S-folds and holographic RG flows on the D3-brane

Type IIB S-folds of the form $\,\textrm{AdS}_{4} \times \textrm{S}^1 \times \textrm{S}^5\,$ are conjectured to correspond to new strongly coupled three-dimensional CFT's on a localised interface of $\,\textrm{SYM}_{4}\,$. In this work we construct holographic RG flows on the D3-brane that generically connect anisotropic deformations of $\,\textrm{SYM}_{4}\,$ in the UV to various S-fold $\textrm{CFT}$'s in the IR with different amounts of supersymmetry and flavour symmetries. Examples of holographic RG flows between \mbox{S-fold} CFT's are also presented. Lastly a geometric interpretation of axion deformations is provided in terms of monodromies on the internal $\,\textrm{S}^{5}\,$ when moving around the $\,\textrm{S}^{1}$. Special attention is paid to the monodromy-induced patterns of symmetry breaking as classified by the mapping torus $T_{h}(\textrm{S}^5)$.


Motivation and outlook
Since its discovery, electromagnetic duality has and continues to provide plenty of intriguing phenomena both in supergravity and quantum field theory. From a string theory perspective, it plays a prominent role amongst the various string/M-theory dualities [1]. Within the realm of four-dimensional maximal gauged supergravity [2], electromagnetic duality was shown to generate new classes of dyonic semi-simple gaugings which hitherto lack a string/M-theory origin. The prototypical example being the SO(8) gauged supergravity which, in its purely electric version [3], arises from the dimensional reduction of eleven-dimensional supergravity on a seven-sphere S 7 [4,5]. The SO (8) supergravity was shown to admit a generalisation to a one-parameter family of dyonic theories based on an electromagnetic deformation parameter originally denoted by c in [6]. However, despite the plethora of new AdS 4 [6,7,8,9], black hole [10,11,12], Janus [13] and domain-wall [14,15] solutions that appear when turning on c , a higher-dimensional interpretation of such parameter (if any) remains elusive and some no-go theorems against such a higher-dimensional geometric origin have been stated [16,17]. After [6], a characterisation and classification of electromagnetic deformations for other gaugings of maximal supergravity was presented in [18] and extended to less supersymmetric theories in [19]. Amongst the non-semisimple gaugings investigated in [18], the ISO(7) and [ SO(1, 1) × SO(6) ] R 12 gauged maximal supergravities have received most of the attention due to their connection with (massive) type IIA [20] and type IIB [21] string theory, respectively.

Type IIA
The dyonic ISO(7) supergravity [22] has been shown to contain a rich structure of AdS 4 vacua [22,23,24,25], asymptotically AdS black hole [26,27], Janus [28] and domain-wall [29,30] solutions. Using the uplift formulae in [31] the AdS 4 vacua have been uplifted to AdS 4 × S 6 backgrounds of massive type IIA supergravity [20,31,32] which further extend to well-defined backgrounds of string theory. This string-theoretic origin permitted the higherdimensional and holographic understanding of the electromagnetic parameter c within the massive IIA context [20]: the parameter c was identified with the Romans mass parameter F 0 [33] in ten dimensions, and interpreted holographically as the Chern-Simons (CS) level k of the dual three-dimensional Chern-Simons-matter theories. Further confirmation came from the successful holographic counting of asymptotically AdS black hole microstates carried out in [34,35].
Getting closer to the goal of this work, domain-wall solutions describing holographic RG flows involing the CFT 3 's dual to the various AdS 4 vacua of the ISO(7) maximal supergravity were obtained in [29,30]. Two types of supersymmetric holographic RG flows were constructed numerically: i) SYM 3 to CFT 3 flows connecting the non-conformal behaviour of the D2-brane in the UV (dual to super-Yang-Mills theory deformed by the CS term) to the various AdS 4 vacua (dual to CFT 3 's) in the IR.
ii) CFT 3 to CFT 3 flows connecting two AdS 4 vacua of the theory.
A summary of such supersymmetric domain-walls is displayed in Figure 1 (left diagram) where we have denoted by N &G 0 an AdS 4 vacuum preserving N supersymmetries and a residual gauge symmetry groupG 0 . The flavour symmetry group G of the dual CFT 3 is then extracted fromG 0 = G × SO(N ) where SO(N ) is the corresponding R-symmetry group.

Type IIB
The dyonic [SO(1, 1) × SO(6)] R 12 maximal supergravity has also been shown to contain a rich structure of AdS 4 vacua [21,36,37]. Unlike for their type IIA counterparts, these AdS 4 vacua organise into multi-parametric families depending on a set of constant axions χ's [37]. Each of these families preserves a fixed amount of supersymmetry and has a fixed AdS 4 radius. Therefore, the axions χ's are interpreted as exactly marginal deformations in the dual CFT 3 's. Within a family of AdS 4 vacua, different values of the axions cause a different breaking of the residual gauge symmetry. More concretely, they induce a breaking of flavour symmetries in the dual CFT 3 's. The AdS 4 vacua with vanishing axions feature the largest possible symmetry within their respective families. Focusing on supersymmetric solutions only, three such AdS 4 vacua have been found preserving various amounts of (super) symmetry: N = 1 & SU(3) [36], N = 2 & SU(2) × U(1) [37] and N = 4 & SO(4) [38]. Upon subtraction of the corresponding R-symmetry factors, these AdS 4 vacua are dual to CFT 3 's with SU(3) , SU(2) and trivial flavour symmetry groups, respectively. These vacua of the [SO(1, 1) × SO(6)] R 12 maximal supergravity have been uplifted to S-fold backgrounds of type IIB supergravity on AdS 4 × S 1 × S 5 that involve non-trivial SL(2, Z) IIB monodromies along S 1 in the hyperbolic class [21,36,37] -commonly denoted by J ∈ SL(2, Z) IIB -and thus the corresponding CFT 3 duals have been referred to as J-fold CFT 3 's in the literature [39]. These J-fold CFT 3 's and their gravity duals have also been investigated using a five-dimensional gauged supergravity approach within the context of holographic interfaces in N = 4 SYM 4 [40,41] [42], and arise as limiting cases of Janus solutions.
In this work we will construct supersymmetric domain-wall (DW 4 ) solutions describing holographic RG flows involving the J-fold CFT 3 's dual to the various AdS 4 vacua of the [SO(1, 1) × SO(6)] R 12 maximal supergravity. As in the type IIA case previously reviewed, two types of holographic RG flows appear in the type IIB context: i) SYM 4 to CFT 3 flows connecting a c-deformation of the D3-brane behaviour in the UV (dual to an anisotropic deformation of SYM 4 ) to the various axion-vanishing AdS 4 vacua (dual to J-fold CFT 3 's) in the IR.
ii) CFT 3 to CFT 3 flows connecting two AdS 4 vacua of the theory, the one at the UV generically involving non-zero axions (equivalently marginal deformations).
A summary of these new supersymmetric domain-walls is also displayed in Figure 1 (right diagram) where the notation N &G 0 has been maintained for the various AdS 4 vacua. The CFT 3 to CFT 3 flow depicted in the diagram corresponds to the case with vanishing axions (no marginal deformations) in the UV, so that the largest possible SU(3) symmetry within the N = 1 family of AdS 4 vacua is realised in the UV. Notwithstanding, CFT 3 to CFT 3 flows also exist that reach N = 1 AdS 4 vacua in the UV with non-vanishing axions so that a smaller SU(2) × U(1) ⊂ SU(3) symmetry is realised in the UV. As we will show in this work, the RG flows of type i) will involve a set of sub-leading corrections in the parameter c of the four-dimensional DW 4 description of the D3-brane solution at c = 0 dual to (isotropic) N = 4 SYM 4 . These corrections will induce anisotropy in the UV. Type IIB examples of such an anisotropic behaviour of SYM 4 have previously been obtained in terms of a charge density of dissolved D7-branes [43] or backreacted geometries corresponding to the intersection of D3-and (smeared) D5-branes along 2 + 1 dimensions [44,45]. The type IIB background in [44,45] involves non-trivial RR fluxes F 3 and F 5 and involves a stack of N c D3-branes and N f flavour D5-branes. Here we obtain anisotropy in a purely closed string setup without sources -sourceless Bianchi identities are satisfied in ten dimensions as the effective four-dimensional supergravity enjoys maximal supersymmetryby implementing the locally geometric SL(2) IIB twist that generates the S-fold background in the IR. As a by-product of the mechanism the results preserve SL(2) IIB covariance.
Another interesting aspect regarding the UV behaviour of the RG flows presented in this work has to do with the role being played by the axions χ's. As we will show explicitly, these axions can be locally reabsorbed in a reparameterisation of the angular coordinates on the five-sphere so that a non-trivial monodromy arises on S 5 × S 1 . We will investigate this issue for the two type IIB backgrounds controlling the UV behaviour of the RG flows: the D3-brane and the N = 1 family of S-folds. We will build upon [46], where this issue has been investigated for the N = 2 family of S-folds, and generalise the results therein to accommodate the two cases of relevance in this work and, as a by-product, also the nonsupersymmetric S-folds in [37]. In this manner we will understand the geometric role played by the axions χ's and relate it to the patterns of (flavour) symmetry breaking that these axions induce as classified by the mapping torus T h (S 5 ).
The paper is organised as follows. In Section 2 we present the supergravity model, the BPS flow equations and the four-dimensional AdS 4 and non-AdS 4 solutions relevant for the work. In Section 3 we numerically construct the holographic RG flows, both SYM 4 to CFT 3 and CFT 3 to CFT 3 flows. In Section 4 we investigate the flows from a ten-dimensional perspective. First we focus on the issue of anisotropy when the UV regime of the flows is controlled by the deformed D3-brane at c = 0 , and then discuss global vs local geometric aspects of the type IIB backgrounds when the axions χ's are activated, as well as the implications they have for symmetry breaking. We conclude and discuss some future lines in Section 5. Some technicalities regarding group theoretical aspects of the four-dimensional supergravity model and its potential oxidation to five dimensions are collected in the Appendix.

