types of four-dimensional solutions, like domain-walls [8,9] or black holes [10,11,12], have also been investigated using instead a phase-like parameterisation ω = arg(1 + i c) ∈ [0 , π/8] of the electromagnetic deformation parameter.However, and despite the rich structure of new solutions at c = 0 , the question about the eleven-dimensional interpretation of the electromagnetic parameter c remains elusive and various no-go theorems have been stated against the existence of such a higher dimensional origin [13,14].Also, for the new supersymmetric AdS 4 vacua at c = 0 , the holographic interpretation of the deformation parameter remains obscure from the perspective of the AdS 4 /CFT 3 correspondence.
Unlike for the SO(8) theory, much more is by now known about the dyonically-gauged ISO (7) supergravity that arises from the reduction of massive IIA supergravity on a sixsphere S 6 [15].In this case the electromagnetic deformation parameter is a discrete (on/off) deformation, namely, it can be set to c = 0 or 1 without loss of generality [16].Various AdS 4 [17,18,19], domain-wall [20,19], and black hole [21,22,23,24] solutions have been constructed which necessarily require a non-zero electromagnetic deformation parameter c .Within this massive IIA context, the electromagnetic parameter is identified with the Romans mass parameter F0 of the ten-dimensional theory [25], and has a holographic interpretation as the Chern-Simons level k of a three-dimensional super-Chern-Simons dual theory [26].
The role of the electromagnetic deformation c has been much less investigated in the context of type IIB supergravity.The relevant dyonically-gauged supergravity in this case is the [SO(1, 1) × SO (6)] ⋉ R 12 theory which arises from the reduction of type IIB supergravity on the product R × S 5 [27].As for the ISO (7) theory, the electromagnetic deformation is again a discrete (on/off) deformation, namely, c = 0 or 1 [16].This four-dimensional supergravity has been shown to contain various types of AdS 4 vacua preserving different amounts of supersymmetry as well as of residual gauge symmetry.In particular, an N = 4 and SO(4) symmetric solution was reported in [28] and subsequently, in [27], uplifted to a class of AdS 4 × S 1 × S 5 S-fold backgrounds of type IIB supergravity using the E 7 (7) exceptional field theory (E 7(7) -EFT).These S-folds involve S-duality twists A (k) (k ≥ 3) that induce SL(2, Z) IIB monodromies M(k) = −S T k of hyperbolic type along S 1 , and can be systematically constructed as quotients of degenerate Janus-like solutions of the type IIB theory [29,30] where the string coupling g s diverges at infinity.Together with the N = 4 & SO(4) solution, additional N = 0 & SO(6) [31] and N = 1 & SU(3) [32] solutions have been found and uplifted to similar S-fold backgrounds of type IIB supergravity with hyperbolic monodromies in [32].From a holographic perspective, these AdS 4 vacua describe new strongly coupled three-dimensional CFT's, referred to as J-fold CFT's in [33] (see also [34,35] and [36]), which are localised on interfaces of N = 4 super-Yang-Mills theory (SYM) [37].In the N = 4 case [33], a hyperbolic monodromy J = −S T k ∈ SL(2, Z) IIB was shown to introduce a Chern-Simons level k in the dual J-fold CFT which, in turn, is constructed from the T (U (N )) theory [38] upon suitable gauging of flavour symmetries.A diagram illustrating this type IIB construction is depicted in Figure 1.On the other hand, a classification of interface SYM theories was performed in [39] (see also [40]) in correspondence to the various amounts of supersymmetry, as well as the largest possible global symmetry, preserved by the interface operators.Three supersymmetric cases were identified: interfaces with N = 4 & SO(4) symmetry, N = 2 & SU(2) × U(1) symmetry and N = 1 & SU(3) symmetry.While the S-folds in [27] and [32] respectively match the symmetries of the N = 4 and N = 1 cases, the gravity duals of the would be N = 2 J-fold CFT's localised on the interface with SU(2) × U(1) symmetry remain missing.In this work we fill this gap and present a new family of AdS 4 × S 1 × S 5 S-folds with N = 2 supersymmetry, SU(2) × U(1) symmetry and, as in the previous cases, involving S-duality twists that induce monodromies of hyperbolic type along S 1 .
The paper is organised as follows.In Section 2 we perform a study of multi-parametric families of AdS 4 vacua in the [SO(1, 1) × SO(6)] ⋉ R 12 maximal supergravity.We find four families of vacua, one of them being N = 2 supersymmetric and containing a vacuum with a residual symmetry enhancement to SU(2)×U (1) .In Section 3, by implementing a generalised Scherk-Schwarz (S-S) ansatz in E 7(7) -EFT, we uplift such an AdS 4 vacuum to a class of AdS 4 × S 1 × S 5 N = 2 S-folds of type IIB supergravity with SU(2) × U(1) symmetry and a non-trivial hyperbolic monodromy along S 1 .In Section 4 we present our conclusions and discuss future directions.
• A two-parameter family of N = 1 & U(1) 2 symmetric AdS 4 vacua with symmetry enhancements to SU(2) × U(1) and SU(3) at specific values of the two arbitrary parameters.
• A one-parameter family of N = 2 & U(1) 2 symmetric AdS 4 vacua with a symmetry enhancement to SU(2) × U(1) at a special value of the arbitrary parameter.
The N = 2 family of AdS 4 vacua is new and we will uplift the solution with SU(2)×U(1) enhanced residual symmetry to a new and analytic family of S-fold backgrounds of type IIB supergravity in Section 3.

