New supersymmetric AdS 5 black strings from 5D N = 4 gauged supergravity

We ﬁnd a large class of new supersymmetric AdS 5 black strings from ﬁve-dimensional N = 4 gauged supergravity coupled to ﬁve vector multi-plets with SO (2) D × SO (3) × SO (3) gauge group. These solutions have near horizon geometries of the form AdS 3 × Σ 2 for Σ 2 being a two-sphere ( S 2 ) or a hyperbolic space ( H 2 ). There are four supersymmetric AdS 5 vacua with N = 4 and N = 2 supersymmetries. By performing topological twists along Σ 2 with SO (2) × SO (2) diag and SO (2) diag gauge ﬁelds, we ﬁnd a number of AdS 3 × Σ 2 ﬁxed points describing near horizon geometries of black strings in asymptotically AdS 5 spaces. Most of the solutions take the form of AdS 3 × H 2 with only one being AdS 3 × S 2 preserving SO (2) diag symmetry. We also give the corresponding black string solutions inter-polating between asymptotically locally AdS 5 vacua and the near horizon AdS 3 × Σ 2 geometries. There are a number of solutions ﬂowing from one, two or three AdS 5 vacua to an AdS 3 × Σ 2 ﬁxed point. These solutions can also be considered as holographic RG ﬂows across dimensions from N = 2 and N = 1 SCFTs in four dimensions to two-dimensional SCFTs with N = (2 , 0) or N = (0 , 2) supersymmetry.


Introduction
Supersymmetric black hole solutions in asymptotically AdS spaces have attracted much attention recently since the groundbreaking work on microstate counting of AdS 4 black holes from supersymmetric localization in [1], see also [2,3,4,5,6,7,8,9,10].This technique has also been extended to supersymmetric AdS 5 black strings in [11,12,13,14].Due to these results, constructing new supersymmetric AdS 5 black string solutions to further test the proposed holographic relation is interesting.Along this line, gauged supergravities in five dimensions provide a very useful tool in which AdS 5 black strings become domain walls interpolating between asymptotically AdS 5 spaces and near horizon geometries of the form AdS 3 × Σ 2 for Σ 2 being a two-sphere (S 2 ) or a two-dimensional hyperbolic space (H 2 ).
On the other hand, supersymmetric AdS 5 black string solutions also describe holographic RG flows from four-dimensional superconformal field theories (SCFTs) in the UV, dual to AdS 5 vacua, to two-dimensional SCFTs in the IR dual to near horizon geometries [15].In this point of view, the SCFTs in four dimensions undergo twisted compactifications on Σ 2 to conformal field theories in two dimensions.Since the pioneering work [15], a number of similar solutions have been found [16,17,18,32,33,21,22,23,24,25,26,27,28].Some of the solutions can be uplifted to string/M-theory, and the SCFTs dual to these solutions have also been identified.
In this paper, we will look for new supersymmetric AdS 5 black string solutions from five-dimensional N = 4 gauged supergravity with SO(2) D × SO(3) × SO(3) gauge group.This gauged supergravity can be constructed from N = 4 supergravity coupled to five vector multiplets and has been studied recently in [29] in which a number of N = 2 and N = 4 supersymmetric AdS 5 vacua and holographic RG flows interpolating among them have been found.Similar solutions within the framework of N = 4 gauged supergravity have also appeared in [27,28], see [30,31,32,33,34,35,36] for AdS black string solutions in higher dimensions.
Unlike the solutions given in [27,28], due to a richer structure of AdS 5 vacua in the gauged supergravity considered in the present paper, there is a large number of new and more interesting AdS 5 black string solutions.In particular, there are solutions connecting one, two or three AdS 5 critical points to AdS 3 × Σ 2 fixed points.A truncation of this SO(2) D × SO(3) × SO(3) gauged supergravity to two or three vector multiplets and SO(2) D × SO(3) gauge group can be embedded in eleven dimensions as shown in [37].Accordingly, both the AdS 5 vacua and AdS 5 black strings within this truncation could be uplifted to higher dimensions leading to new AdS 5 and AdS 3 solutions of string/M-theory.However, apart from the proof of the consistency of the truncation using the powerful framework of exceptional field theories in [37], no complete truncation ansatze in this case have appeared to date.