Four dimensions
Let us consider the maximal supergravity with gauge group [SO(1, 1) × SO(6)] R 12 that arises upon dimensional reduction of type IIB supergravity on S 5 × R [21]. For the sake of tractability, we restrict ourselves to the Z 3 2 -invariant sector of the theory constructed in [37].

Supergravity model and flow equations
This sector describes a minimal N = 1 supergravity coupled to seven chiral multiplets with complex scalars z i and no vector multiplets. We parameterise the complex scalars of the chiral multiplets as The seven complex fields serve as coordinates on a Kähler scalar geometry characterised by a Kähler potential of the form The scalar potential for the Z 3 2 -invariant sector of the theory follows from an holomorphic superpotential of the form which is originated from the gauging in the maximal theory. Truncating away the fermions the Lagrangian acquires an Einstein-scalar form and reads where Σ I = { y 1 , χ 1 , y 2 , χ 2 , . . . , y 7 , χ 7 } (I = 1, . . . , 14) denotes the real and imaginary components of the complex fields z i . The kinetic matrix for these (real) fields reads The scalar potential V can be recovered from the holomorphic superpotential (2.4) using the standard result where D z i W ≡ ∂ z i W + (∂ z i K)W is the Kähler derivative and K z izj is the inverse of the Kähler metric K z izj ≡ ∂ 2 z i ,z j K . This is the N = 1 supergravity model we will investigate during the rest of the work.

First-order flow equations
In order to study RG flows holographically, we will investigate flat-sliced domain-wall (DW 4 ) solutions whose metric takes the form where z ∈ (−∞, ∞) is the coordinate transverse to the domain-wall and A(z) is the scale factor. Asking for the vanishing of the supersymmetry variations of fermions (gravitino and chiralini) in the N = 1 supergravity model, a set of first-order BPS equations consisting of is obtained to which we will refer as flow equations. The real superpotential |W| is constructed from the (complex) gravitino mass term with W in (2.4), and fully specifies the flow equations in (2.9).

AdS 4 vacua: modes and dimensions of dual operators
The simplest solutions to the BPS equations (2.9) are supersymmetric AdS 4 vacua. These solutions have constant scalars and thus satisfy (2.9) provided where |W 0 | −2 = L 2 = −3/V 0 corresponds to values evaluated at the AdS 4 vacuum and C is an arbitrary constant that can be reabsorbed by a rescaling of the coordinates x α in (2.8).
The AdS 4 /CFT 3 holographic dictionary then states that, at an AdS 4 vacuum, scalars fields with a normalised mass m 2 L 2 < 0 correspond to relevant operators, m 2 L 2 = 0 to marginal operators and m 2 L 2 > 0 to irrelevant operators in the dual field theory. Each scalar field comes along with two modes with conformal dimensions The conformal dimension of the dual operator is then identified with ∆ + . The question about which of the two modes is selected by supersymmetry is answered by the diagonalisation of the matrix From a field theory perspective, ∆ − is interpreted as a source for the operator whereas ∆ + is interpreted as a vacuum expectation value (VEV) for the operator. There is an ambiguity with this interpretation whenever the masses lie within the window − 9 4 < m 2 L 2 < − 5 4 , for which an alternative quantisation of the scalar field is possible that interchanges the source and the VEV [47].
In the following we review the three families of N = 1 , N = 2 and N = 4 supersymmetric AdS 4 vacua presented in ref. [37]. These vacua feature a hierarchy of vacuum energies given by (2.14) We will present these families of solutions as well as the spectrum of scalars around the most symmetric vacuum within each family. A summary of such most symmetric AdS 4 vacua can also be found in Table 1.

N = 1 family: vacuum with SU(3) symmetry
There is a two-parameter family of N = 1 supersymmetric AdS 4 solutions that preserves U(1) 2 . It is located at so that |z 4,5,6,7 | = 1 , and is subject to the constraint 1 This family of AdS 4 solutions has a vacuum energy given by and a spectrum of Z 3 2 -invariant normalised scalar masses of the form where L 2 = −3/V 0 is the AdS 4 radius. As discussed in [37], a generic solution in this family preserves U(1) 2 . However, the residual symmetry gets enhanced to SU(2) × U(1) when imposing a pairwise identification between the axions χ 1,2,3 . Finally there is a symmetry enhancement to SU(3) when setting χ 1,2,3 = 0 . This N = 1 & SU(3) symmetric AdS 4 vacuum was uplifted to a family of type IIB S-folds with N = 1 supersymmetry in [36]. Setting the moduli χ 1,2,3 = 0 yields with masses in (2.18) given by By virtue of (2.12), the set of normalised scalar masses in (2.20) implies a set of conformal dimensions ∆ ± for the dual operators given by (2.21) The highlighted conformal dimensions in (2.21) appear as eigenvalues of the matrix (2.13) and will play a role later on when studying holographic RG flows involving this conformal fixed point.

