Holographic RG flows and $AdS_5$ black strings from 5D half-maximal gauged supergravity

We study five-dimensional $N=4$ gauged supergravity coupled to five vector multiplets with compact and non-compact gauge groups $U(1)\times SU(2)\times SU(2)$ and $U(1)\times SO(3,1)$. For $U(1)\times SU(2)\times SU(2)$ gauge group, we identify $N=4$ $AdS_5$ vacua with $U(1)\times SU(2)\times SU(2)$ and $U(1)\times SU(2)_{\textrm{diag}}$ symmetries and analytically construct the corresponding holographic RG flow interpolating between these critical points. The flow describes a deformation of the dual $N=2$ SCFT driven by vacuum expaction values of dimension-two operators. In addition, we study $AdS_3\times \Sigma_2$ geometries, for $\Sigma_2$ being a two-sphere $S^2$ or a two-dimensional hyperbolic space $H^2$, dual to twisted compactifications of $N=2$ SCFTs with flavor symmetry $SU(2)$. We find a number of $AdS_3\times H^2$ solutions preserving eight supercharges for different twists from $U(1)\times U(1)\times U(1)$ and $U(1)\times U(1)_{\textrm{diag}}$ gauge fields. We numerically construct various RG flow solutions interpolating between $N=4$ $AdS_5$ ciritcal points and these $AdS_3\times H^2$ geometries in the IR. The solutions can also be interpreted as supersymmetric black strings in asymptotically $AdS_5$ space. These types of holographic solutions are also studied in non-compact $U(1)\times SO(3,1)$ gauge group. In this case, only one $N=4$ $AdS_5$ vacuum exists, and we give an RG flow solution from this $AdS_5$ to a singular geometry in the IR corresponding to an $N=2$ non-conformal field theory. An $AdS_3\times H^2$ solution together with an RG flow between this vacuum and the $N=4$ $AdS_5$ are also given.


Introduction
AdS 5 /CFT 4 correspondence has attracted much attention since the first proposal of the AdS/CFT correspondence in [1]. Various aspects of the very wellunderstood duality between type IIB theory on AdS 5 × S 5 and N = 4 Super Yang-Mills (SYM) theory in four dimensions are captured by N = 8 SO (6) gauged supergravity in five dimensions which is a consistent truncation of type IIB supergravity on S 5 [2]. One aspect of the AdS/CFT correspondence that has been extensively studied is holographic RG flows. There are many previous works considering these solutions both in N = 8 five-dimensional gauged supergravity and directly in type IIB string theory, see for example [3,4,5,6,7].
Results along this direction with less supersymmetry have also appeared in [8,9,10,11]. In this case, gauged supergravities in five dimensions with N < 8 supersymmetry provide a very useful framework. In this paper, we consider holographic RG flows in half-maximal N = 4 gauged supergravity coupled to vector multiplets. This gauged supergravity has global symmetry SO(1, 1)×SO(5, n), n being the number of vector multiplets. Gaugings of a subgroup G 0 ⊂ SO(1, 1) × SO(5, n) have been constructed in an SO(1, 1) × SO(5, n) covariant manner using the embedding tensor formalism in [12], see also [13]. The resulting solutions should describe RG flows arising from perturbing N = 2 superconformal field theories (SCFTs) by turning on some operators or their expectation values. Holographic solutions describing these N = 2 SCFTs and their deformations are less known compared to the N = 4 SYM. The results of this paper will give more examples of supersymmetric RG flow solutions and should hopefully shed some light on strongly coupled dynamics of N = 2 SCFTs.
We will consider N = 4 gauged supergravity coupled to five vector multiplets. This N = 4 gauged supergravity has a possibility of embedding in ten dimensions since the ungauged supergravity can be obtained via a T 5 reduction of N = 1 supergravity in ten dimensions similar to N = 4 supergravity in four dimensions coupled to six vector multiplets that descends from N = 1 ten-dimensional supergravity compactified on a T 6 . However, it should be emphasized that the gaugings considered here have no known higher dimensional origin todate. We mainly focus on domain wall solutions interpolating between N = 4 AdS 5 vacua or between an AdS 5 vacuum and a singular domain wall corresponding to a non-conformal field theory. These types of solutions have been extensively studied in half-maximal gauged supergravities in various space-time dimensions, see [10,11,14,15,16,17,18,19,20,21] for an incomplete list. The solutions involve only the metric and scalar fields.