The paper is organized as follows.In section 2, we review the construction of five-dimensional N = 4 gauged supergravity with SO(2) D × SO(3) × SO (3) gauge group in the embedding tensor formalism.In sections 3 and 4, we give AdS 3 × Σ 2 geometries with SO(2) diag × SO (2) and SO (2) diag symmetries together with black string solutions interpolating between these geometries and supersymmetric AdS 5 vacua.We end the paper with some conclusions and comments in section 5.
Gaugings of N = 4 supergravity are characterized by the embedding tensor with components ξ M , ξ M N = ξ [M N ] and f M N P = f [M N P ] .These components determine the embedding of gauge groups in the global symmetry group SO(1, 1)×SO (5, n).We are only interested in gaugings with ξ M = 0 which admit supersymmetric AdS 5 vacua as shown in [40].Accordingly, we will set ξ M = 0 which leads to considerable simplification in various expressions.In particular, the quadratic constraints on the embedding tensor simply reduce to Furthermore, for ξ M = 0, the gauge group is embedded solely in SO(5, n) with the corresponding gauge generators in SO(5, n) fundamental representation given by ) being the SO(5, n) invariant tensor.The gauge covariant derivative reads with ∇ µ being a space-time covariant derivative (possibly) including SO(5) × SO(n) composite connection.
The bosonic Lagrangian of a general gauged N = 4 supergravity can be written as where e is the vielbein determinant.L top is the topological term whose explicit form can be found in [38].
The covariant gauge field strength tensors read with In the embedding tensor formalism, the two-form fields B µνM are introduced off-shell.These fields do not have kinetic terms and couple to vector fields via the topological term.It is useful to note the first-order field equations for these two-form fields with M is defined by for The scalar potential is given by where M M N is the inverse of M M N , and M M N P QR is obtained from by raising the indices with η M N .Supersymmetry transformations of fermionic fields are given by in which the fermion shift matrices are defined by is defined in terms of V M m and SO(5) gamma matrices Γ mi j as with Γ ij m = Ω ik Γ mk j .Similarly, the inverse V ij M can be written as We will use the following representation of SO(5) gamma matrices with σ i , i = 1, 2, 3, being the Pauli matrices.The covariant derivative on i is given by with the composite connection defined by In this paper, we will consider N = 4 gauged supergravity coupled to n = 5 vector multiplets with SO(2) D × SO(3) × SO(3) gauge group previously studied in [29].The corresponding embedding tensor is given by, see [29] for more detail, with the coupling constants g 1 , g 2 , h 1 and h 2 .It is useful to note that the first SO(3) ∼ SO(3) R ⊂ SO(5) R is a subgroup of the R-symmetry while the second SO(3) factor is a subgroup of SO(5) symmetry of the vector multiplets.
We end this section by giving an explicit parametrization of the scalar coset SO(5, 5)/SO(5) × SO (5).With SO(5, 5) non-compact generators given by the coset representative can be written as 3 Supersymmetric AdS 5 black strings with SO(2) diag × SO(2) symmetry We now look for supersymmetric AdS 5 black string solutions with a near horizon geometry given by AdS 3 × Σ 2 for Σ 2 = S 2 , H 2 .The metric ansatz is taken to be with , is the flat metric on the two-dimensional Minkowski space.Relevant components of the spin connection are given by Throughout the paper, r-derivatives are denoted by except for f κ (θ) = dfκ(θ) dθ .It is also useful to point out that the metric ansatz (29) leads to solutions interpolating between AdS 5 and AdS 3 × Σ 2 geometries.Near the asymptotic AdS 5 space, we have with L AdS 5 being the corresponding AdS 5 radius while the AdS 3 ×Σ 2 near horizon geometry corresponds to with the AdS 3 radius L AdS 3 .We first consider solutions obtained from a topological twist along Σ 2 by turning on SO(2) diag × SO(2) gauge fields of the form with a 0 , a 3 and a 6 being constants.The SO(2) diag is the diagonal subgroup of SO(2) D × SO(2) R generated by the gauge generators X 0 and From the embedding tensor (24) and the gauge field ansatz (34), we can verify that setting all the two-form fields to zero is a consistent truncation satisfying the two-form field equation (10).The corresponding field strength tensors are then given by in which we have used the relation f κ (θ) = −κf κ (θ).It should also be noted that the relation a 0 = h 1 g 2 a 3 implements the diagonal subgroup of SO(2) D × SO(2) R .Among the 25 scalar fields from the vector multiplets, there are 5 singlets under SO(2) diag × SO(2) corresponding to the following non-compact generators Accordingly, the coset representative can be written as As pointed out in [29], non-vanishing φ 3 and φ 5 break half of the supersymmetry with the corresponding Killing spinors given by 1,3 or 2,4 .