N = 2 family: vacuum with SU(2) × U(1) symmetry
There is a one-parameter family of N = 2 supersymmetric AdS 4 solutions that preserves U(1) 2 . It is located at so that |z 4,6 | = |z 5,7 | = 1 . This family of AdS 4 solutions has a vacuum energy given by and a spectrum of Z 3 2 -invariant normalised scalar masses of the form where L 2 = −3/V 0 is the AdS 4 radius. A generic solution in this family preserves U(1) 2 , but the residual symmetry gets enhanced to SU(2) × U(1) when χ = 0 .
This N = 2 & SU(2) × U(1) symmetric AdS 4 vacuum was uplifted to a family of type IIB S-folds with N = 2 supersymmetry in [37] (see [46] for the uplift including the axion χ ). Setting the modulus χ = 0 yields with masses in (2.24) given by Upon solving (2.12), the set of normalised scalar masses in (2.26) implies a set of conformal dimensions ∆ ± for the dual operators given by (2.27) As in the previous case, some of the conformal dimensions in (2.27) have been highlighted as they will play a role later on when studying holographic RG flows involving this conformal fixed point.

N = 4 vacuum with SO(4) symmetry
There is an N = 4 supersymmetric AdS 4 solution that preserves SO(4) . It is located at so that |z 4,5,6 | = |z 7 | = 1 . This AdS 4 solution has a vacuum energy given by as for the previous solution, and a spectrum of Z 3 2 -invariant normalised scalar masses of the form where L 2 = −3/V 0 is the AdS 4 radius. This N = 4 & SO(4) symmetric AdS 4 vacuum was first reported in [38], and then uplifted to a family of type IIB S-folds with N = 4 supersymmetry in [21]. Solving (2.12) for the set of normalised scalar masses in (2.30) yields a set of conformal dimensions ∆ ± for the dual operators given by m 2 L 2 = −2 (×3) ; 0 (×6) ; 4 (×4) , 10 (×1) , (2.31) As in the previous cases, some of the conformal dimensions in (2.31) have been highlighted as they will play a role later on when studying holographic RG flows involving this conformal fixed point. Table 1: Summary of the AdS 4 supersymmetric vacua with the largest possible residual symmetry within their respective families. The VEVs of z i and the values of the scalar potential at the vacua are provided. In the last line, #∆ J < 0 denotes the number of dual irrelevant operators at each such vacua.

Non-AdS 4 solutions and the D3-brane
In this section we will obtain (semi-)analytic non-AdS 4 solutions of the BPS flow equations (2.9): first in the purely electric case with c = 0 , and then turning on the electromagnetic deformation c .
1,2,3 has an uplift to a ten-dimensional background of type IIB supergravity that is locally equivalent to the D3-brane solution. More concretely, the axions χ (0) 1,2,3 can be locally reabsorbed in a reparameterisation of the angular coordinates θ 1,2,3 along the three commuting translational (shift) isometries on S 5 . This is explicitly shown in Section 4.2 and further discussed in Section 4.4.
It is worth mentioning that the condition (2.33) is required by the BPS equations (2.9) but not by the (local) second-order equations of motion that follow from the Lagrangian (2.5). The scalar potential (2.7) evaluated at the solution (2.32) yields 34) whereas the N = 1 gravitino mass (2.10) reads (2. 35) thus being independent of the arbitrary parameter Φ 0 in (2.32). Lastly, the constraint (2.33) further eliminates the dependence of (2.35) on the axion fields χ (0) 1,2,3 .

Axions and supersymmetry
The amount of four-dimensional supersymmetry preserved by a solution can be assessed by direct evaluation of the eight gravitino masses, namely, the eigenvalues of is the scalar-dependent gravitino mass matrix in the maximal theory [2]. Substituting the analytic BPS solution (2.32) into the expression for A 1 (z i ) one finds a set of (normalised) eigenvalues given by where the ± signs are not correlated. Note that the (+, +, +) and (−, −, −) eigenvalues in (2.36) precisely reproduce the N = 1 gravitino mass (2.35) belonging to the Z 3 2 -invariant sector of the maximal supergravity by virtue of the constraint (2.33). However, such an algebraic constraint does not eliminate the dependence of the six remaining gravitino masses in (2.36) on the axions χ  equations. However, we will be interested in perturbing the analytic solution (2.32) and solve the flow equations (2.9) order by order in powers of the deformation parameter c . We will refer to the resulting power series solution as the deformed D3-brane solution.
At zeroth order the analytic solution in (2.32) is recovered, which depends on the arbitrary parameters (χ (2.37) Following the discussion below (2.36), we will set χ (0) 1,2,3 = 0 in the zeroth order solution so that the largest possible amount of supersymmetry is preserved at this order and the ten-dimensional AdS 5 × S 5 geometry of the D3-brane is globally recovered.
In the following we will analyse in more detail the case where all the integration constants are set to zero, both flavour breaking and SU(3)-preserving constants in (2.40). Then the solution (2.38)-(2.39) acquires a universal form (to first order in the parameter c ) given by which necessarily induces a deviation from (2.32) that is linear in the parameter c and subleading around (gz) → ∞ (UV). However, (2.42) does not capture corrections in Imz 1,2,3 , Imz 4,5,6,7 or the scale factor e A . To get those one must go to higher-orders in c . Finally, it also follows from (2.42) that in contrast to the relation (2.33) obtained at c = 0 .

Higher-order universal corrections
The power series procedure can be iterated to solve the BPS equations (2.9) to any desired order in the deformation parameter c . Setting all the integration constants that appear to zero, the general structure of the universal nth-order solution is (2.44) Note that (2.44) actually becomes an expansion in powers of the quantity around µ = 0 . Therefore, the original expansion in powers of c around c = 0 can be re-interpreted as an expansion in powers of c/(gz) 2 around the UV ( z → ∞ ). Although we are not displaying the µ 2n higher-order corrections with n ≥ 2 (it turns out that some of them vanish identically), we have explicitly computed the solution up to n = 6 . At quadratic order, two corrections of the form µ 2 log (gz) involving non-vanishing functions of Φ 0 appear in the axions Rez 1,2,3 and Rez 4,5,6,7 . At quartic order, three corrections of the form µ 4 log (gz) involving three different non-vanishing functions f 2,1 (Φ 0 ) , g 2,1 (Φ 0 ) and j 2,1 (Φ 0 ) appear in Imz 1,2,3 , Imz 4,5,6,7 and the scale factor e A , respectively. These are the relevant orders at which the logarithms enter the universal solution to (2.9) when c = 0 .
In what follows we will consider a truncation of the universal solution (2.44) to cubic order in the deformation parameter c , namely, This order suffices to capture the first relevant terms in each of the scalar fields as well as in the scale factor for the deformed D3-brane solution. From (2.46) one has that which picks up a dependence on the coordinate z in contrast to the relation (2.43) obtained at linear order in c = 0 .