We will also study solutions with some vector fields non-vanishing. These solutions interpolate between the above mentioned supersymmetric AdS 5 vacua and AdS 3 × Σ 2 geometries in the IR in which Σ 2 is a two-sphere (S 2 ) or a twodimensional hyperbolic space (H 2 ). Holographically, the resulting solutions describe twisted compactifications of the dual N = 2 SCFTs to two-dimensional SCFTs as first studied in [22]. A number of these flows across dimensions have been found within N = 8 gauged supergravity and its truncations in [23,24,25,26,27], see also a universal result in [28] and solutions obtained directly from type IIB theory in [29]. To the best of our knowledge, solutions of this type have not appeared before in the framework of N = 4 gauged supergravity coupled to vector multiplets, see however [30] for similar solutions in pure N = 4 gauged supergravity. Our results should give a generalization of the universal RG flows across dimensions in [28] by turning on the twists from flavor symmetries.
In addition, AdS 3 ×Σ 2 geometries can arise as near horizon limits of black strings. Therefore, flow solutions interpolating between AdS 5 and AdS 3 × Σ 2 should describe black strings in asymptotically AdS 5 space. Similar solutions in N = 2 gauged supergravity have been considered in [31,32,33,34,35]. We will give solutions of this type in N = 4 gauged supergravity. The solutions presented here will provide further examples of supersymmetric AdS 5 black strings and might be useful for both holographic studies of twisted N = 2 SCFTs on Σ 2 and certain dynamical aspects of black strings.
The paper is organized as follow. In section 2, we review N = 4 gauged supergravity in five dimensions coupled to vector multiplets using the embedding tensor formalism. In section 3, a compact U(1) × SU(2) × SU(2) gauge group is considered. Supersymmetric AdS 5 vacua and RG flows interpolating between them are given. A number of AdS 3 × H 2 solutions will also be given along with numerical RG flows interpolating between the previously identified AdS 5 vacua and these AdS 3 × H 2 geometries. In section 4, we repeat the analysis for a noncompact U(1) × SO(3, 1) gauge group. An RG flow from N = 2 SCFT dual to a supersymmetric AdS 5 vacuum to a singular geometry dual to a non-conformal field theory is considered. A supersymmetric AdS 3 × H 2 geometry and an RG flow between this vacuum and the AdS 5 critical point will also be given. We end the paper with some conclusions and comments in section 5.
2 Five dimensional N = 4 gauged supergravity coupled to vector multiplets In this section, we review the structure of five dimensional N = 4 gauged supergravity coupled to vector multiplets. We mainly focus on relevant formulae to find supersymmetric solutions. More details on the construction of N = 4 gauged supergravity can be found in [12] and [13]. In five dimensions, N = 4 gravity multiplet consists of the graviton eμ µ , four gravitini ψ µi , six vectors A 0 and A m µ , four spin-1 2 fields χ i and one real scalar Σ, the dilaton. Space-time and tangent space indices are denoted respectively by µ, ν, . . . = 0, 1, 2, 3, 4 andμ,ν, . . . = 0, 1, 2, 3, 4. The SO(5) ∼ USp(4) R-symmetry indices are described by m, n = 1, . . . , 5 for the SO(5) vector representation and i, j = 1, 2, 3, 4 for the SO(5) spinor or USp(4) fundamental repre-sentation. N = 4 supersymmetry allows the gravity multiplet to couple to an arbitrary number n of vector multiplets. Each vector multiplet contains a vector field A µ , four gaugini λ i and five scalars φ m . The n vector multiplets will be labeled by indices a, b = 1, . . . , n. Components fields in the n vector multiplets are accordingly denoted by (A a µ , λ a i , φ ma ). The 5n scalar fields parametrized the SO(5, n)/SO(5) × SO(n) coset. Combining the gravity and vector multiplets, we have 6 + n vector fields denoted by A M µ = (A 0 µ , A m µ , A a µ ) and 5n + 1 scalars. All fermionic fields are symplectic Majorana spinors subject to the condition with C and Ω ij being the charge conjugation matrix and USp(4) symplectic form, respectively.