We are now in a position to perform a topological twist by canceling component ω θ φ of the spin connection.It turns out that this can be achieved only for g 2 = g 1 or ϕ 3 = ϕ 4 = 0.Both of these possibilities lead to equivalent results, so we will choose to set ϕ 3 = ϕ 4 = 0 in the following analysis.With this, the composite connection along the φ-direction is given by It should be noted that A 6 does not appear in the composite connection since this vector field gauges the SO(2) subgroup of the SO(3) ⊂ SO(5) symmetry of the vector multiplets under which the gravitini are not charged.
From the supersymmetry transformation δψ φi , the twist requires The identity (γ θ φ) 2 = −I 4 imposes the consistency condition This condition can be satisfied by setting g 1 g 2 = 0 or imposing a projector The existence of supersymmetric AdS 5 vacua requires non-vanishing g 1 [40].On the other hand, the SO(2) diag twist requires non-vanishing g 2 , see (34).Therefore, we need to impose the projector (41) on the Killing spinors.Explicitly, the two sign choices of this projector give 2,4 = 0 and 1,3 = 0, respectively.This is precisely in agreement with the N = 2 unbroken supersymmetry preserved by non-vanishing scalars φ 3 and φ 5 as noted before.For definiteness, we will choose the plus sign.
Imposing the projector (41), equation ( 39) leads to a twist condition and a projector

Supersymmetric AdS 5 vacua
We first look at supersymmetric AdS 5 vacua within the aforementioned truncation to SO(2) diag × SO(2) residual symmetry.For ϕ 3 = ϕ 4 = 0, the scalar potential is given by The superpotential is given in terms of the first (α 1 ) or third (α 3 ) eigenvalue of The superpotential admits two supersymmetric AdS 5 critical points.The first one is given by the trivial critical point at the origin of the scalar manifold SO(5, 5)/SO(5) × SO( 5) V 0 is the cosmological constant, and the AdS 5 radius is given by L = − 6 V 0 .We can also choose The second critical point is N = 2 supersymmetric and given by , We have taken g 2 > g 1 for simplicity.For φ 5 = 0, we recover the results of [41] and [29] with Similar to the result of [29], the gauge group is broken to SO(2 symmetry is enhanced to SO(3) ⊂ SO(5) due to the vanishing of φ 1 .We also note that for φ 3 = 0, we have which shows a symmetric role between φ 3 and φ 5 .Indeed, all values of φ 5 lead to physically equivalent N = 2 AdS 5 vacua with SO(2) diag generator given by different linear combinations of X 0 and X 3 .

Supersymmetric AdS 3 × Σ 2 fixed points
With the twist condition (42) and the projector (43), the supersymmetry transformations δψ φi and δψ θi lead to the same BPS equation as usually the case in performing topological twists.Setting 2 = 4 = 0, we arrive at the following BPS equations As in [29], we have also used the γ r-projector of the form for It can be verified that these equations imply the second-order field equations.
