Supersymmetric $AdS_5$ black holes and strings from 5D $N=4$ gauged supergravity

We study supersymmetric $AdS_3\times \Sigma_2$ and $AdS_2\times \Sigma_3$ solutions, with $\Sigma_2=S^2,H^2$ and $\Sigma_3=S^3,H^3$, in five-dimensional $N=4$ gauged supergravity coupled to five vector multiplets. The gauge groups considered here are $U(1)\times SU(2)\times SU(2)$, $U(1)\times SO(3,1)$ and $U(1)\times SL(3,\mathbb{R})$. For $U(1)\times SU(2)\times SU(2)$ gauge group admiting two supersymmetric $N=4$ $AdS_5$ vacua, we identify a new class of $AdS_3\times \Sigma_2$ and $AdS_2\times H^3$ solutions preserving four supercharges. Holographic RG flows describing twisted compactifications of $N=2$ four-dimensional SCFTs dual to the $AdS_5$ vacua to the SCFTs in two and one dimensions dual to these geometries are numerically given. The solutions can also be interpreted as supersymmetric black strings and black holes in asymptotically $AdS_5$ spaces with near horizon geometries given by $AdS_3\times \Sigma_2$ and $AdS_2\times H^3$, respectively. These solutions broaden previously known black brane solutions including half-supersymmetric $AdS_5$ black strings recently found in $N=4$ gauged supergravity. Similar solutions are also studied in non-compact gauge groups $U(1)\times SO(3,1)$ and $U(1)\times SL(3,\mathbb{R})$.


Introduction
Black branes of different spatial dimensions play an important role in the develoment of string/M-theory. They lead to many insightful results such as the construction of gauge theories in various dimensions and the celebrated AdS/CFT correspondence [1]. According to the latter, black branes in asymptotically AdS spaces are of particular interest since they are dual to RG flows across dimensions from superconformal field theories (SCFTs) dual to the asymptotically AdS spaces to lower-dimensional fixed points dual to the near horizon geometries [2]. Recently, a new approach for computing microscopic entropy of AdS 4 balck holes has been introduced based on twisted partition functions of three-dimensional SCFTs [3,4,5,6,7,8,9]. This has also been applied to AdS black holes in other dimensions [10,11,12,13,14].
In this paper, we are interested in supersymmetric black holes and black strings in asymptocally AdS 5 spaces from five-dimensional N = 4 gauged supergravity coupled to vector multiplets [15,16]. These solutions have near horizon geometries of the forms AdS 2 × Σ 3 and AdS 3 × Σ 2 , respectively. We will consider Σ 3 in the form of a three-sphere (S 3 ) and a three-dimensional hyperbolic space (H 3 ). Similarly, Σ 2 will be given by a two-sphere (S 2 ) and a two-dimensional hyperbolic space (H 2 ), or a Riemann surface of genus g > 1. Similar solutions have previously been found in minimal and maximal gauged supergravities, see for example [17,18,19,20,21,22,23,24]. This type of solutions has also appeared in pure N = 4 gauged supergravity in [25], and recently, half-supersymmetric black strings with hyperbolic horizons have been found in matter-coupled N = 4 gauged supergravity with compact U(1) × SU(2) × SU (2) and non-compact U(1) × SO(3, 1) gauge groups [26].
We will look for more general solutions of AdS 5 black strings with both hyperbolic and spherical horizons and preserving 1 4 of the N = 4 supersymmetry in five dimensions. The solutions interpolate between N = 4 supersymmetric AdS 5 vacua of the gauged supergravity and near horizon geometries of the form AdS 3 × Σ 2 . In addition, we will look for supersymmetric black holes interpolating between AdS 5 vacua and near horizon geometries AdS 2 × Σ 3 . According to the AdS/CFT correspondence, these solutions describe RG flows across dimensions from the dual N = 2 SCFTs to two-and one-dimensional SCFTs in the IR. The IR SCFTs are obtained via twisted compactifications of N = 2 SCFTs in four dimensions. Many solutions of this type have been found in various space-time dimensions, see [27,28,29,30,31,32,33,34,35,36,37,38] for an incomplete list.
