Holographic RG flows and symplectic deformations of N=4 gauged supergravity

We study four-dimensional $N=4$ gauged supergravity with $SO(4)\times SO(4)\sim SO(3)\times SO(3)\times SO(3)\times SO(3)$ gauge group in the presence of symplectic deformations. There are in general four electric-magnetic phases corresponding to each $SO(3)$ factor, but two phases of the $SO(3)$ factors embedded in the $SO(6)$ R-symmetry are fixed. One phase can be set to zero by $SL(2,\mathbb{R})$ transformations. The second one gives equivalent theories for any non-vanishing values and can be set to $\frac{\pi}{2}$ resulting in gauged supergravities that admit $N=4$ supersymmetric $AdS_4$ vacua. The remaining two phases are truely deformation parameters leading to different $SO(4)\times SO(4)$ gauged supergravities. As in the $\omega$-deformed $SO(8)$ maximal gauged supergravity, the cosmological constant and scalar masses of the $AdS_4$ vacuum at the origin of the scalar manifold with $SO(4)\times SO(4)$ symmetry do not depend on the electric-magnetic phases. We find $N=1$ holographic RG flow solutions between $N=4$ critical points with $SO(4)\times SO(4)$ and $SO(3)_{\textrm{diag}}\times SO(3)\times SO(3)$ or $SO(3)\times SO(3)_{\textrm{diag}}\times SO(3)$ symmetries. We also give $N=2$ and $N=1$ RG flows from these critical points to various non-conformal phases. However, contrary to the $\omega$-deformed $SO(8)$ gauged supergravity, there exist non-trivial supersymmetric $AdS_4$ critical points only for particular values of the deformation parameters within the scalar sectors under consideration.

An extension of symplectic deformations to N ≥ 2 gauged supergravities has been considered in [16] in which some examples of symplectically deformed N = 2 and N = 4 gauged supergravities have been given. In the present paper, we are interested in symplectic deformations of N = 4 gauged supergravity with SO(4) × SO(4) ∼ SO(3) × SO(3) × SO(3) × SO(3) gauge group. Generally, there can be four deformation parameters or electric-magnetic phases for the four SO(3) factors, see also [17,18] for an eariler construction of N = 4 gauged supergravity with these phases. In the notation of [16], these phases are denoted by α 0 , α, β 1 and β 2 . α 0 can be set to zero by SL(2, R) transformations. In addition, all values of α > 0 lead to equivalent gauged supergravities and can be set to π 2 . The remaining phases β 1 and β 2 constitute free deformation parameters of the SO(4) × SO(4) gauged supergravity. With these two phases, we expect to find a rich structure of vacua and other interesting holographic solutions as in the ω-deformed SO(8) gauged supergravity.
It turns out that, unlike the ω-deformed SO(8) gauged supergravity, there do not exist any supersymmetric AdS 4 critical points apart from those identified previously in [20] or critical points related to these at least within the aforementioned scalar sectors. However, we do find new classes of holographic RG flows with N = 1 and N = 2 supersymmetries. In particular, some of the N = 1 solutions describe RG flows between N = 4 critical points with SO(4) × SO(4) and SO (3) (3) symmetries. To the best of the author's knowledge, these are the first examples of holographic RG flows between conformal fixed points preserving N = 1 supersymmetry in the framework of N = 4 gauged supergravity, see [32] for examples of N = 1 RG flows to non-conformal phases, and should further extend the list of known N = 4 and N = 2 solutions given in [20] and [21], see [12,13,14,32,33,34,35,36,37,38,39,40,41,42,43,44,45] for an incomplete list of similar solutions in other four-dimensional gauged supergravities.
It should be pointed out that the N = 4 gauged supergravity under consideration here has currently no known higher dimensional origins as in the case of ω-deformed SO(8) gauged supergravity. Accordingly, the complete holographic interpretation in string/M-theory framework is unavailable. However, it is still useful to have holographic solutions in lower dimensional gauged supergravities, and with recent developments in double field theory formalism, particularly the result of [46], the embedding of SO(4) × SO(4) gauged supergravity in higher dimensions could be achieved.