The full global symmetry of N = 4 supergravity coupled to n vector multiplets is SO(1, 1) × SO(5, n). The SO(1, 1) ∼ R + factor is identified with the coset space described by the dilaton Σ. Gaugings can be efficiently described, in an SO(1, 1) × SO(5, n) covariant manner, by using the embedding tensor formalism. N = 4 supersymmetry allows three components of the embedding tensor ξ M , The existence of supersymmetric AdS 5 vacua requires ξ M = 0, see [36] for more detail. Since, in this paper, we are only interested in supersymmetric AdS 5 vacua and solutions interpolating between these vacua or solutions asymptotically approaching AdS 5 , we will restrict ourselves to the gaugings with ξ M = 0.
With ξ M = 0, the gauge group is entirely embedded in SO(5, n). The gauge generators in the fundamental representation of SO(5, n) can be written in terms of the SO(5, n) generators (t M N ) As a result, the covariant derivative reads where ∇ µ is the usual space-time covariant derivative including the spin connection. It should be noted that the definition of ξ M N and f M N P includes the gauge coupling constants. Furthermore, SO(5, n) indices M, N, . . . are lowered and raised by η M N and its inverse η M N . In order to define a consistent gauge group, generators X M = (X 0 , X M ) must form a closed subalgebra of SO(5, n). This requires ξ M N and f M N P to satisfy the quadratic constraints The first condition is the usual Jacobi identity. From the result of [36], gauge groups that admit N = 4 supersymmetric AdS 5 vacua are generally of the form is a compact group gauged by vector fields in the vector multiplets. H 0 is a noncompact group gauged by three of the graviphotons and dim H 0 − 3 vectors from the vector multiplets. In addition, H 0 must contain an SU(2) subgroup. For simple groups, H 0 can be SU(2) ∼ SO(3), SO(3, 1) and SL(3, R).
The bosonic Lagrangian of a general gauged N = 4 supergravity coupled to n vector multiplets can be written as where e is the vielbein determinant. L top is the topological term which we will not give the explicit form here due to its complexity. The covariant gauge field strength tensors read where The two-form fields do not have kinetic terms and satisfy the first-order field equation with H (3) defined by and These two form fields arise from vector fields that transform non-trivially under the U(1) part of the gauge group. The scalar potential is given by where M M N is the inverse of M M N , and M M N P QRS is obtained from by raising the indices with η M N . Fermionic supersymmetry transformations are given by In the above equations, the fermion shift matrices are defined by where Γ ij m = Ω ik Γ mk j and Γ mi j are SO(5) gamma matrices. Similarly, the inverse V ij M can be written as The covariant derivative on ǫ i is given by where the composite connection is defined by Before considering various supersymmetric solutions, we note here the relation between the scalar potential and the fermion shift matrices A 1 and A 2 Rasing and lowering of indices i, j, . . . by Ω ij and Ω ij are also related to complex conjugation for example We begin with a compact gauge group U(1) × SU(2) × SU (2). In order to gauge this group, we need to couple the gravity multiplet to at least three vector multiplets. Components of the embedding tensor for this gauge group are given by where g 1 , g 2 and g 3 are the corresponding coupling constants for each factor in To parametrize the scalar coset SO(5, n)/SO(5) × SO(n), we introduce a basis for GL(5 + n, R) matrices in terms of which SO(5, n) non-compact generators are given by Y ma = e m,a+5 + e a+5,m , m = 1, 2, . . . , 5, a = 1, 2, . . . , n .
We will mainly consider the case of n = 5 vector multiplets, but the results can be straightforwardly extended to the case of n > 5.