The N = 8 theory: gauging and scalar potential
We follow the conventions and notation of [32], which slightly differ from those of [27], to describe the dyonically-gauged maximal supergravity with gauge group G in (2.1).For the purposes of this work, i.e. the study of AdS 4 vacua, we set to zero all the vector and (auxiliary [41]) tensor fields of the theory, so that the bosonic Lagrangian reduces to the following one which describes the scalar fields M M N coupled to Einstein gravity in the presence of a scalar potential.The scalar fields serve as coordinates on the coset space of maximal supergravity with M = 1, . . ., 56 being a fundamental index of E 7 (7) .The coset representative V is constructed by direct exponentiation of the 70 non-compact generators t A B (with t A A = 0 ) and t ABCD = t [ABCD] generators of E 7 (7) in the SL(8) basis 1 .The scalar potential in (2.2), which survives our truncation to the Einstein-scalar sector, is induced by the gauging of the group G in (2.1) within the maximal theory and has the following general form: which depends on the gauge coupling g , the scalar matrix M M N (and its inverse M M N ) and on a constant embedding tensor X M N P living in the 912 of E 7(7) [43].This tensor codifies how the gauge group G is embedded into the E 7 (7) duality group of maximal supergravity.Moreover, it also specifies the gauge connection which involves both electric and magnetic vector fields transforming under the Sp(56) group of electromagnetic transformations of the theory (for reviews see [44,45]).