The paper is organized as follows. In section 2, we review the general structure of N = 4 gauged supergravity in the embedding tensor formalism together with symplectic deformations of SO(4) × SO(4) gauge group. The truncations to SO(3) diag × SO(3) diag , SO(2) × SO(2) × SO(2) × SO(2) and SO(3) diag × SO(3) singlet scalars are considered in sections 3, 4 and 5, respectively. In these scalar sectors, we will focus on AdS 4 vacua and possible RG flow solutions between these vacua and RG flows to singular geometries. We end the paper by giving some conclusions and comments on the results in section 6. Useful formulae and details on relevant BPS equations can be found in the appendix.

Matter-coupled N = 4 gauged supergravity
We first review the general structure of N = 4 gauged supergravity coupled to vector multiplets in the embedding tensor formalism [47], see also [48] for an eariler construction. In four dimensions, there are two types of N = 4 supermultiplets, the gravity and vector multiplets, with the following field content and The component fields in the gravity multiplet are given by the graviton eμ µ , four gravitini ψ i µ , six vectors A m µ , four spin-1 2 fields χ i and one complex scalar τ while those in a vector multiplet are given by a vector field A µ , four gaugini λ i and six scalars φ m .
All fermionic fields and supersymmetry parameters transform in fundamental representation of SU (4) R ∼ SO(6) R and are subject to the chirality projections while those transforming in anti-fundamental representation of SU (4) R satisfy The complex scalar τ consists of the dilaton φ and the axion χ which parametrize SL(2, R)/SO(2) coset manifold. This SL(2, R)/SO(2) can be described by the coset representative V α of the form with Similarly, the 6n scalars φ ma parametrize SO(6, n)/SO(6) × SO(n) coset manifold with the coset representative denoted by V A M . Under the global SO(6, n) and local SO(6) × SO(n) symmetries, V A M transforms by left and right multiplications, respectively. Accordingly, the SO(6) × SO(n) index A can be split as A = (m, a) resulting in the following components of the coset representative The matrix V A M satisfies the relation with η M N = diag(−1, −1, −1, −1, −1, −1, 1, . . . , 1) being the SO(6, n) invariant tensor. The inverse of V A M will be denoted by V A M = (V m M , V a M ). All possible gaugings of the aforementioned matter-coupled N = 4 supergravity are encoded in the embedding tensor [47]. N = 4 supersymmetry allows only two non-vanishing components of the embedding tensor denoted by ξ αM and f αM N P . A given gauge group G 0 ⊂ SL(2, R) × SO(6, n) can be embedded in both SL(2, R) and SO(6, n) and can be gauged by either electric or magnetic vector fields or combinations thereof. We also note that each magnetic vector field must be accompanied by an auxiliary two-form field in order to remove the extra degrees of freedom. The embedding tensor also needs to satisfy the quadratic constraint in order for the resulting gauge generators to form a closed subalgebra of SL(2, R) × SO(6, n).
In this paper, we are mainly interested in gauge groups admitting supersymmetric AdS 4 vacua. As shown in [49], see also [17,18] for an earlier result, this requires the gauge groups to be embedded solely in SO(6, n) and gauged by both electric and magnetic vector fields. This implies that both electric and magnetic components of f αM N P must be non-vanishing and ξ αM = 0. Accordingly, we will set ξ αM to zero from now on. Furthermore, since we will study supersymmetric AdS 4 vacua and domain wall solutions that involve only the metric and scalar fields, we will also set all vector and fermionic fields to zero.