We mainly consider N = 4 gauged supergravity coupled to five vector multiplets with gauge groups entirely embedded in the global symmetry SO (5,5). We will also restrict ourselves to gauge groups that lead to supersymmetric AdS 5 vacua. These gauge groups have been shown in [39] to take the form of U(1) × H 0 × H with the U(1) gauged by the graviphoton that is a singlet under USp(4) ∼ SO(5) R-symmetry. The H ⊂ SO(n + 3 − dim H 0 ) is a compact group gauged by vector fields in the vector multiplets, and H 0 is a non-compact group gauged by three of the graviphotons and dim H 0 −3 vectors from the vector multiplets. The remaining two graviphotons in the fundamental representation of SO(5) are dualized to massive two-form fields. In addition, H 0 must contain an SU (2) subgroup. For the case of five vector multiplets, possible gauge groups that admit supersymmetric AdS 5 vacua and can be embedded in SO (5,5) are U(1) × SU(2) × SU(2), U(1) × SO(3, 1) and U(1) × SL(3, R). We will look for AdS 5 black string and black hole solutions in all of these gauge groups.
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, we find supersymmetric AdS 3 × Σ 2 solutions preserving four supercharges and give numerical RG flow solutions interpolating between these geometries and supersymmetric AdS 5 vacua. An AdS 2 × H 3 solution together with an RG flow interpolating between AdS 5 vacua and this geometry will also be given. In section 4 and 5, we repeat the same analysis for non-compact U(1) × SO(3, 1) and U(1) × SL(3, R) gauge groups. Since the U(1) × SL(3, R) gauge group has not been studied in [26], we will discuss its construction and supersymmetric AdS 5 vacuum in detail. The full scalar mass spectrum at this critical point will also be given. This should be useful in the holographic context since it contains information on dimensions of operators dual to supergravity scalars. We end the paper with some conclusions and comments in section 6.
2 Five dimensional N = 4 gauged supergravity coupled to vector multiplets In this section, we briefly review the structure of five dimensional N = 4 gauged supergravity coupled to vector multiplets with the emphasis on formulae relevant for finding supersymmetric solutions. The detailed construction of N = 4 gauged supergravity can be found in [15] and [16]. The 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 representation. The gravity multiplet can 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, and the components fields within these vector multiplets will be denoted by (A a µ , λ a i , φ ma ). From both gravity and vector multiplets, there are in total 6 + n vector fields which will be denoted by . The 5n scalar fields from the vector multiplets parametrize the SO(5, n)/SO(5)× SO(n) coset. To describe this coset manifold, we introduce a coset representative V A M transforming under the global SO(5, n) and the local SO(5) × SO(n) by left and right multiplications, respectively. We use indices M, N, . . . = 1, 2, . . . , 5 + n for global SO(5, n) indices. The local SO(5) × SO(n) indices A, B, . . . will be split into A = (m, a). We can accordingly write the coset representative as The matrix V A M is an element of SO(5, n) and satisfies the relation with η M N = diag(−1, −1, −1, −1, −1, 1, . . . , 1) being the SO(5, n) invariant tensor. Equivalently, the SO(5, n)/SO(5) × SO(n) coset can also be described in term of a symmetric matrix which is manifestly invariant under the SO(5) × SO(n) local symmetry. Gaugings promote a given subgroup G 0 of the full global symmetry SO(1, 1)× SO(5, n) of N = 4 supergravity coupled to n vector multiplets to be a local symmetry. These gaugings are efficiently described by using the embedding tensor formalism. N = 4 supersymmetry allows three components of the embedding tensor ξ M , ξ M N = ξ [M N ] and f M N P = f [M N P ] . The first component ξ M describes the embedding of the gauge group in the SO(1, 1) ∼ R + factor identified with the coset space parametrized by the dilaton Σ. From the result of [39], the existence of N = 4 supersymmetric AdS 5 vacua requires ξ M = 0. In this paper, we are only interested in solutions that are asymptotically AdS 5 , so we will restrict ourselves to the gaugings with ξ M = 0.
For ξ M = 0, the gauge group is entirely embedded in SO(5, n) with the gauge generators given by where ∇ µ is the usual space-time covariant derivative. We use the convention that the definition of ξ M N and f M N P includes the gauge coupling constants. Note also that SO(5, n) indices M, N, . . . are lowered and raised by η M N and its inverse η M N , respectively.