Z 3 2 invariant sector
In order to efficiently search for extrema of the scalar potential (2.4), we will now construct a Z 3 2 invariant sector of the [SO(1, 1) × SO( 6)] ⋉ R 12 maximal supergravity.This sector can be recast as a minimal N = 1 supergravity coupled to seven chiral multiplets z i with i = 1, . . ., 7 .The same invariant sector has recently been explored in the dyonically-gauged ISO(7) theory [19] and the purely electric SO(8) theory [46], and it originally appeared in the context of type II orientifold compactifications with generalised fluxes [47,48].
To describe this sector of the maximal theory, we first focus on a four-element Klein subgroup of SL (8) .Its action on the fundamental index A is given by (2.7) together with the remaining generators I and 2 .In addition, we will also require invariance under an extra Z * 2 generator acting as The resulting Z 3 2 invariant sector describes N = 1 supergravity coupled to seven chiral multiplets (and no vector multiplets) (2.9) The fourteen real spinless fields are associated with generators t A B (scalars) and t [ABCD] (pseudo-scalars) of E 7 (7) in the SL(8) basis.The former have associated generators of the form whereas the latter correspond with generators given by Exponentiating (2.10) and (2.11) with coefficients ϕ i and χ i as yields a parameterisation of an 7 subspace of the coset space in (2.3).The kinetic terms in the resulting N = 1 sector follow from (2.2) and (2.12), and are given by (2.13) These match the standard kinetic terms L kin = −(∂ 2 z i ,z j K) dz i ∧ * dz j for a set of seven chiral fields z i with Kähler potential (2.14) Lastly, when restricted to the Z 3 2 invariant sector entering (2.12), the scalar potential, as computed from (2.4), can be recovered from a holomorphic superpotential using the standard N = 1 formula where is the Kähler derivative and K z i zj is the inverse of the Kähler metric K z i zj ≡ ∂ 2 z i ,z j K .Note that only the last term in the superpotential (2.15) turns out to be sensitive to the electromagnetic parameter c .

New families of AdS 4 vacua
A thorough study of the structure of extrema of the scalar potential (2.4), restricted to the Z 3  2 invariant sector, reveals a rich structure of (fairly) symmetric AdS 4 vacua.We find four families of vacua preserving N = 0, 1, 2 or 4 supersymmetry as well as various residual gauge symmetries ranging from U(1) 2 to SO(6) ∼ SU(4) .The three supersymmetric families are also supersymmetric within the N = 1 model with seven chirals presented in the previous section, and therefore satisfy the F-flatness conditions that follow from the superpotential (2.15) and Kähler potential (2.14).Importantly, all the AdS 4 vacua we will present in this section are genuinely dyonic, namely, they disappear if taking the limit c → 0 to a purely electric gauging of G in (2.1).
There is a three-parameter family of N = 0 solutions that preserves U(1) 3 and is located at with χ 1,2,3 being arbitrary (real) parameters.This family of solutions has a vacuum energy given by and a spectrum of normalised scalar masses of the form where χ ≡ χ 1 + χ 2 + χ 3 and L 2 = −3/V 0 is the AdS 4 radius.This family of solutions is perturbatively unstable due to the mass eigenvalue −3 lying below the Breitenlohner-Freedman bound for stability in AdS 4 [49] .The computation of the vector masses yields (2.21) Note that a generic solution in this family preserves a U(1) 3 symmetry as three vectors are generically massless.Therefore, out of the 28 massless scalars in (2.20),only 3 of them correspond to physical directions in the scalar potential.Imposing a pairwise identification between the free axions χ 1,2,3 results in a symmetry enhancement to SU(2) × U(1) 2 .A further identification χ 1 = χ 2 = χ 3 = 0 implies a symmetry enhancement to SU(3) × U(1) .Lastly, setting χ 1,2,3 = 0 enhances the symmetry to SU(4) ∼ SO(6) .This SO(6) symmetric solution was originally studied in [29] from a ten-dimensional perspective and, more recently, connected with a family of type IIB S-fold backgrounds in [32].
There is a two-parameter family of N = 1 supersymmetric AdS 4 solutions that preserves U(1) 2 and is located at subject to the constraint This family of AdS 4 solutions has a vacuum energy given by and a spectrum of normalised scalar masses of the form where L 2 = −3/V 0 is the AdS 4 radius.The computation of the vector masses yields (2.26) Note that a generic solution in this family preserves U(1) 2 as only two vectors are generically massless.Therefore, out of the 28 massless scalars in (2.25),only 2 of them correspond to physical directions in the potential.The residual symmetry gets enhanced to SU(2) × U(1) when imposing a pairwise identification between the axions χ 1,2,3 so that a total of four vectors become massless.Finally there is a symmetry enhancement to SU(3) when setting χ 1,2,3 = 0 so that a total of eight vectors become massless.The SU(3) symmetric solution was recently uplifted to a ten-dimensional family of type IIB S-fold backgrounds in [32].