Holographic RG flows
In this section we numerically construct BPS domain-wall solutions that interpolate between the supersymmetric AdS 4 vacua of Section 2.2 in the IR ( z → −∞ ) and the non-AdS 4 solution of Section 2.3 with c = 0 in the UV ( z → ∞ ). We will also present an example of a domain-wall that interpolates between the AdS 4 vacuum with N = 2 & SU(2) × U(1) symmetry in the IR and the AdS 4 vacuum with N = 1 & SU(3) symmetry in the UV. All these domain-walls in supergravity correspond to holographic RG flows in the field theory side.

Boundary conditions
Let us start discussing the system of first-order and non-linear differential equations in (2.9). The set of equations for the scalars can be solved independently of the one for the scale factor, which can be readily integrated once the profiles for the scalars are known. This means that we must set one boundary condition per (real) scalar field. Moreover, it proves more efficient to numerically shoot from the IR and flow up to the UV. Perturbing around an AdS 4 configuration dual to the J-fold CFT 3 in the deep IR with scalar VEVs Σ (0)I translates into a choice of boundary conditions of the form 3 restricted to the set of modes with ∆ J < 0 in the AdS 4 spectrum, as demanded by regularity of the flow in the deep IR (z → −∞). After imposing (3.1), the BPS equations will determine the set of permitted λ I J . However, generic values of the permitted λ I J will end up in a singular flow for which some scalars diverge at a finite radial distance. We will thus have to perform a scanning of the λ I J parameter space in order to determine the region yielding regular flows between the IR and the UV. In summary, a given choice of parameters λ I J in the IR boundary conditions (3.1) will translate into a specific choice of parameters (χ

SYM
We will solve the BPS equations (2.9) numerically by perturbing around the N = 1 & SU(3) AdS 4 vacuum (2.15) in the IR (z → −∞). This will generate generic flows towards a nonconformal behaviour in the UV (z → ∞).

IR boundary conditions
Around the N = 1 & SU(3) solution, there are two irrelevant modes in the spectrum (2.21) that are compatible with regularity of the flows in the IR ( z → −∞ ). The linearised BPS equations then allow for two arbitrary real parameters (Λ , λ) specifying the IR boundary conditions (3.1), which read Note that, whenever non-vanishing, one of the parameters Λ or λ can be set at will by a shift on the coordinate z . We will set Λ = −1 without loss of generality 4 , which translates into a one-dimensional parameter space to be scanned.

Behaviour of the flows
Fixing λ = 0 implies Rez 1,2,3 = 0 in the IR boundary conditions (3.3). In this case we obtain the numerical flow 5 depicted in Figure 2 that approaches the deformed D3-brane solution in the UV (z → ∞). As previously discussed in Section 2.3.2, the UV behaviour of this flow is understood as a sub-leading correction in the electromagnetic deformation c of the D3-brane solution in (2.32) with Rez 1,2,3 = 0 Activating the parameter λ makes the axions Rez 1,2,3 run along the flow. In this case, the UV region is reached with This agrees with (2.43) obtained at first-order in the deformation parameter c . One such generic flows is depicted in Figure 3.

Study of the parameter space
We have performed a numerical scanning of values for the parameter λ , and found regular flows only within the interval Outside this range, singular flows occur with some scalar fields diverging at a finite radial distance.

Gravitino masses and supersymmetry
The For generic BPS flows with the parameter λ within the range (3.6), an explicit evaluation of the eight gravitino masses, namely the eigenvalues of A IJ A JK , shows that only one of them is compatible with the IR value of the N = 1 gravitino mass (2.10) belonging to the Z 3 2 -invariant sector. More concretely, the eight gravitino masses of the maximal theory turn out to split as λ = 0 :

SYM
Let us now solve the BPS equations (2.9) by perturbing around the N = 2 & SU(2) × U(1) AdS 4 vacuum (2.22) in the IR (z → −∞). This will cause again the appearance of generic flows towards a non-conformal behaviour in the UV (z → ∞).

IR boundary conditions
that are compatible with regularity of the flows in the IR ( z → −∞ ). The linearised BPS equations then allow for four parameters (Λ 1 , Λ 2 ) and (λ 1 , λ 2 ) specifying the IR boundary conditions (3.1) which read (3.10) As before, and whenever non-vanishing, one of the parameters Λ 1,2 or λ 1,2 can be set at will by a shift on the coordinate z . We will set Λ 1 = −1 without loss of generality 6 , which leaves us this time with a three-dimensional parameter space to be scanned.

Behaviour of the flows
Fixing Λ 2 = 0 and λ 1,2 = 0 implies Rez 1,2,3 = 0 and Rez 4,6 = 0 in the IR boundary conditions (3.10). In this case we obtain the flow depicted in Figure 4. The UV ( z → ∞ ) behaviour of this flow is again understood as a sub-leading correction in the electromagnetic deformation c of the D3-brane solution in (2.32) with Now we can explore the UV behaviour of the flows when activating the parameters Λ 2 and λ 1,2 in the boundary conditions (3.10). As it can be directly seen from there, the parameter λ 1 controls the pair-wise identified perturbation of Rez 1 = Rez 3 whereas Λ 2 subsequently controls the perturbation of Rez 2 . Turning on just the parameter Λ 2 generates flows reaching the UV with in agreement with (2.43). Turning on the remaining parameters λ 1,2 makes more scalars run along the flow. Flows of these types are presented in Figure 5.

Study of the parameter space
This time we must perform a numerical scan of flows in a three-dimensional parameter space (Λ 2 ; λ 1 , λ 2 ) . Various sections of the parameter space can be taken which are depicted in Figure 6. The three parameters Λ 2 and λ 1,2 control the values of the axions Rez 1,2,3 when generically approaching the D3-brane solution in the UV. Let us discuss in more detail some features of the parameter space depicted in Figure 6. The borders of the various sections delimit the region of the three-dimensional parameter space producing regular flows. Outside this region the flows have some scalar field diverging at a finite radial distance. Importantly, the region around the upper corner in the (λ 1 , λ 2 )projection is very special as it produces flows passing arbitrarily close to the N = 1 J-fold CFT 3 's before continue flowing to SYM 4 in the deep UV. This limiting CFT 3 to CFT 3 holographic RG flows are presented separately in Section 3.4.

Gravitino masses and supersymmetry
For the N = 1 & SU(2) × U(1) AdS 4 vacuum in the IR the flavour symmetry group realised in the dual J-fold CFT 3 is SU (2) . Under this SU(2) symmetry the eight gravitini of the maximal theory decompose this time as in (A.10), namely, For generic BPS flows with parameters Λ 2 and (λ 1 , λ 2 ) in the regions shown in Figure 6, the evaluation of the eight eigenvalues of A IJ A JK shows that, as before, only one of them is generically compatible with the IR value of the N = 1 gravitino mass (2.10) belonging to the Z 3 2 -invariant sector. However, specific choices of the parameters (λ 1 , λ 2 ) this time yield different splittings of the eight gravitino masses λ 2 1 + λ 2 2 = 0 : 8 → 4 + 1 + 1 + 1 + 1 ,  when evaluated along the numerical flows. In (3.14) we have boxed the N = 2 supersymmetry realised at the AdS 4 vacuum in the deep IR and highlighted (in blue) those gravitino masses with respect to which the numerical flows are BPS.