Generators X M = (X 0 , X M ) of a consistent gauge group must form a closed subalgebra of SO(5, n). This requires ξ M N and f M N P to satisfy the quadratic constraints Gauge groups that admit N = 4 supersymmetric AdS 5 vacua generally take the form of U(1) × H 0 × H, see [39] for more detail. The U(1) is gauged by A 0 µ while H ⊂ SO(n + 3 − dim H 0 ) is a compact group gauged by vector fields in the vector multiplets. H 0 is a non-compact group gauged by three of the graviphotons and dim H 0 − 3 vectors from the vector multiplets. H 0 must also contain an SU(2) subgroup. For simple groups, H 0 can be SU(2) ∼ SO(3), SO(3, 1) and SL(3, R).
In the embedding tensor formalism, there are two-form fields that are introduced off-shell. These two-form fields do not have kinetic terms and coupled to vector fields via a topological term. In all of the solutions considered here, the two-form fields can be consistently truncated out. We will accordingly set all the two-form fields to zero from now on. 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 whose explicit form will not be given here since, given our ansatz for the gauge fields, it will not play any role in the present discussion. With vanishing two-form fields, the covariant gauge field strength tensors read where 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 . All fermionic fields are described by symplectic Majorana spinors subject to the following condition with C and Ω ij being respectively the charge conjugation matrix and USp(4) symplectic form. Supersymmetry transformations of fermionic fields (ψ µi , χ i , λ a i ) are given by (15) in which the fermion shift matrices are defined by In these equations, V ij M is defined in term of V M m as where Γ ij m = Ω ik Γ mk j and Γ mi j are SO(5) gamma matrices. Similarly, the inverse element V ij M can be written as In the subsequent analysis, we use the following explicit choice of SO(5) gamma matrices Γ mi j given by where σ i , i = 1, 2, 3 are the usual Pauli matrices. The covariant derivative on ǫ i reads where the composite connection is defined by In this work, we mainly focus on the case of n = 5 vector multiplets. To parametrize the scalar coset SO(5, 5)/SO(5) × SO(5), it is useful to introduce a basis for GL(10, R) matrices (e M N ) P Q = δ M P δ N Q (22) in terms of which SO(5, 5) non-compact generators are given by For a compact U(1) × SU(2) × SU(2) gauge group, components of the embedding tensor are given by where g 1 , g 2 and g 3 are the coupling constants for each factor in U(1) × SU(2) × SU(2). The scalar potential obtained from truncating the scalars from vector multiplets to U(1) × SU(2) diag ⊂ U(1) × SU(2) × SU(2) singlets has been studied in [26]. There is one U(1) × SU(2) diag singlet from the SO(5, 5)/SO(5) × SO(5) coset corresponding to the following SO(5, 5) non-compact generator With the coset representative given by the scalar potential can be computed to be The potential admits two N = 4 supersymmetric AdS 5 critical points given by i : ii : In critical point i, we have set g 2 = − √ 2g 1 to make this critical point occur at Σ = 1. However, we will keep g 2 explicit in most expressions for brevity. Critical point i is invariant under the full gauge symmetry U(1) × SU(2) × SU(2) while critical point ii preserves only U(1)×SU(2) diag symmetry due to the non-vanising scalar φ. V 0 denotes the cosmological constant, the value of the scalar potential at a critical point.

Supersymmetric black strings
We now consider vacuum solutions of the form AdS 3 ×Σ 2 with Σ 2 being S 2 or H 2 . A number of AdS 3 × H 2 solutions that preserve eight supercharges together with RG flows interpolating between them and supersymmetric AdS 5 critical points have already been given in [26]. In this section, we look for more general solutions that preserve only four supercharges.
We begin with the metric ansatz for the Σ 2 = S 2 case where dx 2 1,1 is the flat metric in two dimensions. For Σ 2 = H 2 , the metric is given by To preserve some amount of supersymmetry, we perform a twist by cancelling the spin connection along the Σ 2 by some suitable choice of gauge fields. We will first consider abelian twists from the U(1) × U(1) × U(1) subgroup of the U(1) × SU(2) × SU(2) gauge symmetry. The gauge fields corresponding to this subgroup will be denoted by (A 0 , A 5 , A 8 ). The ansatz for these gauge fields will be chosen as for the S 2 case and for the H 2 case.