(1) symmetry
There is a one-parameter family of N = 2 supersymmetric AdS 4 solutions that preserves U(1) 2 and is located at This family of AdS 4 solutions has a vacuum energy given by and a spectrum of normalised scalar masses of the form ) where L 2 = −3/V 0 is the AdS 4 radius.The computation of the vector masses yields (2.30) Note that a generic solution in this family preserves U(1) 2 as only two vectors are generically massless.Therefore, out of the 30 massless scalars in (2.29),only 4 of them correspond to physical directions in the scalar potential.However, the residual symmetry gets enhanced to SU(2) × U(1) when χ = 0 and two additional vectors become massless.This special AdS 4 vacuum will be uplifted to a ten-dimensional family of type IIB S-fold backgrounds in Section 3.

N = 4 vacuum with SO(4) symmetry
There is an N = 4 supersymmetric AdS 4 solution that preserves SO(4) and is located at This AdS 4 solution has a vacuum energy given by as for the previous solution, and a spectrum of normalised scalar masses of the form where L 2 = −3/V 0 is the AdS 4 radius.The computation of the vector masses yields thus reflecting the SO(4) residual symmetry at the AdS 4 solution.Therefore, out of the 48 massless scalars in (2.33),only 26 of them correspond to physical directions in the scalar potential.This N = 4 solution was first reported in [28], and then uplifted to a tendimensional family of type IIB S-fold backgrounds in [27].

S-folds with N = 2 supersymmetry
From this moment on we will set without loss of generality.From (2.18), (2.22), (2.27) and (2.31) it becomes clear that varying c amounts to a rescaling of the vacuum expection values of z 1,2,3 ∝ c at the AdS 4 vacua.After c has been set to unity, varying g simply corresponds to a rescaling of the vacuum energy V 0 ∝ g 2 c −1 and thus to a redefinition of the AdS 4 radius L 2 = −3/V 0 .Let us emphasise again that all the AdS 4 vacua in Section 2.3 are genuinely dyonic as they do not survive the limit c → 0 to implement a purely electric gauging.In this limit one has that Im(z 1,2,3 ) → 0 or, by virtue of (2.9), a runaway behaviour towards the boundary of moduli space ϕ 1,2,3 → ∞ .Going back to the goal of this section, the N = 2 family of solutions in Section 2.3.3 is new and preserves a U(1) 2 symmetry.It is a one-parameter family of AdS 4 vacua and, in the special case of the parameter vanishing χ = 0 , there is an enhancement of symmetry to SU(2) × U(1) .Following [27], and implementing a generalised S-S ansatz in E 7(7) -EFT [50], we will uplift such an N = 2 & SU(2) × U(1) symmetric AdS 4 vacuum to a class of ten-dimensional S-fold backgrounds of type IIB supergravity of the form AdS 4 × S 1 × S 5 with an S-duality hyperbolic monodromy along S 1 .