RG flows between N = 4 supersymmetric AdS 5 critical points
We will consider scalar fields that are singlets of , the scalars transform as (5,5). With the above embedding of the gauge group in SO(5, 5), the scalars transform under U(1) × SU(2) × SU(2) gauge group as and transform under U(1) × SU(2) diag as where the subscript denotes the U(1) charges. According to this decomposition, there is one singlet corresponding to the following SO(5, 5) non-compact generator Using the coset representative parametrized by we find the scalar potential for φ and Σ as follow This potential admits two N = 4 supersymmetric AdS 5 critical points. The first one is given by This critical point is invariant under the full gauge symmetry U(1) × SU(2) × SU(2) since Σ is a singlet of the whole SO(5, 5) global symmetry. Furthermore, we can rescale Σ, or equivalently set g 2 = − √ 2g 1 to bring this critical point located at Σ = 1. The cosmological constant, the value of the scalar potential at the critical point, is Another supersymmetric critical point is given by This critical point also preserves the full N = 4 supersymmetry but has only U(1) × SU(2) diag symmetry due to the non-vanising scalar φ. The cosmological constant for this critical point is This second N = 4 AdS 5 critical point has been shown to exist in [11] when an additional SU(2) dual to a flavor symmetry of the dual N = 2 SCFT is present. We now analyze the BPS equations arising from setting supersymmetry transformations of fermions to zero. We first define V M ij with the following explicit choice of SO(5) gamma matrices Γ mi where σ i , i = 1, 2, 3 are the usual Pauli matrices.
Since we are interested only in solutions with only the metric and scalars non-vanishing, we will set all the vector and two-form fields to zero. The metric ansatz is given by the standard domain wall with dx 2 1,3 being the metric on four dimensional Minkowski space. In addition, the scalars Σ and φ as well as the Killing spinors ǫ i are functions of only the radial coordinate r.
We begin with the δψ µi = 0 conditions for µ = 0, 1, 2, 3 which lead to where ′ denotes the r-derivative. Multiply this equation by A ′ γ r and iterate, we find where W will be identified with the superpotential. When substitute this result in equation (41), we find On the other hand, equation (40) leads to the projection condition on ǫ i where I i j is defined via The condition δψr i = 0 gives the usual r-dependent Killing spinors of the form ǫ i = e A 2 ǫ 0i for constant spinors ǫ 0i satisfying (44). Using the projector (44) in conditions δχ i = 0 and δλ a i = 0, we can derive the first order flow equations for Σ and φ.
For the coset representative in (32), we find the superpotential The matrix I i j in the γ r projection is given by The scalar kinetic term reads The scalar potential (33) can be written in term of the superpotential as By choosing the sign choice such that the U(1) × SU(2) × SU (2) is identified with r → ∞, the BPS conditions from δχ i and δλ a i reduce to the following equations It can be readily verified that the critical points given in (34) and (36) are also critical point of W and solve equations (50) and (51). These critical points are then N = 4 supersymmetric. Together with the A ′ equation we have the full set of BPS equations to be solved for RG flows interpolating between the two supersymmetric AdS 5 critical points. It can be verified that these BPS equations imply the second-order field equations.
By treating φ as an independent variable, we can solve for Σ(φ) and A(φ) as follow We have neglected an irrelevant additive integration constant in A. The constant C will be chosen in such a way that Σ approaches the second AdS 5 vacuum. This leading to the final form of the solution for Σ Finally, the solution for φ(r) is given by where the new radial coordinate ρ is defined by dρ dr = Σ −1 . This solution is the same as that obtained in [11] and has a very similar structure to solutions obtained from half-maximal gauged supergravities in seven and six dimensions [14,16].
Near the UV N = 4 critical point, we find where the AdS 5 radius is given by This behavior implies that the RG flow dual to this solution is driven by vacuum expectation values of operators with dimension ∆ = 2. Near the IR point, we find where The operator dual to φ becomes irrelevant in the IR with dimension ∆ = 6 while the operator dual to Σ has dimension ∆ = 2 as in the UV. For completeness, we give masses for all scalars at both critical points in table 1 and 2. In these tables, the singlets (1, 1) 0 and one of the 1 0 with m 2 L 2 = −4 in table 2 corresponds to Σ. The massless scalars 3 0 in table 2 are Goldstone bosons corresponding to the symmetry breaking SU(2) × SU(2) → SU(2) diag . The massless scalars 5 0 are dual to marginal operators in the dual N = 2 SCFT. It should also be noted that most of the results in this section have already been found in [11]. However, the full scalar mass spectra are new results that have not been studied in [11].