With all these, the bosonic Lagrangian can be written as where e = √ −g is the vielbein determinant. The scalar potential is given in terms of the scalar coset representative and the embedding tensor by We also note that f αM N P include the gauge coupling constants. The symmetric matrix M M N is defined by with M M N denoting its inverse. The tensor M M N P QRS is obtained from by raising indices with η M N . Finally, M αβ is the inverse of the symmetric 2 × 2 matrix M αβ defined by Fermionic supersymmetry transformations are given by with the fermion shift matrices defined by V ij M and V ij M are defined in terms of the 't Hooft symbols G ij m as and The explicit representation of G ij m used in this paper is given in the appendix. It is also useful to note that upper and lower i, j, . . . indices are related by complex conjugation.

SO(4) × SO(4) gauge group and symplectic deformations
In this work, we only consider SO ( In general, each SO(3) factor can acquire a non-trivial SL(2, R) phase resulting in symplectic deformations of a particular SO(4) × SO(4) gauging such as purely electric gauged SO(4) × SO(4) gauge group [16], see also [17,18] for an earlier consideration of this deformation. To give an explicit form of the embedding tensor, it is convenient to split the SO(6, 6) fundamental index as M = (m,m,â,ã) form,m,â,ã = 1, 2, 3. The embedding tensor for symplectically deformed SO(4) × SO(4) gauging as given in [46] can be written, in the notation of [16], as The constants α 0 , α, β 1 and β 2 are the electric-magnetic phases while g 0 , g, h 1 and h 2 are the corresponding gauge coupling constants for each SO(3) factor. A particular case of α 0 = β 1 = 0 and α = β 2 = π 2 , after a redefinition of gauge coupling constants, has been considered in [19,20,21]. For later convenience, we will call the SO(4) × SO(4) gauge group with this particular choice of phases "undeformed" SO(4) × SO(4) gauge group. It has been pointed out in [16] that by gauge fixing the SL(2, R) symmetry, we can set α 0 = 0. In addition, all the gaugings with α > 0 are equivalent to the gauge group with α = π 2 up to a shift of gauge invariant theta terms and a redefinition of the axion. We will set α 0 = 0 but keep α generic to keep track of the effects of symplectic deformations. We also note that if all the phases are not 0 or π 2 , see [16] for possible ranges of these phases, all four SO(3) factors are dyonically gauged by both electric and magnetic vector fields since all f ±M N P are non-vanishing. For convenience, we will introduce the notation SO (3)  We also note that for particular values of the electric-magnetic phases with ω ∈ [0, π 8 ] and g = −g 0 = h 1 = h 2 , the resulting N = 4 gauged supergravity is a truncation of the ω-deformed SO(8) maximal gauged supergravity constructed in [1].

Parametrization of scalar manifold and BPS equations
Since we are mainly interested in holographic RG flow solutions in the form of supersymmetric domain walls, an explicit parametrization of the scalar manifold SO(6, 6)/SO(6) × SO(6) is crucial. To give the SO(6, 6)/SO(6) × SO(6) coset representative, we first define SO(6, 6) generators in the fundamental representation by The SO(6, 6) non-compact generators are then given by To make things more manageable, we will only consider particular truncations of the full 36-dimensional coset to submanifolds with a few scalars non-vanishing. The truncations we will consider contain singlet scalars under To find supersymmetric domain wall solutions, we use the standard metric ansatz with dx 2 1,2 being the metric on three-dimensional Minkowski space. The only remaining non-vanishing fields are given by scalars. To preserve the isometry of dx 2 1,2 , scalar fields can depend only on the radial coordinate r. Supersymmetric solutions can be found by considering solutions to the BPS equations obtained by setting fermionic supersymmetry transformations to zero. With the metric ansatz (24), the variations of gravitini along µ = 0, 1, 2 directions give To proceed, we will use Majorana representation for space-time gamma matrices with all γμ real and γ 5 purely imaginary. Left and right chiralities of fermions are then related to each other by complex conjugation. The symmetric matrix A ij 1 can be diagonalized with eiganvalues denoted by A i . In general, in the presence of unbroken supersymmetry, some or all of these eigenvalues will give rise to the superpotential in terms of which the scalar potential can be written. Letα be the eigenvalue of the Killing spinors î corresponding to the unbroken supersymmetry. We can rewrite the above equation as To proceed further, we impose the following projector with an r-dependent phase Λ. This projector relates the two chiralities of î breaking half of the supersymmetry. A domain wall solution is then half-supersymmetric. By defining the superpotential we obtain the BPS condition and for W = |W|.