Type IIB uplift using E 7(7) -EFT
Generalised Scherk-Schwarz (S-S) reductions of exceptional field theory (EFT) have proved a very efficient method to perform consistent truncations of eleven-dimensional and type IIB supergravity on spheres and hyperboloids [51].Here we are interested in the uplift of an AdS 4 vacuum of a four-dimensional gauged maximal supergravity, which thus selects the E 7(7) -EFT of [50] as the natural framework to carry out this mission.
The E 7(7) -EFT lives in an extended space-time that consists of an external four-dimensional space with coordinates x µ ( µ = 0, . . ., 3 ) and a 56-dimensional generalised internal space with coordinates Y M ( M = 1, . . ., 56 ) in the fundamental representation 56 of E 7 (7) , subject to the action of the E 7(7) -covariant generalised diffeomorphisms.In order to uplift an AdS 4 vacuum amongst those in Section 2.3 to a ten-dimensional background of type IIB supergravity, the relevant field content of E 7(7) -EFT reduces to the external metric g µν (x, Y ) and the internal generalised metric M M N (x, Y ) (vector and tensor fields are consistently set to zero).These are connected with the metric g µν (x) and the scalar fields M M N (x) of the four-dimensional maximal supergravity in (2.2) via a generalised S-S ansatz [51] The entire dependence on the Y M coordinates is then encoded in a twist matrix U M K (Y ) and a scaling function ρ(Y ) satisfying where X M N K is the embedding tensor specifying the gauging in the four-dimensional supergravity, ϑ M is a constant scaling tensor and | 912 denotes projection onto the 912 irreducible representation of E 7 (7) where the embedding tensor lives.
For the dyonic gauging of G ⊂ SL(8) in (2.1) the non-vanishing components of the embedding tensor were given in (2.5) and the tensor ϑ M vanishes identically.The generalised S-S ansatz depends on six physical coordinates (y i , ỹ) ∈ Y M : five of them are electric y i (i = 2, . . ., 6) and one is magnetic ỹ .Considering the electric-magnetic splitting of generalised coordinates Y M = (Y AB , Y AB ) under SL( 8) ⊂ E 7 (7) , one has with components and

.9)
The twist matrix in (3.9) also depends on a function K(y i ) which is given in this case by a hypergeometric function [27] Using the dictionary between the fields of type IIB supergravity and those of E 7(7) -EFT [52,53], together with the S-S ansatz (3.2) involving generalised twist parameters (3.5)-(3.9),one arrives at the final uplift formulae for the purely internal components of the type IIB fields: (inverse) metric G mn , two-form potentials B α = (B 2 , C 2 ) with α = 1, 2 , four-form potential C 4 and axion-dilaton m αβ .
The various blocks M mn , M p nβ , M p lmn and M mα nβ entering the r.h.s of (3.11) can be extracted from the internal generalised metric M M N (x, y i , ỹ) by performing the grouptheoretical decomposition that is relevant for the embedding of type IIB supergravity into E 7(7) -EFT: The physical coordinates are identified as y m = (y i , ỹ) , with m = (i, 7) and i = 2, . . ., 6 , which implies a further group-theoretical branching GL(6) → GL(1) × GL( 5) compatible with the R (or S 1 ) × S 5 factorisation of the geometry we are behind of.The various mappings between coordinates discussed above are summarised as We refer the reader to the original works [52,53] (and also [27,32]) for more details on the generalised S-S reductions of E 7( 7) -EFT and their connection with the gauged maximal supergravities.
We now move to the uplift the AdS 4 vacuum with N = 2 & SU(2) × U(1) symmetry discussed in Section 2.3.3 to a ten-dimensional background of type IIB supergravity using (3.11).We have explicitly verified that the ten-dimensional equations of motion and Bianchi identities of type IIB supergravity are satisfied2 .