Supersymmetric RG flows from N = 2 SCFTs to two dimensional SCFTs
We now consider another type of solutions namely solutions interpolating between supersymmetric AdS 5 vacua identified previously and AdS 3 × Σ 2 geometries. In the present consideration, Σ 2 is a two-sphere (S 2 ) or a two-dimensional hyperbolic space (H 2 ). We begin with the metric ansatz for the Σ 2 = S 2 case ds 2 = e 2f (r) dx 2 1,1 + dr 2 + e 2g(r) (dθ 2 + sin 2 θdφ 2 ) where dx 2 1,1 is the flat metric in two dimensions. It is useful to note the components of the spin connection with the obvious choice of vielbein eμ = e f dx µ , er = dr, eθ = e g dθ, eφ = e g sin θdφ (62) forμ = 0, 1.
To preserve some amount of supersymmetry, we impose a twist condition by cancelling the spin connection on S 2 with some gauge connections. We will consider abelian twists from U(1) × U(1) × U(1) ⊂ U(1) × SU(2) × SU(2) and its U(1) × U(1) diag subgroup. The corresponding gauge fields are denoted by (A 0 , A 5 , A 8 ). Note that turning on A 0 and A 5 correspond to a twist along the R-symmetry U(1) × SU(2) of the dual N = 2 SCFTs while a non-vanishing A 8 is related to turning on the gauge field of SU(2) flavor symmetry. The latter cannot be used as a twist since the Killing spinor is neutral under this symmetry.
An effect of the twisting procedure is to cancel ωθφ on S 2 . The BPS conditions δψ iθ = 0 and δψ iφ = 0 then lead to the same BPS equation. In order to achieve this, we consider the ansatz for the gauge fields We consider two type of solutions with unbroken gauge symmetry U(1) × U(1) × U(1) and U(1) × U(1) diag . We begin with a simpler case of U(1) × U(1) × U(1) invariant sector consisting of four singlet scalars Σ and ϕ i , i = 1, 2, 3. The latter correspond to the SO(5, 5) non-compact generators Y 53 , Y 54 and Y 55 . The coset representative is then given by A straightforward computation gives relevant components of the covariant derivative on the Killing spinors ǫ i In order to cancel the spin connection, we need to impose the conditions Consistency with (iγθφ) 2 = I 4 requires the conditions (g 1 a 0 ) 2 + (g 2 a 5 ) 2 = 1 and g 1 g 2 a 0 a 5 = 0 .
The second condition implies, for non-vanishing g 1 and g 2 , either a 0 = 0 or a 5 = 0 for which the first condition gives g 2 a 5 = ±1 or g 1 a 0 = ±1, respectively. These two possibilities correspond respectively to the α-twist and β-twist studied in [37], see also a discussion in [28]. For a 0 = 0 and g 2 a 5 = ±1, the condition (66) becomes a projector For a 5 = 0 and g 1 a 0 = ±1, we find It should be noted that we can make a definite sign choice for the twist condition and the γθφ projector. The other possiblility can be obtained by changing the sign of a 0 or a 5 together with a sign change in the γθφ projector. In the remaining part of this paper, we will choose the twist conditions and γθφ projector with the upper sign.
For the U(1)×U(1) diag sector with the U(1) diag being a diagonal subgroup of U(1) × U(1) ⊂ SU(2) × SU(2), we have five singlets from the vector multiplet scalars corresponding to the following non-compact generators of SO(5, 5) giving rise to the coset representative The result of the analysis is the same as in the previous case but with an additional condition imposed on the flux parameters a 5 and a 8 g 2 a 5 = g 3 a 8 implementing the U(1) diag gauge symmetry. It turns out that, in both cases, all two-form fields can be consistently set to zero provided that A 1 and A 2 vanish.
For the H 2 case, we simply change sin θ to sinh θ in the metric (60) and take the gauge fields to be A M = a M cosh θdφ. The twist procedure works as in the S 2 case. However, due to the opposite sign in the covariant field strengths H M = dA M , the resulting BPS equations for the two cases are related to each other by a sign change in the twist parameters a M .

Flow solutions with U(1) × U(1) × U(1) symmetry
With the coset representative (64), the scalar potential and the superpotential are given respectively by and The scalar kinetic term reads The scalar potential can also be written in term of the superpotential as It can be easily checked that setting ϕ 2 = ϕ 3 = 0 is a consistent truncation. Moreover, the result with non-vanishing ϕ 2 and ϕ 3 is not significantly different from that with ϕ 2 = ϕ 3 = 0. Therefore, we will further simplify the computation by using this truncation and set ϕ 1 = ϕ.