There are three singlets from the SO(5, 5)/SO(5) × SO(5) coset corresponding to the SO(5, 5) non-compact generators Y 53 , Y 54 and Y 55 . However, these can be consistently truncated to only a single scalar with the coset representative given by We now begin with the analysis for Σ 2 = S 2 . With the relevant component of the spin connection ωφθ = e −g cot θeφ, we find the covariant derivative of ǫ i along theφ direction where . . . refers to the term involving g ′ that is not relevant to the present discussion. Note also that a 8 does not appear in the above equation since A 8 is not part of the R-symmetry under which the gravitini and supersymmetry parameters are charged.
For half-supersymmetric solutions considered in [26], it has been shown that the twists from A 0 and A 5 can not be performed simultaneously, and there exist AdS 3 × H 2 solutions. However, if we allow for an extra projector such that only 1 4 of the original supersymmetry is unbroken, it is possible to keep both the twists from A 0 and A 5 non-vanishing. To achieve this, we note that We then impose the following projector to make the two terms with a 0 and a 5 in To cancel the spin connection, we then impose another projector and the twist condition It should be noted that the condition (41) reduces to that of [26] for either a 0 = 0 or a 5 = 0. However, the solutions in this case preserve only four supercharges, or N = 2 supersymmetry in three dimensions, due to the additional projector (39).
To setup the BPS equations, we also need the γ r projection due to the radial dependence of scalars. Following [26], this projector is given by with I i j defined by The covariant field strength tensors for the gauge fields in (34) can be straightforwardly computed, and the result is For Σ 2 = H 2 , the cancellation of the spin connection ωφθ = e −g coth θeφ is again achieved by the gauge field ansatz (35) using the conditions (39), (40) and (41). On the other hand, the covariant field strengths are now given by which have opposite signs to those of the S 2 case. This results in a sign change of the parameter (a 0 , a 5 , a 8 ) in the corresponding BPS equations. With all these, we obtain the following BPS equations In these equations, κ = 1 and κ = −1 refer to Σ 2 = S 2 and Σ 2 = H 2 , respectively. It can also be readily verified that these equations also imply the second order field equations. We now look for AdS 3 solutions from the above BPS equations. These solutions are characterized by the conditions g ′ = ϕ ′ = Σ ′ = 0 and f ′ = 1 We find the following AdS 3 solutions.
• For ϕ = 0, AdS 3 solutions only exist for a 8 = 0 and are given by .
This should be identified with similar solutions of pure N = 4 gauged supergravity found in [25]. Since a 8 and ϕ vanish in this case, the AdS 3 solution has a larger symmetry U(1) × U(1) × SU (2). Note also that unlike half-supersymmetric solutions that exist only for Σ 2 = H 2 , both Σ 2 = S 2 and Σ 2 = H 2 are possible by appropriately chosen values of a 0 , a 5 and g 1 , recall that g 2 = − √ 2g 1 . • For ϕ = 0, we find a class of solutions Note that when a 8 = 0, we recover the AdS 3 solutions in (50). As in the previous solution, it can also be verified that these AdS 3 solutions exist for both Σ 2 = S 2 and Σ 2 = H 2 .
Examples of numerical solutions interpolating between N = 4 AdS 5 vacuum with U(1) × SU(2) × SU(2) symmetry to these AdS 3 × Σ 2 are shown in figure 1 and 2. At large r, the solutions are asymptotically N = 4 supersymmetric AdS 5 critical point i given in (30). It should also be noted that the flow solutions preserve only two supercharges due to the γ r projector imposed along the flow.
As pointed out in [26], there are five singlets from the vector multiplet scalars but these can be truncated to three scalars corresponding to the following noncompact generators of SO(5, 5) The coset representative is then given by To implement the U(1) diag gauge symmetry, we impose an additional condition on the parameters a 5 and a 8 as follow We can repeat the previous analysis for the U(1) × U(1) × U(1) twists, and the result is the same as in the previous case with the twist condition (41) and projectors (39), (40) and (42).
With the same procedure as in the previous case, we obtain the following BPS equations From these equations, we find the following AdS 3 × Σ 2 solutions.