Ten-dimensional metric
We adopt the conventions of [32] to describe the geometry of the round five-sphere S 5 .Using coordinates y i ( i = 2, . . ., 6 ) to parameterise S 5 , the metric and its inverse are given by Ĝij = δ ij + δ ik δ jl y k y l 1 − y m δ mn y n and Ĝij = δ ij − y i y j .
However it will also be convenient to introduce a set of embedding coordinates Y m on R 6 ( m = 2, . . ., 7 ) of the form so that the Killing vectors on S 5 are constructed as Following the derivation of [27], the internal part of the ten-dimensional metric has components in (3.11) given by where M ij 18 = 0 as a consequence of having set χ = 0 in the N = 2 AdS 4 vacuum, and where we have defined The warping factor ∆ in (3.17) is nowhere vanishing and reads ∆ = (det G) The six-dimensional internal metric becomes more transparent if first introducing a new variable for the magnetic coordinate and then a set of angular variables for S 5 of the form with ranges given by In this manner, and upon introducing a set of SU(2) left-invariant one-forms 2 (cos α dβ + sin α sin β dγ) , the internal six-dimensional metric takes a simple R × S 5 form with a warping factor ∆ −1 = 6 − 2 cos(2θ) and where we have introduced S 2 and (squashed) S3 metrics to describe the deformation of the internal S 5 .These metrics are explicitly given by Bringing together (3.24) and the external AdS 4 part of the geometry, one obtains a tendimensional metric of the form 3 This metric has an SU(2) × U(1) φ × U(1) σ symmetry, where U(1) σ acts as a rotation on the (σ 1 , σ 3 )-plane.Finally, our choice of undeformed frames for the metric (3.27) is The two-form potentials B α = (B 2 , C 2 ) in (3.11) transform as a doublet under the global S-duality group SL(2, R) IIB of type IIB supergravity.An explicit computation along the lines of [27] shows that in terms of a local SO(1, 1) ⊂ SL(2, R) IIB twist matrix This matrix encodes the dependence of the two-form potentials on the direction η .Using the scalar block M kl mγ at the N = 2 AdS 4 vacuum under consideration, and using differential form notation, one finds with (3.32)The two-form potentials in (3.32) preserve SU(2) × U(1) σ but break the U(1) φ factor due to the explicit dependence on the coordinate φ .

C 4 potential
The internal component of the four-form potential C 4 can be explicitly obtained from the third uplift formula in (3.11).Computing the associated (purely internal) five-form field strength, and imposing ten-dimensional self-duality, one gets where denotes the volume of the deformed five-sphere.Note that U(1) φ is also broken by F 5 due to its explicit dependence on the coordinate φ .

Axion-dilaton
The axion-dilaton matrix m αβ can be obtained from the last equation in (3.11).Transforming linearly under S-duality, a direct computation shows an explicit dependence of m αβ on the A-twist in (3.30) of the form with τ = C 0 + i e −Φ and

S-fold interpretation
The dependence of the full type IIB solution on the coordinate η along the R direction of the geometry (3.27) is totally encoded in the local SL(2, R) IIB A-twist in (3.30).This twist matrix is of hyperbolic type and thus induces a non-trivial monodromy when forcing the η coordinate to be periodic η → η + T with period T , namely, when replacing R → S 1 in the geometry.Generalising the A-twist in (3.30) to a discrete k-family ( k ∈ N with k ≥ 3 ) of new ones the monodromy (3.37) gets generalised to a k-family of SL(2, Z) IIB hyperbolic monodromies with T (k) = ln(k + √ k 2 − 4) − ln(2) and Tr M(k) > 2 .Therefore, as discussed in [27] (see also [32]), these backgrounds can be interpreted as locally geometric compactifications on S 1 × S 5 involving a k-family of S-duality monodromies (3.39).These monodromies can be written as and thus define a k-family of S-fold backgrounds.Moreover, the argument wielded in [27] for the straightforward uplift of the four-dimensional supersymmetries to ten dimensions relied on the monodromy (3.37) being in the hyperbolic conjugacy class of SL(2, R) IIB .This is still our case, so the S-folds presented here preserve N = 2 supersymmetry.Lastly, various holographic aspects of both N = 4 [27] and N = 1 [32,36] S-folds with hyperbolic monodromies have respectively been investigated in [33,34,35] and [36] within the context of three-dimensional quiver theories involving N = 4 T (U (N )) theories [38], and their potential generalisation to N = 1 SCFT's.It would be interesting to extend these holographic studies to the N = 2 S-folds with hyperbolic monodromies (3.39) presented in this work.