We first consider the case with a 0 = 0. By using the γr projection (44) with the matrix I i j given in (47), we find the following BPS equations The sign choices κ = +1 and κ = −1 correspond to Σ 2 = S 2 and Σ 2 = H 2 , respectively. We will use this convention throughout the paper. The AdS 3 ×Σ 2 vacua are characterized by the conditions g ′ = ϕ ′ = Σ ′ = 0 and f ′ = 1 L AdS 3 . It turns out that the above equations admit any AdS 3 solutions only for a 8 = 0 and κ = −1. In this case, we find that any constant value of ϕ leads to an AdS 3 × H 2 solution of the form where ϕ 0 is a constant. This solution preserves eight supercharges or N = 4 in three dimensions due to the γθφ projector. On the other hand, the entire flow solution will preserve only four supercharges due to an additional γr projector. For ϕ 0 = 0, the solution has U(1) × U(1) × SU(2) symmetry due to the vanishing A 8 while the solution with ϕ 0 = 0 has smaller symmetry U(1) × U(1) × U(1). The resulting AdS 3 × H 2 geometry should be dual to a two dimensional N = (2, 2) SCFT with SU(2) or U(1) flavor symmetry depending on the value of ϕ 0 . An asymptotic analysis near the AdS 3 × H 2 critical point shows that ϕ is dual to a marginal operator in the two-dimensional SCFT. The central charge of the dual SCFT can also be computed by [38] which is independent of ϕ 0 . g 0 is the value of g(r) at the AdS 3 critical point. For H 2 being a genus g > 1 Riemann surface, we have vol(H 2 ) = 4π(g − 1). Examples of numerical flow solutions interpolating between N = 4 supersymmetric AdS 5 and N = 4 supersymmetric AdS 3 × H 2 with different values of ϕ 0 are given in figure 1. The solution with ϕ 0 = 0 is effectively the same as that studied in [28] which is in turn obtained from the solutions in [30] by turning off the U(1) gauge field. In this case, the matter multiplets can be decoupled. Solutions with ϕ 0 = 0 are only possible in the matter-coupled gauged supergravity and have not previously appeared.
For the case of a 5 = 0, we also find that the BPS conditions require a 8 = 0. The resulting BPS equations read Note also that the BPS equation for ϕ does not involve a 0 since ϕ is neutral under A 0 . In this case, AdS 3 vacua do not exist. For ϕ ′ = g ′ = 0, we find a singular behavior of Σ at a finite value of r for some contant C. This has also been pointed out in [28]. singlet scalars, and the computation is much more complicated than the previous case. We will again make a truncation by setting φ 4 = φ 5 = 0 in the following analysis. The scalar potential with this truncation is given by

Flow solutions with
This can be written in term of the superpotential as where the superpotential in this case is given by It can be verified that this superpotential admits two critical points given in equations (34) and (36). When φ 1 = φ 3 = 0, this is the U(1) × U(1) × U(1) invariant sector. For φ 3 = 0 and φ 1 = φ 2 , we reobtain the U(1) × SU(2) diag invariant scalars which admit the second N = 4 AdS 5 critical point with U(1) × SU(2) diag symmetry. We firstly consider the twist from A 0 gauge field. For a 5 = 0, the U(1) diag symmetry also demands a 8 = 0. The BPS equations for φ 1 , φ 2 and φ 3 will not depend on the twist parameter a 0 since they are not charged under A 0 . Therefore, the only possibility to have AdS 3 vacua is to set these scalars to their values at the two AdS 5 critical points. Setting all φ i = 0 for i = 1, 2, 3 dose not lead to any AdS 3 solutions as in the previous case. The other choice namely φ 3 = 0 and φ 1 = φ 2 = 1 2 ln g 3 −g 2 g 3 +g 2 does not give rise to any AdS 3 vacua either. Therefore, we will not give the explicit form of the BPS equations in this case.
We now consider the twist from A 5 and A 8 gauge fields. In this case, we do find some AdS 3 solutions. The BPS equations read for which there is a relation g 2 a 5 = g 3 a 8 to be imposed. We find that these equations admit AdS 3 × Σ 2 solutions only for κ = −1. The AdS 3 × H 2 solutions are given as follow.