• For φ 1 = φ 3 = 0, there is a family of AdS 3 solutions given by , We refrain from giving the explicit form of L AdS 3 at this vacuum due to its complexity.
• For φ 3 = 0, we find • Finally, for φ 1 = 0, we find Unlike the previous case, at large r, we find that solutions to these BPS equations can be asymptotic to any of the two N = 4 supersymmetric AdS 5 vacua i and ii given in (30) and (31). Therefore, we can have RG flows from the two AdS 5 vacua to any of these AdS 3 × Σ 2 solutions. Some examples of these solutions for Σ 2 = S 2 are given in figures 3, 4, 5 and 6.

Supersymmetric black holes
We now move to another type of solutions, supersymmetric AdS 5 black holes. We will consider near horizon geometries of the form AdS 2 × Σ 3 for Σ 3 = S 3 and Σ 3 = H 3 . The twist procedure is still essential to preserve supersymmetry. For the S 3 case, we take the metric to be ds 2 = −e 2f (r) dt 2 + dr 2 + e 2g(r) dψ 2 + sin 2 ψ(dθ 2 + sin 2 θdφ 2 ) .
we obtain non-vanishing components of the spin connection ωφθ = e −g cot θ sin ψ eφ, ωφψ = e −g cot ψeφ, ωθψ = e −g cot ψeθ . (66) We then turn on gauge fields corresponding to the U(1) × SU(2) diag ⊂ U(1) × SU(2) × SU(2) symmetry and consider scalar fields that are singlet under U(1) × SU(2) diag . Using the coset representative (28), we find components of the composite connection that involve the gauge fields (67) The components of the spin connection on S 3 that need to be cancelled are ωφθ, ωφψ and ωθψ. To impose the twist, we set A 0 = 0 and take the SU(2) diag gauge fields to be A 3 = a 3 cos ψdθ, A 4 = a 4 cos θdφ, A 5 = a 5 cos ψ sin θdφ (68) together with A 3+m = g 2 g 3 A m for m = 3, 4, 5. By considering the covariant derivative of ǫ i along θ and φ directions, we find that the twist is achieved by imposing the following conditions g 2 a 3 = g 2 a 4 = g 2 a 5 = 1 (69) and projectors Note that the last projector is not independent of the first two. Therefore, the AdS 2 solutions preserve four supercharges of the original supersymmetry. Condition (69) also implies a 3 = a 4 = a 5 . We will then set a 4 = a 5 = a 6 = a from now on. Using the definition (8), we find the gauge covariant field strengths and H 3+m = g 2 g 3 H m for m = 3, 4, 5. For Σ 3 = H 3 , we use the metric ansatz ds 2 = −e 2f dt 2 + dr 2 + e 2g y 2 (dx 2 + dy 2 + dz 2 ) with non-vanishing components of the spin connection where various components of the vielbein are given by Since there are only two components, ωxŷ and ωẑŷ, of the spin connection to be cancelled in the twisting process, we turn on the following SU(2) gauge fields and A m+3 = g 2 g 3 A m , for m = 3, 4, 5. Repeating the same analysis as in the S 3 case, we find the twist conditions and projectors The last projector is not needed for the twist with A 4 = 0. In addition, it follows from the first two projectors as in the S 3 case. The twist condition (76) again implies thatã = a, and the covariant field strengths in this case are given by and H m+3 = g 2 g 3 H m , for m = 3, 4, 5. Note that although A 4 = 0, we have nonvanishing H 4 due to the non-abelian nature of SU(2) field strengths.
With all these ingredients, the following BPS equations are straightforwardly obtained As in the AdS 3 solutions, κ = 1 and κ = −1 corresponds to Σ 3 = S 3 and Σ 3 = H 3 , respectively. It turns out that only κ = −1 leads to an AdS 2 solution given by This solution preserves N = 4 supersymmetry in two dimensions and U(1) × SU(2) diag symmetry. As r → ∞, f ∼ g ∼ r, solutions to the above BPS equations are asymptotic to either of the N = 4 AdS 5 vacua in (30) and (31). RG flow solutions interpolating between these AdS 5 vacua and the AdS 2 × H 3 solution in (83) are shown in figure 7 and 8. In particular, the flow in figure 8 connects three critical points similar to the solution given in the previous section. We end this section by a comment on the possibility of turning on the twist from A 0 along with those from the SU(2) diag gauge fields. As in the previous section, if we impose an additional projector the projection matrix of the A 0 term in the composite connection (67) will be proportional to that of A 3 . We can take the ansatz for A 0 to be and proceed as in the A 0 = 0 case. This results in the projectors given in (77) and the twist conditions g 2 a 4 = g 2 a 5 = 1 and g 1 a 0 + g 2 a 3 = 1 .