Repeating the same procedure for δψ î r , we find a differential condition on the Killing spinors 2∂ r î − Wγr î = 0 .
With the condition (26), we find for constant spinors î 0 . Finally, using the γr projector (27) in the variations δχ i and δλ i a , we can determine all the BPS equations for scalars. In subsequent sections, we will find explicit solutions for various residual symmetries and different numbers of unbroken supersymmetries.

SO(3) diag × SO(3) diag sector
We begin with a simple case of SO(3) diag × SO(3) diag singlet scalars. This sector has been considered in the undeformed SO(4)×SO(4) gauge group in [20]. In this work, we will consider effects of arbitrary electric-magnetic phases. There are two singlets from SO(6, 6)/SO(6) × SO(6) coset corresponding to the non-compact generatorsŶ Accordingly, the coset representative can be written as The dilaton and axion are also SO(3) diag × SO(3) diag singlets since these scalars are singlets under the full SO(6, 6) global symmetry. Therefore, the SO(3) diag × SO(3) diag sector consists of 4 scalars.

Supersymmetric AdS 4 vacua
We first look at possible supersymmetric AdS 4 vacua within the SO(3) diag × SO(3) diag sector. The scalar potential is given by As in the undeformed SO(4) × SO(4) gauge group, it turns out that A ij 1 tensor is proportional to the identity matrix with the four equal eigenvalues given by We now look for possible supersymmetric AdS 4 vacua. We begin with a simple case of φ 1 = φ 2 = 0. It can be straightforwardly verified that this choice satisfies all the BPS conditions provided that This leads to a supersymmetric AdS 4 vacuum preserving N = 4 supersymmetry and the full SO(4) × SO(4) gauge symmetry with the corresponding cosmological constant given by The AdS 4 radius is given by It should be noted that V 0 < 0 since the reality condition on φ implies gg 0 sin α < 0. We also note that for α = 0, the AdS 4 vacuum does not exist. This is in agreement with the fact that α = 0 together with the previous choice of α 0 = 0 imply that the SO(3) 0 and SO(3) α are both electrically gauged leading to no supersymmetric AdS 4 vacua [49]. We can bring this SO(4) × SO(4) vacuum to the origin of the scalar manifold by shifting the dilaton and axion or equivalently choosing This simply realizes the general result of [16] that any values of α > 0 lead to physically equivalent theories up to a redefinition of the axion. Therefore, this N = 4 AdS 4 vacuum is the same as that of the undeformed SO(4) × SO(4) gauged supergravity considered in [20]. Moreover, the scalar masses turn out to be independent of all electric-magnetic phases with all scalar masses equal m 2 L 2 = −2. This result is similar to the maximally supersymmetric AdS 4 vacuum at the origin of the scalar manifold in ω-deformed SO(8) N = 8 gauged supergravity.