• I. The simplest solution is obtained by setting φ i = 0, i = 1, 2, 3 and (97) • II. One of the AdS 3 × H 2 solutions with vector multiplet scalars nonvanishing is given by • III. There is another AdS 3 × H 2 solution located at All of these solutions preserve eight supercharges corresponding to N = 4 supersymmetry in three dimensions or equivalently N = (2, 2) in the dual two dimensional SCFTs. It should also be noted that critical points II and III appear to be related by a permutation of φ i . However, the solution with φ 2 = 0 does not exist since this also requires φ 1 = φ 3 = 0 and a 8 = a 5 = 0. The next step is to find RG flow solutions interpolating between N = 4 supersymmetric AdS 5 critical points and the above AdS 3 × H 2 geometries. We first consider a simple case of the flow to AdS 3 × H 2 critical point I with φ 1 = φ 2 = φ 3 = 0. The BPS equations simplify considerably to We can partially solve these equations analytically and find a relation between solutions of g and Σ of the form However, the complete solutions can only be found numerically. In this case, the solutions reduce to the universal flows across dimensions considered in [28]. An example of these solutions is given in figure 2. AdS 3 × H 2 critical point II is more interesting in the sense that it can be connected to both of the N = 4 AdS 5 vacua. In order to obtain RG flow solutions, we set φ 3 = 0 which is a consistent truncation. An example of flows from AdS 5 with U(1) × SU(2) × SU(2) symmetry to AdS 3 × H 2 critical point II is given in figure 3. With suitable boundary conditions, we can find a solution that flows from AdS 5 with U(1) × SU(2) × SU(2) symmetry and approaches AdS 5 with U(1) × SU(2) diag symmetry before reaching the AdS 3 × H 2 critical point II. A solution of this type is shown in figure 4.   Similarly, we can set φ 1 = 0 and find a numerical solution interpolating between AdS 5 vacuum with U(1) × SU(2) × SU(2) symmetry and AdS 3 × H 2 critical point III. The result is shown in figure 5. We have also verified that all of these solutions satisfy the corresponding field equations.
4 Supersymmetric RG flows in U (1) × SO (3,1) gauge group In this section, we consider a non-compact gauge group U(1) × SO(3, 1) with the embedding tensor At the vacuum, the U(1) × SO(3, 1) gauge group will be broken down to it maximal compact subgroup U(1)×SO(3) ⊂ U(1)×SO (3,1). Under this unbroken symmetry, there is one scalar singlet from SO(5, 5)/SO(5)×SO(5) corresponding to the non-compact generator With the usual parametrization of the coset representative of the form the scalar potential is given by This potential admits only one N = 4 supersymmetric AdS 5 vacuum due to the absence of flavor symmetry in the dual N = 2 SCFT in agreement with the result of [11]. This critical point is located at As in the U(1)×SU(2)×SU(2) gauge group, we can rescale Σ such that Σ = 1 at the AdS 5 vacuum. Equivalently, we can choose the value of g 2 to be g 2 = − √ 2g 1 . With this choice, the cosmological constant and AdS 5 radius are given by

BPS equations and holographic RG flow solutions
Since there is only one supersymmetric AdS 5 critical point, supersymmetric RG flows between AdS 5 critical points do not exist. We will look for solution describing a domain wall with one limit being the AdS 5 critical point identified above and another limit being a singular geometry dual to an N = 2 non-conformal field theory.
With the same procedure as before, the superpotential in this case reads It can be easily verified that W has only one critical point. The potential can be written in term of the superpotential as The BPS equations for this gauge group are given by By combining these equations, the flow equations for the warp factor A and the dilaton Σ can be written as The solution for Σ can be readily obtained To make the flow approach the AdS 5 critical point, we choose the constant C 1 to be . This leads to Finally, by redefining the radial coordinate r to ρ via dρ dr = Σ −1 , we find the solution for φ(ρ) where an additive integration constant has been discarded.
As r → ∞, we find The operator dual to φ is irrelevant as indicated by the value of m 2 L 2 = 12 in table 3. From the solution (119), φ → ±∞ at a finite value of ρ. Explicitly, we find that, as φ → ±∞, for some constant C. In both cases, Σ → 0 and V → ∞. As a result, these singularities are unphysical by the criterion of [39].