We can see that at this stage the parameter a 3 needs not be equal to a 4 and a 5 . However, consistency of the BPS equations from δλ a i conditions require a 3 = a 4 = a 5 and hence a 0 = 0 by the conditions in (86). This is because A 0 does not appear in δλ a i variation. The resulting BPS equations then reduce to those of the previous case with A 0 = 0. So, we conclude that the A 0 twist cannot be turned on along with the SU(2) diag twists.

U (1) × SO(3, 1) gauge group
For non-compact U(1) × SO(3, 1) gauge group, components of the embedding tensor are given by This gauge group has already been studied in [26]. The scalar potential admits one supersymmetric N = 4 AdS 5 vauum at which all scalars from vector multiplets vanish and Σ = 1 after choosing g 2 = − √ 2g 1 . At the vacuum, the gauge group is broken down to its maximal compact subgroup U(1) × SO(3). A holographic RG flow from this critical point to non-conformal field theory in the IR and a flow to AdS 3 × H 2 vacuum preserving eight supercharges have also been studied in [26]. In this case, AdS 3 × S 2 solutions do not exist.
In this section, we will study AdS 3 × Σ 2 and AdS 2 × Σ 3 solutions preserving four supercharges. The analysis is closely parallel to that performed in the previous section, so we will give less detail in order to avoid repetition.

Supersymmetric black strings
We will use the same metric ansatz as in equations (32) and (33) and consider the twist from U(1) × U(1) gauge fields. The second U(1) is a subgroup of the SO(3) ⊂ SO(3, 1). There are in total five scalars that are singlet under this U(1) × U(1), but as in the compact U(1) × SU(2) × SU(2) gauge group, these can be truncated to three singlets corresponding to the following SO(5, 5) noncompact generators With the embedding tensor (88), the compact SO(3) symmetry is generated by X 3 , X 4 and X 5 generators.
Using the coset representative of the form we can repeat all the analysis of the previous section by using the ansatz for the gauge fields A 0 = a 0 cos θdφ and A 5 = a 5 cos θdφ, for Σ 2 = S 2 and A 0 = a 0 cosh θdφ and A 5 = a 5 cosh θdφ, for Σ 2 = H 2 . The result is similar to the compact case with the projectors (39) and (40) and the twist condition (41).
As in the compact case, Σ 2 can be either S 2 or H 2 , depending on the values of a 5 , a 0 , g 1 and g 2 such that the twist condition (41) is satisfied. This is in contrast to the half-supersymmetric solution found in [26] for which only Σ 2 = H 2 is possible.
To find a domain wall interpolating between the AdS 5 vacuum to this AdS 3 × Σ 2 solution, we further truncate the BPS equations by setting φ i = 0 for i = 1, 2, 3. The resulting equations are given by An example of numerical solutions is shown in figure 9.

Supersymmetric black holes
We now consider AdS 2 × Σ 3 solutions within this non-compact gauge group. We will look for solutions with U(1) × SO(3) ⊂ U(1) × SO(3, 1) symmetry. There is one U(1) × SO(3) singlet from the SO(5, 5)/SO(5) × SO(5) coset corresponding to the non-compact generator The coset representative can be written as Using the metric ansatz (64) and (72) together with the gauge fields (68) and (75), we find that the twist can be implemented by using the projectors given in (70). Furthermore, the twist condition also implies that a 3 = a 4 = a 5 = a with g 2 a = 1, and the twist from A 0 cannot be turned on. The AdS 2 × Σ 3 solutions preserve four supercharges.