To look for other supersymmetric vacua, it is more convenient to first analyze the resulting BPS conditions. The conditions arising from δλ i a reduce to the following two equations In these equations and in the following analysis, we choose an upper sign choice in (30) and (31) for definiteness. This also identifies the trivial SO(4) × SO(4) critical point in the limit r → ∞ in the RG flow solutions. At the vacua with constant φ 1 and φ 2 , we have φ 1 = φ 2 = 0, and consistency of the above two equations imposes the following conditions (44) after setting α = π 2 . All these conditions imply that for φ 1 = 0 or φ 2 = 0, AdS 4 vacua are possible only for Accordingly, non-trivial N = 4 supersymmetric AdS 4 vacua with at least SO(3) diag × SO(3) diag symmetry only exist in the undeformed SO(4) × SO(4) gauge group considered in [20]. We also note that for both φ 1 and φ 2 non-vanishing, the residual symmetry is give by SO (3) (4) symmetries. Further analysis also shows that consistency of the full BPS equations from δλ a i = 0 conditions with r-dependent scalars also requires (45) for any values of α = 0, π 2 . This also implies that apart from the solutions found in [20] no supersymmetric domain walls or RG flows with at least SO(3) diag ×SO(3) diag symmetry exist in the symplectically deformed SO(4) × SO(4) gauge group. Therefore, in SO(3) diag × SO(3) diag sector, no new AdS 4 vacua and holographic RG flows interpolating between them exist apart from those already given in [20].
Combining A and φ equations with χ equation, we obtain the solutions for A and φ as functions of χ with an integration constant C 0 . We have neglected an additive integration constant for A which can be removed by rescaling coordinates of dx 2 1,2 . Finally, by changing to a new radial coordinate ρ defined by we obtain the solution for χ of the form with another integration constant ρ 0 which can also be removed by shifting the coordinate ρ. For α = π 2 , this is the holographic RG flow from a three-dimensional N = 4 SCFT to a non-conformal field theory in the IR given in [20]. However, the χ(ρ) solution has not been given. Accordingly, the present result should fill this gap.
For α = 0, the BPS equations simplify considerably to together with χ = 0. The solution takes a simple form φ = 2 ln 1 2 g 2 + g 2 0 (r − r 0 ) and with constant χ = χ 0 . This gives a half-supersymmetric domain wall vacuum of The SO(2) × SO(2) × SO(2) × SO(2) sector of the undeformed SO(4) × SO(4) gauged supergravity has been considered recently in [21] in which a number of holographic RG flows and Janus solutions have been found. In the present paper, we will consider the same sector with electric-magnetic phases. As shown in [21], there are four SO(2) × SO(2) × SO(2) × SO(2) singlet scalars from SO(6, 6)/SO(6) × SO(6) corresponding to non-compact generators Y 33 , Y 36 , Y 63 and Y 66 in terms of which the coset representative can be written as Together with the dilaton and axion from the gravity multiplet, there are six scalars in the SO(2) × SO(2) × SO(2) × SO(2) sector. The kinetic term for these six scalars takes the form in which we have introduced a symmetric matrix G rs and a notation Φ r = (φ, χ, φ 1 , φ 2 , φ 3 , φ 4 ), with r, s = 1, 2, . . . , 6 for later convenience.
The resulting scalar potential is given by In this case, the phases β 1 and β 2 do not appear in the scalar potential. In addition, as in the undeformed SO(4) × SO(4) gauge group, the potential admits only a trivial AdS 4 critical point at φ = χ = φ 1 = φ 2 = φ 3 = φ 4 = 0 for α = π 2 and g 0 = −g.