Connection with Janus-like solutions
The type IIB solution with N = 2 & SU(2) × U(1) symmetry we just obtained can be mapped to a new (but equivalent) solution with a linear dilaton profile along the coordinate η upon performing a global Λ ∈ SL(2, R) IIB transformation, equivalently a change of duality frame, based on the matrix element The composed action of Λ A −1 (η) on (3.36) yields a shift of the form Φ → Φ−2η .Therefore, a degenerate Janus-like behaviour with a linear dilaton Φ running from −∞ to ∞ becomes manifest giving rise to a varying string coupling g s that interpolates between the singular values 0 and ∞ .Upon performing the Λ ∈ SL(2, R) IIB transformation (3.41) on the original solution found in Section 3.1, a new type IIB background is generated.The metric and self-dual five-form flux are SL(2, R) IIB singlets and are not affected by the transformation.Therefore, they take the same form as in (3.27) and (3.33), namely,  (3.46) The nowhere vanishing warping factor still reads ∆ −4 = 6 − 2 cos(2θ) . (3.47) In the asymptotic region at η → −∞ one has that g s in (3.42) diverges (strong coupling) and B 2 dominates over other gauge potentials, e.g., C 0 → 0 and C 2 → 0 .On the contrary, in the asymptotic region at η → ∞ , the solution becomes dominated by C 0 and C 2 whereas g s → 0 (weak coupling) and B 2 → 0 .At intermediate values of the coordinate η one has an interpolating behaviour between these two regimes.Finally, it is also worth noticing that, unlike for the N = 4 [27] and N = 1 [32] S-folds, there is no SL(2, R) IIB frame in which the axion C 0 (and thus the dual θ-angle) vanishes identically or becomes independent of the coordinate η .

Conclusions
In this work we have extended the study of AdS 4 vacua in [31,28,32] for the dyonicallygauged [SO(1, 1) × SO(6)] ⋉ R 12 maximal supergravity and found multi-parametric families of new AdS 4 vacua.Within one such families, all the solutions preserve the same amount of supersymmetry but, importantly, residual symmetry enhancements occur at particular values of the parameters.The previously known N = 0 & SO(6) [31], N = 1 & SU(3) [32] and N = 4 & SO(4) [28] AdS 4 vacua are shown to correspond to the points of largest symmetry enhancement within their respective families.This is in line with the analysis of (global) symmetry breaking patterns of three-dimensional interface SYM theories presented in [39].
In the second part of the paper we focused on the new family of N = 2 supersymmetric AdS 4 vacua and, more concretely, on the vacuum within this family featuring the largest possible residual symmetry, which turns to be SU(2) × U(1) .By implementing a generalised S-S ansatz in E 7( 7 where we have shifted the coordinate φ → ψ + π 4 .However, since these terms are generated by the generalised S-S ansatz discussed in Section 3.1, we will retain them here. background fluxes break the U(1) φ factor explicitly by introducing a dependence on the coordinate φ .In many aspects, the realisation of symmetries is much alike the AdS 5 × S 5 background by Pilch and Warner [54] that uplifts the N = 2 and SU(2) × U(1) symmetric AdS 5 vacuum of the five-dimensional SO(6) maximal supergravity presented in [55].
Finally it would be interesting to investigate the brane setups underlying the families of S-folds presented here (and in [32]), especially due to the non-trivial SL(2, Z) IIB hyperbolic monodromies M(k) = −S T k .It would also be interesting to investigate holographic aspects of such N = 2 and N = 1 S-folds (in the spirit of the J-fold CFT's of [33,34,35][36] with J = −S T k ), as well as to study holographic RG flows by explicitly constructing domain-wall solutions interpolating between the various families of AdS 4 vacua presented in this work.Lastly, since the S-folds here and in [32] display SU(2) isometries in the internal geometry, it would also be interesting to apply non-abelian T-duality in order to generate new analytic type IIA backgrounds.We plan to address these and related issues in the future.