Using the projector (42), we can derive the following BPS equations These equations admit one AdS 2 × H 3 solution given by while AdS 2 × S 3 solutions do not exist. By setting φ = 0, we find a numerical solution to the above BPS equations as shown in figure 10. Components of the embedding tensor for this gauge group are given by

Supersymmetric AdS 5 vacuum
The SL(3, R) factor is embedded in SO(3, 5) ⊂ SO (5,5) such that its adjoint representation is identified with the fundamental representation of SO (3,5). The SO(3) ⊂ SL(3, R) is embedded in SL(3, R) such that 3 → 3. Decomposing the adjoint representation of SO (3,5) to SL(3, R) and SO(3), we find that the 25 scalars transform under SO(3) ⊂ SL(3, R) as Unlike the U(1) × SO(3, 1) gauge group, there is no singlet under the compact SO(3) symmetry. Taking into account the embedding of the U(1) factor in the gauge group as described in (110), we find the transformation of the scalars under with the subscript denoting the U(1) charges. It can be readily verified by studying the corresponding scalar potential or recalling the result of [39] that this U(1) × SL(3, R) gauge group admits a supersymmetric N = 4 AdS 5 vacuum at which all scalars from vector multiplets vanish with Σ = 1 and We have, as in other gauge groups, set g 2 = − √ 2g 1 to bring this vacuum to the value of Σ = 1. All scalar masses at this vacuum are given in table 1
These equations admit one supersymmetric AdS 3 × Σ 2 solution given by φ 2 = φ 3 = 0, Σ = √ 2κ a 5 g 1 , and a domain wall interpolating between this critical point and the supersymmetric AdS 5 is shown in figure 12. It should also be noted that this AdS 3 × Σ 2 solution is the same as in the U(1) × SO(3, 1) gauge group.

Supersymmetric black holes
We end this section with an analysis of AdS 2 × Σ 3 solutions and domain walls connecting these solutions to the supersymmetric AdS 5 . In order to preserve supersymmetry, SO(3) ⊂ SL(3, R) gauge fields must be turned on. However, in the present case, there is no SO(3) singlet scalar from the vector multiplets. After using the twist condition g 2 a = 1 and projectors in (70) and (77) together with the ansatz for the gauge fields in (68) and (75), we obtain the BPS equations These equations turn out to be the same as in the SO(3, 1) case after setting all the scalars from vector multiplets to zero. A single AdS 2 × H 3 critical point is again given by (109).

Conclusions and discussions
We have found a new class of supersymmetric black strings and black holes in asymptotically AdS 5 space within N = 4 gauged supergravity in five dimensions coupled to five vector multiplets with gauge groups U(1) × SU(2) × SU(2), U(1) × SO(3, 1) and U(1) × SL(3, R). These generalize the previously known black string solutions preserving eight supercharges by including more general twists along Σ 2 . Furthermore, unlike the half-supersymmetric solutions which only exhibit hyperbolic horizons, the 1 4 -supersymmetric black strings can have both S 2 and H 2 horizons. On the other hand, the AdS 5 black holes only feature H 3 horizons.
For U(1) × SU(2) × SU(2) gauge group, we have identified a number of AdS 3 × Σ 2 solutions preserving four supercharges. The solutions have U(1) × U(1) ×U(1) and U(1) ×U(1) diag symmetries and correspond to N = (0, 2) SCFTs in two dimensions. We have given many examples of numerical RG flow solutions from the two supersymmetric AdS 5 vacua to these AdS 3 × Σ 2 geometries. We have also found a supersymmetric AdS 2 × H 3 solution describing the near horizon geometry of a supersymmetric black hole in AdS 5 . For U(1) × SO(3, 1) and U(1) × SL(3, R) gauge groups, all AdS 3 × Σ 2 and AdS 2 × Σ 3 solutions exist only for vanishing scalar fields from vector multiplets and have the same form for both gauge groups.
It would be interesting to compute twisted partition functions and twisted indices in the dual N = 2 SCFTs compactified on Σ 2 and Σ 3 . These should provide a microscopic description for the entropy of the aforementioned black strings and black holes in AdS 5 space. On the other hand, it is also interesting to find supersymmetric rotating AdS 5 black holes similar to the solutions found in minimal and maximal gauged supergravities [40,41] or black holes with horizons in the form of a squashed three-sphere [42,43,44]. Furthermore, embedding these solutions in string/M-theory is of particular interest and should give a full holograpic interpretation for the RG flows across dimensions identified here.