In this case, the A ij 1 tensor is diagonal and takes the form of with the two eigenvalues given by A − and A + eigenvalues correspond to unbroken N = 2 supersymmetry with the Killing spinors given by 1,4 and 2,3 , respectively. The two choices are equivalent, and we will choose 1,4 as Killing spinors for definiteness. With 2,3 = 0 and the superpotential of the form we find that all the BPS equations can be written collectively as G rs is the inverse of the scalar matrix G rs which is in turn given by We also note that the scalar potential can be written as For α = π 2 and g 0 = −g, the explicit form of the BPS equations reads We numerically solve these equations with some examples of possible solutions given in figure 1. The solutions interpolate between the supersymmetric AdS 4 vacuum with SO(4) × SO(4) symmetry and singular geometries with diverging scalars. In all solutions, we see that V → −∞ implying that all singularities are physical by the criterion given in [50]. Indeed, we find the scalar potential, for α = π 2 and g 0 = −g, which is always bounded from above. Therefore, for any diverging behaviors of scalar fields, all possible solutions in SO(2) × SO(2) × SO(2) × SO(2) truncation are physically acceptable and describe RG flows from the dual N = 4 SCFT to various non-conformal phases in the IR. Using the scalar masses m 2 L 2 = −2 and asymptotic behaviors near the AdS 4 critical point

SO(3) diag × SO(3) sector
In this section, we look at another scalar sector with residual symmetry SO(3) diag × SO (3). Since this sector has not previously been studied, we will give the construction in more detail than the previous two cases. The Accordingly, there are two SO(3) diag × SO(3) singlets corresponding to the two (1, 1) representations. These two singlets correspond to the following SO(6, 6) non-compact generatorŝ If we consider an even smaller SO(3) diag residual symmetry, there are two additional singlets obtained from the last representation (3,3) in (76). These singlets correspond to the non-compact generatorŝ We also note thatŶ 1 andŶ 4 are SO(3) diag × SO(3) diag singlets considered in section 3. The coset representative for SO(3) diag sector can then be written as However, it turns out that the resulting scalar potential and BPS equations are highly complicated. Therefore, we refrain from giving the complete analysis of this sector but simply note that the A ij 1 tensor takes the form (A, B, B, B).
To make the analysis more traceable, we will perform further truncation to SO(3) diag × SO(3) singlet scalars by setting φ 2 = φ 4 = 0. Although this subtruncation leads to simpler expressions for the results, there are still some new interesting features. For simplicity of the results, in this section, we will set α = π 2 and g 0 = −g. The A ij 1 tensor for the subtruncation still takes the form (80) with the eigenvalues given by We find that the first eigenvalue A gives rise to the superpotential of the form Accordingly, RG flow solutions will preserve only N = 1 supersymmetry. It should be noted that for φ 3 = 0 or φ 1 = 0, the two eigenvalues A and B are equal leading to an enhanced N = 4 supersymmetry. The scalar potential can be written in terms of the superpotential as follows The explicit form of V is given in the appendix. We note that only β 1 appears in the results. We can also make another subtruncation by setting φ 1 = φ 3 = 0 in which only β 2 appears. This gives similar results with (φ 1 , φ 3 ) and (φ 2 , φ 4 ) together with β 1 and β 2 interchanged. On the other hand, if we consider the full SO(3) diag sector, both β 1 and β 2 appear in the scalar potential and the superpotential. From the superpotential given above, we have not found any non-trivial supersymmetric AdS 4 critical points for arbitrary values of β 1 . However, there are two AdS 4 vacua for particular values of β 1 = 0 and β 1 = π 2 . These are given by and ii : By the same procedure as in the previous sections, we find the BPS equations of the form The explicit form of these equations is rather long and given in the appendix.

N = 4 holographic RG flows
We begin with holographic RG flow solutions preserving N = 4 supersymmetry obtained by truncating out the axion χ together with one of the two scalars φ 1 and φ 3 . Although the solution interpolating between the trivial critical point and critical point i has already been given in [20], it is useful to repeat it here in the present convention for the sake of comparison with the N = 1 solutions given later on.