SYM 4 to CFT 3 with N = 4
Lastly we will solve the BPS equations (2.9) by perturbing around the N = 4 & SO(4) AdS 4 vacuum (2.28) in the IR (z → −∞). This will trigger again the appearance of generic flows towards a non-conformal behaviour in the UV (z → ∞).

IR boundary conditions
Around the N = 4 & SO(4) solution, there are again four irrelevant modes in the spectrum (2.31)   that are compatible with regularity of the flow in the IR ( z → −∞ ). The linearised BPS equations then allow for four parameters Λ and λ i , with i = 1, 2, 3 , specifying the IR boundary conditions (3.1). These read  Im(z 5,6 ) D3-brane with λ ≡ λ 1 + λ 2 + λ 3 . Note that the parameters λ i enter the IR boundary conditions (3.16) in a symmetric manner and fully specify Rez i . As before, one of the parameters, either Λ or λ i , can be set at will by a shift on the coordinate z . We will set Λ = −1 without loss of generality 7 , which leaves us also this time with a three-dimensional parameter space to scan.

Behaviour of the flows
The IR boundary conditions (3.16) are highly symmetric. Fixing λ i = 0 in (3.16) implies Rez 1,2,3 = 0 . In this case we obtain the flow depicted in Figure 7. The UV ( z → ∞ ) behaviour of this flow is again understood as a sub-leading correction in the electromagnetic deformation c of the D3-brane solution in (2.32) with Rez 1,2,3 = 0 and Φ 0 = 0 .   Figure 8.

Study of the parameter space
The fact that λ 1,2,3 enter the IR boundary conditions (3.16) symmetrically renders the three parameters completely interchangeable as far as the induced flows are concerned. In Figure 9 the section of the parameter space allowing for regular holographic RG flows with λ 1 = 0 is depicted. Similar figures are obtained upon setting λ 2 = 0 or λ 3 = 0 . Finally, within our numerical precision, we do not observe flows reaching the N = 1 family of AdS 4 vacua (2.15) in the UV. The three parameters λ 1,2,3 control the values of the axions Rez 1,2,3 when approaching the D3-brane solution in the UV.

Gravitino masses and supersymmetry
The N = 4 & SO(4) AdS 4 vacuum in the IR realises a trivial flavour symmetry group in the dual J-fold CFT 3 . For generic BPS flows with parameters λ i in the parameter space of Figure 9, the evaluation of the eight eigenvalues of A IJ A JK shows that, as in the previous cases, only one of them is generically compatible with the IR value of the N = 1 gravitino mass (2.10) belonging to the Z 3 2 -invariant sector. Specific choices of the parameters λ i yield

CFT 3 to CFT 3
We now present an example of CFT 3 to CFT 3 holographic RG flow that connects the J-fold CFT 3 with N = 1 & SU(3) symmetry in the UV to the J-fold CFT 3 with N = 2 & SU(2) symmetry in the IR (see Figure 1). This flow requires an extreme fine tuning of the IR boundary conditions in (3.10) and is depicted in Figure 10 as shown in Figure 11. Therefore, the UV symmetry enhancement to SU(3) does not take place. From a ten-dimensional perspective, having the pair-wise identification (3.20) between axions translates into reaching the N = 1 type IIB S-folds with SU(2) × U(1) ⊂ SU(3) symmetry [37] in the UV. The precise manner in which the axions Rez 1,2,3 trigger such a symmetry breaking geometrically is investigated in Section 4.4 by looking at the local coordinate redefinitions they induce on the internal geometry. Finally, a more detailed study of CFT 3 to CFT 3 holographic RG flows including the axions Rez 1,2,3 dual to exactly marginal deformations in the corresponding J-fold CFT 3 's goes beyond the scope of this work, and will be presented elsewhere together with their type IIB embedding. in the IR. As shown in [21,36,37], this amounts to having a factorised internal geometry of the form M 6 = S 1 × S 5 . However, the axions Rez 1,2,3 were shown in the previous section to generically run over the RG flow and to reach the deep UV with a non-zero value. On the gravity side, the above RG flows are described by ten-dimensional type IIB backgrounds with an interpolating geometry of the form sketched in Figure 12. In the UV, the solution asymptotes to a c-dependent subleading correction of the AdS 5 × S 5 geometry (only locally if the axions Rez 1,2,3 = 0 ) whereas, in the IR, an S-fold geometry emerges [21,36,37]. The function ∆ IR is the warping function at the S-fold solution in the deep IR, possibly depending on the S 5 coordinates. This is to be distinguished from the z-dependent scaling behaviour of ∆ UV in the deep UV which is generically governed, as we saw in Section 3, by the deformed D3-brane solution in Section 2.3.2.

Uplifting the IR: S-folds
In the deep IR, the RG flows reach the type IIB S-fold solutions originally presented in [36] (N = 1), [37] (N = 2) and [21] (N = 4). We concentrate on the S-fold solutions with Rez 1,2,3 = 0 allowing for the largest possible flavour symmetries (no exactly marginal deformations) in the dual J-fold CFT 3 's. In this case, the function f (z i ) in Figure 12 turns out to take the simple form and depends only on the scalars Imz 1,2,3 . In the deep IR these scalars must be evaluated at their VEVs for the three S-folds under consideration (see Table 1), so that Therefore, the electromagnetic deformation c is essential for the existence of the J-fold CFT 3 's serving as IR fixed points in the RG flows from the deformed N = 4 SYM 4 .

Uplifting the UV: D3-brane at c = 0
In this section we uplift the four-dimensional solution (2.32) obtained at c = 0 to tendimensional type IIB supergravity and connect it (locally) to the D3-brane solution.

Vanishing axions
Let us first set the three axions χ (0) 1,2,3 = 0 so that the largest amount of supersymmetry is preserved (see discussion below (2.36)). Then the four-dimensional solution contains two arbitrary parameters (g, Φ 0 ) and uplifts to a ten-dimensional type IIB background with a factorised internal geometry of the form S 1 × S 5 in the limit of an infinite radius for S 1 so that S 1 → R . The five-sphere is round and displays its largest possible SO(6) symmetry. The various ten-dimensional fields are given by where ds 2 DW 4 is the domain-wall metric displayed in (2.8) and vol 5 = g −5v ol 5 . The metric on the round S 5 of unit radius is given by and is normalised as R ij = 4 G ij . Lastly, the warping function ∆(z) takes the simple form At first sight the DW 4 metric in (4.4) seems to break conformal invariance. However, the five-dimensional piece of the metric (4.4) spanned by the domain-wall and the coordinate η can be recast as an AdS 5 metric upon a change of coordinates thus recovering the maximally supersymmetric AdS 5 × S 5 near-horizon geometry of the D3brane with L AdS 5 = g −1 . This is nothing but the holographic dual of N = 4 SYM 4 .