RG flows to AdS 3 × Σ 2 geometries
We now restrict ourselves to scalars which are invariant under SO(2) ⊂ SO(3) ⊂ SO(3, 1) whose generator is (T 5 ) M N = f 5M N . There are in total five singlets from SO(5, 5)/SO(5) × SO(5). However, as in the case of U(1) × SU(2) × SU(2) gauge group, we can truncate this set to just three singlets corresponding to the following non-compact generators The coset representative is given by and the potential reads which admits only a single supersymmetric critical point at which all vector multiplet scalars vanish.
The metric ansatz is still given by (60). We will consider the twists obtained from turning on U(1) × U(1) ⊂ U(1) × SO(3, 1) gauge fields along Σ 2 . These gauge fields will be denoted by A 0 and A 5 . As in the previous section, the twists from A 0 and A 5 cannot be turned on simultaneously. Furthermore, the A 0 twist does not lead to AdS 3 × Σ 2 solutions. We will therefore consider only the twist from A 5 . It turns out that the two-form fields can also be consistently set to zero provided that we set the gauge fields A 1 = A 2 = 0.

Conclusions and discussions
We have studied gauged N = 4 supergravity in five dimensions coupled to five vector multiplets with compact and non-compact gauge groups U(1)×SU(2)×SU(2) and U(1) × SO(3, 1). For U(1) × SU(2) × SU(2) gauge group, we have recovered two supersymmetric N = 4 AdS 5 vacua with U(1) × SU(2) × SU(2) and U(1) × SU(2) diag symmetries together with the RG flow interpolating between them found in [11]. However, we have also given the full mass spectra for scalar fields at both critical points which have not been studied in [11]. These should be useful in the holographic context since it provides information about dimensions of operators dual to the supergravity scalars. For U(1) × SO(3, 1) gauge group, there is only one N = 4 supersymmetric AdS 5 critical point with vanishing vector multiplet scalars. We have given an RG flow solution from an N = 2 SCFT dual to this vacuum to a non-conformal field theory dual to a singular geometry. However, this singularity is unphysical within the framework of N = 4 gauged supergravity. It would be interesting to embed this solution in ten or eleven dimensions and further investigate whether this singularity is resolved or has any physical interpretation in the context of string/M-theory. We have also considered AdS 3 × Σ 2 solutions by turning on gauge fields along Σ 2 . We have found that in order to preserve eight supercharges, the twists from the U(1) factor in the gauge group and the Cartan U(1) ⊂ SU(2), denoted by the parameters a 0 and a 5 , cannot be performed simultaneously. It should also be noted that for less supersymmetric solutions, both a 0 and a 5 can be non-vanishing such as 1 4 -BPS solutions found in [30] for pure N = 4 gauged supergravity with U(1) × SU(2) gauge group. It would also be interesting to look for more general solutions of this type.
For U(1) × SU(2) × SU(2) gauge group, we have identified a number of AdS 3 × H 2 solutions preserving eight supercharges. We have given numerical RG flow solutions from the two AdS 5 vacua to these AdS 3 × H 2 geometries. For U(1) × SO(3, 1) gauge group, there is one AdS 3 × H 2 solution when all scalars from vector multiplets vanish. The solution preserves eight supercharges similar to the solutions in the compact gauge group. A numerical RG flow between this solution and the N = 4 AdS 5 vacuum has also been given. All of these solutions describe twisted compactifications of N = 2 SCFTs on H 2 and should be of interest in holographic studies of N = 2 SCFTs in four dimensions and in the context of supersymmetric black strings. It is noteworthy that the space of AdS 5 and AdS 3 solutions in the compact gauge group is much richer than that of the non-compact gauge group. This is in line with similar studies of half-maximal gauged supergravities in other dimensions.
There are a number of future works extending our results presented here. It is interesting to consider flow solutions with non-vanishing two-form fields similar to the recently found solutions in seven and six dimensions in [40,41,42]. These solutions will also give a description of conformal defects in the dual N = 2 SCFTs. Furthermore, finding Janus solution within this N = 4 gauged supergravity is also of particular interest. This can be done by an analysis similar to that initiated in [43] and [44]. Up to now, this type of solutions has only appeared in N = 8 and N = 2 gauged supergravities, see for example [45,46].