With φ 3 = χ = 0, the BPS equations reduce to The solution can be found by first combining φ and φ 1 equation and finding the solution for φ as a function of φ 1 . The result is given by We readily see that φ → 0 as φ 1 → 0. To make the solution end at the IR fixed point given by critical point i, we choose the integration constant to be Similarly, we find the solution for A as follows Finally, using the previous results and changing to a new radial coordinate ρ given by dρ dr = e − φ 2 , we find with ρ 0 being an integration constant. Near the UV and IR fixed points, the asymptotic behaviors of the scalar fields are given respectively by and Accordingly, the flow is driven by relevant operators of dimensions ∆ = 1, 2. In the IR, the operator dual to φ 1 becomes irrelevant with dimension ∆ = 4 while that dual to φ is still relevant. Similarly, by the same procedure, we can find a flow solution interpolating between the trivial AdS 4 critical point and ciritcal point ii. In this case, the BPS equations read with the solution given by with ρ defined as in the previous case. The asymptotic behaviors and holographic interpretations are also similar. Furthermore, the solution for φ 3 can be rewritten in a similar form as (95) by changing to another radial coordinate η given by dη dρ = e φ resulting in In these two solutions, we see that the operators dual to φ and φ 1 break conformal symmetry but preserve N = 4 Poincare supersymmetry in three dimensions.

N = 1 holographic RG flows
We now consider holographic RG flows in the full SO(3) diag × SO(3) sector. We first point out that setting χ = 0 still gives A = B resulting in N = 1 supersymmetry. However, truncating out only χ is not consistent with the BPS equations given in (87) unless φ 1 = 0 or φ 3 = 0. Accordingly, N = 1 RG flow solutions to critical points i or ii involve all scalars in the SO(3) diag × SO (3) sector. This makes finding the solutions more difficult, so we will numerically give some examples of possible solutions.
Using the BPS equations given in the appendix, we find an RG flow solution interpolating between the trivial N = 4 fixed point with SO(4) × SO(4) symmetry and critcal point i as shown in figure 2. Since all scalars have the same mass m 2 L 2 = −2 at the SO(4) × SO(4) critical point, the flow is again driven by relevant operators of dimensions ∆ = 1, 2. In the IR, using the scalar masses given in [20], we find that φ 1 is dual to an irrelevant operator of dimension ∆ = 4, but φ, χ and φ 3 are dual to relevant operators of dimensions ∆ = 1, 2. Unlike the N = 4 solutions given above, in addition to breaking conformal symmetry, turning on the operators dual to χ and φ 3 along the flow further breaks the N = 4 Poincare supersymmetry to N = 1. However, at the IR fixed point, the conformal symmetry is restored, and the supersymmetry is enhanced to N = 4 due to the vanishing of χ and φ 3 .
A similar N = 1 flow solution from the SO(4) × SO(4) fixed point to critical point ii can also be found. This is shown in figure 3. For other values of the phase β 1 , we have not found any non-trivial AdS 4 critical points. Examples of RG flows from the SO(4) × SO(4) fixed point to non-conformal phases are given in figure 4. There are also RG flows from AdS 4 critical points i and ii to non-conformal phases. Examples of these solutions are given in figures 5 and 6. Unlike the N = 2 RG flows given in the previous section, these N = 1 RG flows turn out to be unphysical according to the criterion of [50] due to V → ∞ as seen from the figure. It could be interesting to see whether these singularities are physical in the (if any) uplifted solutions to ten or eleven dimensions.

Conclusions and discussions
In this paper, we have studied symplectically deformed N = 4 gauged supergravity with SO(4) × SO(4) ∼ SO(3) × SO(3) × SO(3) × SO(3) gauge group with two independent electric-magnetic phases. We have considered three scalar (4). Similar to the ω-deformed SO(8) maximal gauged supergravity, for the trivial supersymmetric AdS 4 vacuum at the origin of the scalar manifold, the cosmological constant and scalar masses are independent from the electric-magnetic phases. However, unlike the ωdeformed SO(8) gauged supergravity, it turns out that other AdS 4 critical points are the same or related to those identified previously in [20] for the "undeformed" SO(4) × SO(4) gauge group. Although we have not found any genuinely new supersymmetric AdS 4 vacua, we have given a large number of new holographic RG flows preserving N = 2 and N = 1 supersymmetries.