Non-vanishing axions = local coordinate redefinitions
Turning on a constant value for the axions χ (0) 1,2,3 = 0 in (2.32) does not affect the metric on the round S 5 so it is still given byĜ ij in (4.5). However, the G ηη component of the metric reads (4.9) and the G iη components of the metric take the form in terms of the embedding coordinates Y m on R 6 ( m = 2, . . . , 7 ) for the five-sphere of unit radius Therefore, the reconstruction of a direct product metric AdS 5 × S 5 , as performed when χ (0) 1,2,3 = 0 , is no longer obvious due to the axion-induced terms in (4.9). Nonetheless, for arbitrary (constant) values of χ (0) 1,2,3 , the ten-dimensional metric with ∆ given in (4.6), can be related to the one with χ (0) 1,2,3 = 0 by a local change of coordinates as we show now.

(4.19)
As a result, the ten-dimensional geometry reduces locally to AdS 5 × S 5 and the type IIB backgrounds are still given by (4.4) in terms of the S 5 redefined volume vol 5 . Consequently, the resulting type IIB backgrounds with χ

Uplifting the UV: deformed D3-brane at c = 0
In this section we investigate various aspects of the Z 2 × SU(3) invariant sector of the [SO(1, 1) × SO(6)] R 12 gauged supergravity which is obtained upon identifying the scalar fields in the Z 3 2 invariant sector as z 1 = z 2 = z 3 ≡ z 1,2,3 and z 4 = z 5 = z 6 = z 7 ≡ z 4,5,6,7 . (4.20) This sector of the maximal theory is of special interest (see Appendix A for details on its group theoretical embedding). Perturbing the four-dimensional incarnation of the D3-brane solution in (2.32), and solving the BPS equations perturbatively in the parameter c , we found the universal solution in (2.44) which, as already emphasised, is compatible with (4.20). It will also provide us with a starting point to discuss axion deformations of N = 1 S-folds later on in Section 4.4.

Ten-dimensional metric
The ten-dimensional metric takes the form in terms of a four-dimensional space-time given by ds 2 DW 4 in (2.8) and an internal space M 6 = S 1 η S 5 with S 5 = CP 2 S 1 . We refer the reader to Appendix A of [36] for a detailed description of the SU(2)-structure on the five-sphere S 5 = CP 2 S 1 when viewed as a Sasaki-Einstein manifold. As we will discuss in detail in Section 4.4, a non-trivial monodromy on M 6 is induced by the set of non-zero (constant) axions Rez 1,2,3 in the type IIB backgrounds so that the real one-form η in (4.21) is given by η = dβ + Rez 1,2,3 dη + A 1 .

(4.26)
This assignment of frames on CP 2 is compatible with a choice of embedding coordinates Y m on the five-sphere of the form Note that the axions Rez 1,2,3 enter the ten-dimensional geometry (4.21) exclusively through the one-form η in (4.22).

Recovering IR and UV geometries
In order to recover the ten-dimensional geometries for the S-folds and the D3-brane, we will express the metric (4.21) as with α = 0, 1, 2 , in terms of a rescaled coordinate w = 2 −3 η , a new radial coordinate ρ defined as  Then the expected AdS 5 × S 5 local geometry with a round S 5 metric ds 2 S 5 = ds 2 CP 2 + η 2 is recovered. Note also that, since Rez 1,2,3 take constant values in the D3-brane solution (2.32), one has that (4.22) can be re-expressed as η = d(β + Rez 1,2,3 η) + A 1 , thus redefining the coordinate β along the U(1) fiber in S 5 = CP 2 S 1 as β → β + Rez 1,2,3 η . (4.44) This coordinate redefinition is the analogue of (4.18) and, therefore, the same global vs local issues regarding the periodicity of the new angular variable apply here. This will be discussed in more detail in Section 4.4.

Deformed D3-brane and anisotropic SYM 4
Let us investigate the ten-dimensional type IIB uplift of the domain-walls constructed in Sections 3.1, 3.2 and 3.3 when approaching the UV ( z → ∞ ). We will holographically relate such a UV behaviour to having an anisotropic deformation of SYM 4 .

Five-dimensional picture
Setting c = 0 modifies the AdS 5 metric in (4.7) so that it acquires a dependence on the scalar fields 45) in terms of the functions F (z i ) and H(z i ) in (4.28) and the warping factor ∆ in (4.29). Consistently, F (z i ) = 1 and H(z i ) = ∆ 2 when c = 0 so that the undeformed AdS 5 metric in (4.7) is recovered. Note that the deformed AdS 5 metric in (4.45) singles out the coordinate η which, in the deep IR, will span the S 1 factor of the AdS 4 × S 1 × S 5 S-fold geometry [21,36,37]. This is the same coordinate along which the type IIB fields acquire a dependence on as a consequence of the SL(2) IIB monodromy in (4.31). However, the RG flows constructed in Section 3 occur along the radial direction gz . Therefore, the dependence of the type IIB fields m αβ and B α on the η coordinate induced by the S-folding (equivalently by the parameter c ) is ultimately connected to having an anisotropic deformation of N = 4 SYM 4 (see, e.g. [43,50]). Let us look at this issue in more detail from a ten-dimensional perspective.

Ten-dimensional picture
In order to present the ten-dimensional metric (4.21) in a suitable form to be compared with previous results in the literature regarding holographic anisotropy in SYM 4 (see [44,45]), it is convenient to first perform a redefinition of the radial coordinate as so that (4.21) conforms with (4.48) It then becomes transparent that the function e 2m in (4.48) will be responsible for anisotropy in the dual 4D field theory. Using  Note that, upon dimensional reduction on the five-sphere S 5 , the three-form field strengths H α in (4.34) give rise to an SL(2) IIB doublet of one-form field strengths along the dη direction in the (external) geometry (4.45). From the expressions of b α in (4.33), and using the four-dimensional solution in (2.46), one finds . . ImΩ .

(4.53)
10 This is also the case for the non-universal solution in (2.38)-(2.39) computed at linear order in c provided the identifications in (4.20) hold. 11 Similarly, the functions occurring in front of the internal S 5 metric in (4.48) are given by Importantly, the dependence on the coordinate η through the twist matrix A(η) will factorise out due to the contraction with the axion-dilaton matrix m αβ in the ten-dimensional kinetic term m αβ H α ∧ H β . Similar one-form terms along the dη direction in the (external) geometry (4.45) appear from the axion-dilaton matrix in (4.37) with m γδ = m + 0 0 m − and m ± = e ∓Φ 0 1 ± 32 sinh(2Φ 0 ) c 2 (g ζ) 2 + . . . , (4.54) and also from F 5 in (4.38). From the axion-dilaton one finds whereas from the five-form field strength one gets (4.56) We have verified the stability of the above results using the four-dimensional solution computed up to twelfth order in the parameter c .
Other type IIB constructions have been put forward to generate holographic anisotropy in SYM 4 . For instance, [44,45] employ backreacted geometries involving D3-and (smeared) D5branes along 2 + 1 dimensions. Our setup involves only closed strings (without source terms) and differs from the one in [44,45] in that it generates anisotropy in a purely geometric manner by implementing the locally geometric SL(2) IIB twist A(η) in (4.31). As a consequence of the mechanism here, the various one-form sources of anisotropy, namely, (F (1) α , F (1)αβ , F (1) ) , organise themselves into SL(2) IIB multiplets. The SL(2) IIB covariance here could explain the difference between the ζ-powers appearing in (4.49)-(4.50) and in [45]. Also along these lines, it would be interesting to characterise the operators triggering SYM 4 anisotropy in an SL(2) IIB covariant setup of the type investigated here. To this end, it would be helpful to have the oxidation of the RG flows presented in this work to five dimensions. Some group theoretical considerations on the 4D ↔ 5D dictionary are presented in Appendix A.