The N = 2 solutions decribe holographic RG flows from the dual N = 4 CSM theory to non-conformal phases in the IR driven by relevant operators of dimensions ∆ = 1, 2. The SO(4) × SO(4) symmetry is broken down to SO(2) × SO(2)×SO(2)×SO(2) along the flows through the IR phases. We have found that      symmetry in the UV to non-conformal phases in the IR for different values of the electric-magnetic phase β 1 = 0 (red), π 6 (green), π 4 (blue), π 3 (purple), π 2 (pink) with g = 1 and h 1 = 2.    all these singular solutions describe physical RG flows in the dual field theories since the singularities are physically acceptable by the criterion of [50]. Moreover, we have also shown that all non-conformal flows within this sector are physical in the sense that all types of singularities lead to the scalar potential that is bounded from above. These solutions also generalize those given recently in [21] within an SO(2) × SO(2) × SO(3) × SO(2) subtruncation. However, in this SO(2)×SO(2)×SO(2)×SO(2) sector, no nontrivial AdS 4 critical points appear, so there are no RG flows between confornal fixed points. In addition, no nontrivial electric-magnetic phases appear in the analysis. Given that all values of the phase α > 0 give rise to equivalent gauged supergravities, this sector is essentially the same as the undeformed SO(4) × SO(4) gauged supergravity considered in [20] and [21].
For N = 1 solutions within SO(3) diag × SO(3) sector, the phase β 1 for an SO(3) factor in the vector multiplets appears. We have found three N = 4 supersymmetric AdS 4 critical points with one of them being the trivial AdS 4 critical point. The remaining two non-trivial critical points exist for particular values of β 1 = 0 and β 1 = π 2 . The first one preserves SO(3) diag × SO(3) × SO(3) symmetry identified in [20]. The second one with SO(3) × SO(3) diag × SO(3) symmetry is very similar to the first critical point and should be related by electric-magnetic duality. We have studied N = 1 supersymmetric RG flows between the trivial AdS 4 vacuum to these two non-trivial critical points similar to the N = 4 RG flows given previously in [20]. These flows are driven by relevant operators of dimensions ∆ = 1, 2 and preserve N = 1 supersymmetry along the flows. At both the UV and IR fixed points, the supersymmetry enhances to N = 4. For other values of the phase β 1 , we have not found any non-trivial AdS 4 critical point. An intensed numerical search suggests that there are no other supersymmetric AdS 4 vacua in this sector. We have also given a number of holographic RG flows from AdS 4 critical points to various types of non-conformal phases. Unlike the N = 2 solutions, it turns out that all these flows are unphysical.
Similar to the ω-deformed SO(8) gauged supergravity, the N = 4 gauged supergravity considered here currently has no higher dimensional origin. It would be interesting to find the embedding of this gauged supergravity in ten or eleven dimensions. The relevant consistent truncation ansatze could be obtained by using double field theory at SL(2) angles developed in [46] similar to the embedding of half-maximal gauged supergravities in higher dimensions studied in [51,52,53,54,55,56]. These could be used to uplift the solutions given here to ten/eleven dimensions resulting in a complete AdS 4 /CFT 3 holography in the framework of string/M-theory. In particular, the unphysical singularities of N = 1 non-conformal RG flows might be resolved in ten/eleven dimensions and give rise to genuine gravity duals of three-dimensional field theories. It would also be interesting to identify the dual N = 4 SCFTs and relevant deformations dual to the solutions given in this paper. In addition, the SO(3) diag sector in which both the phases β 1 and β 2 appear deserves further study and might lead to new AdS 4 vacua. Finally, other types of solutions such as Janus solutions and AdS 4 black holes are also worth considering.

Acknowledgement
This work is funded by National Research Council of Thailand (NRCT) and Chulalongkorn University under grant N42A650263. The author would like to thank G. Inverso for a useful correspondence and T. Assawasowan for collaboration in a related project.

A Useful formulae
In this appendix, we give some formulae used in the main text in particular the convention on 't Hooft matrices and explicit forms of the scalar potential and BPS equations in SO(3) diag × SO(3) sector.