Axions and symmetry breaking
The UV behaviour of the RG flows presented in Section 3 was shown to be controlled by the asymptotic values of the axions Rez 1,2,3 . This was so for both SYM 4 and N = 1 J-fold CFT 3 cases. In order to have a ten-dimensional geometric description of such asymptotics, we must uplift four-dimensional solutions with arbitrary (constant) axions χ 1,2,3 . To this end we will follow the geometric construction in [46], and provide the uplift of a generic N = 1 S-fold with U(1) 2 symmetry. This will lead us to propose a generalisation of the mechanism in [46] which allows us to understand the holographic breaking of a symmetry group G : G = SU (3) for the N = 1 S-folds, G = SO(6) for the D3-brane (and also for the non-supersymmetric S-folds in [37]), and G = SU (2) for the N = 2 S-folds. This last case was the one studied in detail in [46], where the implications for the structure of the Kaluza-Klein spectrum were also analysed. It would then be interesting to perform a similar analysis of the KK spectrum for the N = 1 and non-supersymmetric S-folds.

N = 1 S-folds and SU(3) symmetry
The mechanism builds upon the morphism S 5 ∼ = SU(3)/SU(2) and the largest possible flavour symmetry group G = SU(3) of the corresponding N = 1 J-fold CFT 3 's. Choosing a point p 0 ∈ S 5 ⊂ C 3 we can map SU(3) → S 5 : g → g(p 0 ) with isotropy group SU (2) . This map allows us to construct a right-action of SU(3) over S 5 and to relate a coordinate system on SU(3) to a coordinate system on S 5 . There is also, a priori, a left-action of U(1) -the commutant of SU(2) in SU(3) -on S 5 .
When the axions are set to zero, χ 1,2,3 = 0 , the internal space is the direct product M 6 = S 5 × S 1 . The coordinate η along S 1 will then be periodic with η ∈ [0, T ) . On the contrary, when χ i = 0 , the axions induce a fibration of S 5 over S 1 characterised by a non-trivial monodromy. In order to identify such a monodromy, let us first introduce a twist element provided (4.58) holds. Note that (4.58) precisely matches the constraint (2.16) found for the axions in the N = 1 family of S-folds.
To each point η ∈ S 1 we will now associate an S 5 (η) parameterised by an element g ∈ SU(3)/SU(2) of the formĝ = h(η) · g . (4.60) This allows us to define an application that is globally well-defined if we have the equivalence relation ∼ on S 5 × S 1 defined as Since h(T ) ∈ G , we can use this application to map a solution with χ i = 0 to a new solution with generic (constant) axions χ i . Even if we can reabsorb the axions locally, there is an obstruction to do it globally. Having a non-trivial monodromy h(T ) in (4.59) reduces the action of G to G 0 ⊂ G on S 5 × S 1 / ∼ provided G 0 commutes with h(T ) . For the various type IIB backgrounds appearing in this work, and also for the non-supersymmetric S-folds in [37] 12 , we have explicitly verified that the coordinate redefinitions induced by φ h in (4.61) precisely recovers the dependence of the ten-dimensional metric on the axions entering h(η) Type IIB on Local change of coordinates in (4.57). The latter can be readily obtained by employing E 7(7) -ExFT techniques, as it was done in Section 4.3.1.
As an example, let us consider the N = 1 S-fold with the largest possible symmetry G = SU(3) and χ i = 0 . Turning on two axions χ 1 = −χ 3 = n 2π T with n ∈ N yields h(T ) = I so that G 0 = SU(3) . However, turning on three axions χ 1 = χ 2 = − 1 2 χ 3 as in (3.20) yields G 0 = SU(2) × U(1) . It then becomes clear that a general axion configuration will yield G 0 = U(1) 2 so that G = SU(3) gets broken to its Cartan subgroup. Note also that specific choices of χ i might be of special interest, like for example fractional multiples 2π kT , as these generate Z k monodromies.
Using the embedding coordinates Y m in (4.27) the action of φ h on the angular variables entering (4.22)-(4.25) reads in perfect agreement with the ten-dimensional result obtained in (4.44). It now becomes clear that SU (3) invariance restricts the coordinate redefinitions to those inducing a monodromy only on the U(1) fiber in S 5 = CP 2 S 1 . Finally, note also that the change of coordinates in (4.63) is globally well-defined if χ i = n i 2π T with i n i = 0 . This will be shown in a moment to be equivalent to the case χ i = 0 .

Monodromies, mapping torus and symmetry breaking
Let us start with a background involving an internal manifold of the form M n ×S 1 where M n is an homogeneous space for a compact group G and S 1 is a T -periodic circle parameterised by a coordinate η . We will show how to modify this factorised geometry by introducing a fibration of M n over S 1 in such a way that the original background is mapped to a new one. This new background is parameterised by an element h ∈ G which allows us to define the mapping torus T (M) h as the quotient for some χ i , T ∈ R where {h i } denotes a normalised basis of the r-dimensional Cartan subalgebra h of g . We must then find a map from the original background with M n × S 1 to a new one corresponding to a different element in T (M) h in a consistent manner. In other words, as we go around S 1 , we want to smoothly transport points on M n by elements of the isometry group, h(η) , such that h(T ) p = h p for p ∈ M n . A way to do this is by choosing so that the application consistently maps fields on the original manifold to fields on T (M) h . Some comments are now in order: • The symmetry G of the original background has been reduced to its subgroup G 0 that commutes with the element h(T ) for its action to be globally well-defined. This can break the symmetry G down to a direct product of rank(G) abelian U(1) factors. • Locally, every operation can be reversed by a local change of coordinates so that the new background is also a solution of the (local) equations of motion. • The relevant information is encoded in the choice of h(T ) . In particular, this means that the axions χ i have periodicity 2π T . Moreover, the mapping torus could be trivial in the sense that T (M ) h ∼ = M n × S 1 as for example when h(T ) = I .
Let us show how this construction works in simpler terms. We start by quotienting M n by the U(1) r ⊂ G generated by {h i } . In this way we obtain M n−r = M n /U(1) r so that M n can be recast as T r M n−r . By choosing an appropriately normalised basis for the Cartan subalgebra, it is always possible to assign coordinates θ i ( i = 1, . . . , r ) on T r with periodicities of 2π such that the generators h i act on them as translations (or shifts). Then, the map φ h (θ i ) = θ i + χ i η , (4.68) can locally be seen as change of variables on M n × S 1 . Note that (4.68) is precisely the local coordinate transformations that appeared in (4.18) and (4.63). Focussing now on the bigger torus T r S 1 , we can identify it as R r+1 /Λ where Λ is the lattice Here e i are vectors with a non-zero entry only in the i th coordinate and e η can be thought of as e r+1 . This illustrates how n-tuples of χ i producing the same lattice will give rise to the same monodromy and therefore equivalent fibrations and geometries. Examples are given by those changes of basis parameterised by elements of SL(r + 1, Z) . Since we want to keep the general form of the basis in order to be able to identify the axions, we must restrict to the subgroup Z r ∈ SL(r + 1, Z) containing elements of the form These elements act on the axions χ i as χ i → χ i + n i 2π T which is in line with the periodic nature of the axions when viewed as ingredients of the internal geometry. Note that the change of coordinates in (4.70) is always well-defined. Then, the fibration allows to replace dθ i → dθ i + χ i dη in the original background with vanishing axions and vice-versa.

D3-brane and SO(6) symmetry
As we showed in Section 4.2, the D3-brane at c = 0 features an SO(6) symmetry that suggests this time to think of S 5 as the quotient SO(6)/SO (5) . The relevant symmetry group is now G = SO (6) . We will investigate the action of SO(6) on the geometry and the relation with turning on the axions χ (0) 1,2,3 . Note that the construction in Section 4.4.2 can be formally extended to the case T → ∞ , i.e. S 1 → R , in order to study the patterns of symmetry breaking for the D3-brane background. However, the axions χ (0) i are no longer periodic in this case as 2π T → 0 . Turning on the axions translated into the coordinates redefinition in (4.18), thus inducing a non-trivial monodromy on S 5 × S 1 by virtue of (4.9). Using the embedding coordinates Y m (4.13) on R 6 , the action of (4.18) is encoded into a matrix element h(η) that belongs to the SO(2) 3 Cartan subalgebra of SO(6) , namely, Observe that no constraint on the axions is needed in this case as G = SO(6) ∼ = SU(4) has rank 3. Looking at Section 4.2 we identify the T 3 with the angles θ i parameterising independent rotations in the planes (Y 2 , Y 3 ) , (Y 4 , Y 5 ) and (Y 6 , Y 7 ) . In (4.18) we provided the change of variables making the axions disappear locally. Up to a conventional sign for χ 3 , this agrees with the general result in (4.68). Note also that the absence of an algebraic constraint for the axions χ i when G = SO (6) is precisely what allowed for in the deformed D3-brane solution controlling the UV behaviour of the RG flows in Section 3. This generically causes an axionic breaking of the SO(6) symmetry down to a subgroup of SU(3) . In addition, we have also verified that this construction reproduces the correct number of axions, coordinate redefinitions and patterns of symmetry breaking for the nonsupersymmetric family of S-folds in [37]. We will close this section by emphasising the importance of h being an element of the symmetry group G . For example, if we do not enforce that h ∈ SU(3) for the N = 1 S-folds, the coordinate combination 3β + ψ maps to Since the two-form potentials B α have an explicit dependence on (3β + ψ) they will gain an explicit dependence on η through the local coordinate transformation in (4.73), thus clashing with the equations of motion. Note that this is not a problem for the D3-brane as B α vanish identically and we can safely take h ∈ SO(6) .

Conclusions
In this work we have investigated various aspects of type IIB S-fold backgrounds by using the [ SO(1, 1) × SO(6) ] R 12 gauged maximal supergravity as a four-dimensional effective description thereof and the E 7(7) -ExFT as a tool to uplift four-dimensional solutions of said supergravity to ten dimensions. The focus of the paper has been three-fold: • Firstly we have constructed new holographic RG flows on the D3-brane that connect an anisotropic deformation of SYM 4 in the UV to supersymmetric J-fold CFT 3 's in the IR.
In addition, we have also provided examples of CFT 3 to CFT 3 flows between N = 1 J-fold CFT 3 's with either SU(3) or SU(2) × U(1) flavour symmetry in the UV and the N = 2 J-fold CFT 3 with SU(2) flavour symmetry in the IR. The role played by the axions as moduli fields dual to marginal deformations was analysed when characterising the UV behaviour of such RG flows. • Secondly the ten-dimensional holographic description of such RG flows was investigated.
The anisotropic deformation of SYM 4 was interpreted in a purely geometric manner and connected to the locally geometric SL(2) IIB twist matrix A(η) generating the S-fold. In this manner we identified a set of five-dimensional one-forms (F (1) α , F (1)αβ , F (1) ) as the sources of anisotropy which came into SL(2) IIB representations. As mentioned in the main text, it would be interesting to characterise the operators triggering such a SYM 4 anisotropy in an SL(2) IIB covariant setup of the type investigated here.
• Thirdly we investigated the type IIB geometric origin of the four-dimensional axions dual to marginal deformations of the S-fold CFT 3 's. In all the cases analysed (which also include the D3-brane and the non-supersymmetric S-folds in [37]), and upon uplift to ten dimensions, the axions were shown to be locally reabsorbable in a reparameterisation of the angular coordinates on the five-sphere. However this reparameterisation induces a non-trivial h-monodromy on S 5 × S 1 which is responsible for the various patterns of flavour symmetry breaking observed in the dual S-fold CFT 3 's. Building upon the recent results in [46], the possible patterns of flavour symmetry breaking induced by the h-monodromy were connected to the mapping torus T (S 5 ) h .
Let us conclude by emphasising that, although a solid proof of a precise correspondence between 4D axions and monodromy-induced patterns of symmetry breaking in 10D is still lacking, the mechanism proposed in this work (see Figure 13) correctly reproduces the known results (number of axions, patterns of flavour symmetry breaking, etc.) for all the S-folds captured by the Z 3 2 -invariant sector of the [SO(6) × SO(1, 1)] R 12 maximal supergravity, as well as for the D3-brane. It would also be interesting to assess whether the R-symmetry groups can be incorporated to the story, perhaps opening up new possibilities for RG flows possibly in different sectors of the theory. We hope to come back to this and related issues in the near future.

A.1 5D supergravity: scalar fields and dual operators
Let us consider the five-dimensional SO (6) corresponding to the following holographic operators: gauge coupling g ∈ 1 4 , theta-angle θ ∈ 1 −4 , fermion bilinears ψ ψ ∈ 10 −2 and their conjugatesψψ ∈ 10 2 and, lastly, scalar bilinears φ φ ∈ 20' 0 . As a result, the SO(6) ∼ = SU(4) invariant sector of the 5D theory contains 2 scalars associated with the two singlets 1 ±4 . Lastly, the eight gravitini of the theory branch as USp (8) 13 The various entries in the right-most column of (A.5) are in one-to-one correspondence with the set of possible bosonic deformations of N = 4 SYM4. The latter are parameterised by the bosonic auxiliary fields of the off-shell N = 4 conformal supergravity [51] (see also [52]), which transform in the displayed irreps of the (global) SU(4) R-symmetry group of undeformed N = 4 SYM4.

SU(2) × U(1) and SO(4) invariant sectors
These two sectors of the theory also play an important role in this work. The SU(2) × U(1) invariant sector is defined through the group theoretical embeddings and the corresponding gravitini decompositions thus being a non-supersymmetric sector.