, φ 5 = φ * 5 = constant, It turns out that all of these fixed points only exist for κ = −1 leading to AdS 3 × H 2 geometries.For critical points 2 and 3, this is obviously seen by the reality of G and the twist condition (42).It should also be pointed out that at critical point 2, a 6 = 0 is required by consistency of equation (53) for φ 1 = 0.Although in this work, we are mainly interested in the cases of Σ 2 = S 2 and Σ 2 = H 2 , it could be useful to point out the possibility of obtaining black string solutions with AdS 3 × R 2 near horizon geometry.Since in this case, the Σ 2 = R 2 is flat, there is no need to perform a twist.A similar analysis leading to the twist condition (42) would require that a 3 = 0 which also implies a 0 = 0. Therefore, the possible solutions will not be charged under any subgroup of the SU (4) R ∼ SO(6) R R-symmetry.In this case, only A 6 gauge field corresponding to an SO(2) subgroup of the SO(3) flavor symmetry in the dual four-dimensional SCFTs has a non-trivial magnetic flux along Σ 2 = R 2 .Setting a 3 = 0 in the above construction of solutions with SO(2) diag × SO(2) twist, we have not found any AdS 3 × R 2 vacua from the resulting BPS equations.It is possible that different twists such as turning on SO(2 gauge fields could lead to AdS 3 × R 2 solutions.However, we will not further consider this type of solutions in the present paper. Before giving explicit solutions interpolating between supersymmetric AdS 5 vacua and these AdS 3 × H 2 fixed points, we first note the unbroken supersymmetry of the solutions.Due to the projectors ( 41), ( 43) and (56), the black string solutions preserve 16  2 3 = 2 supercharges.Using the relation among SO(1, 4) gamma matrices i = γ 0γ 1γ rγ θγ φ, we find the chirality matrix on the two-dimensional Minkowski space in which we have used the projector ( 43) and an explicit form of the γ r-projector (56), γ r i = ±(σ 2 ⊗ σ 3 ) i j j .We then see that the Killing spinors 1 and 3 have definite two-dimensional chiralities, so the flow solutions will preserve N = (2, 0) or N = (0, 2) Poincare supersymmetry in two dimensions.At the AdS 3 ×H 2 fixed points, the supersymmetry is enhanced to 4 supercharges since the γ r-projector is not necessary for constant scalars.This results in N = (2, 0) or N = (0, 2) superconformal symmetry in two dimensions.Accordingly, the aforementioned AdS 3 × H 2 fixed points are dual to two-dimensional N = (2, 0) or N = (0, 2) SCFTs.

Supersymmetric black string solutions
We now find solutions to the BPS equations that interpolate between supersymmetric AdS 5 critical points and AdS 3 × H 2 fixed points identified previously.We begin with a simple solution interpolating between the N = 2 AdS 5 vacuum to AdS 3 × H 2 fixed point 2. It should be noted that Σ, φ 1 and φ 3 take the same values at both of these critical points.We can then truncate these scalar fields out by setting them to the values at the critical points together with a 6 = 0.For κ = −1, the remaining BPS equations are simply given by F = (g 2 − g 1 ) 3 Accordingly, the flow from N = 2 AdS 5 vacuum to AdS 3 × H 2 critical point 2 is driven only by the metric function G(r).The above equations can be readily solved by

2
(64) in which we have removed an additive integration constant in F (r).The constant r 0 can also be removed by shifting the radial coordinate r.From this solution, we immediately see that as r → ∞, which gives locally asymptotically N = 2 AdS 5 critical point.On the other hand, for r → −∞, we find that the solution becomes AdS 3 × H 2 fixed point 2 For more general solutions with non-vanishing scalars, we need to solve the BPS equations numerically to find the solutions.We follow [29] to fix numerical values of various parameters by defining Examples of solutions for φ 1 = 0, a 6 = 0,  the second type of solutions interpolating between N = 4 and N = 2 AdS 5 vacua and AdS 3 × H 2 geometries in the IR.
We now consider solutions flowing to AdS 3 × H 2 critical point 1 with only φ 1 non-vanishing.For φ 3 = φ 5 = 0, the truncated BPS equations only admit the N = 4 AdS 5 critical point as an asymptotic geometry.Therefore, in this case, there are only solutions interpolating between this AdS 5 critical point and the AdS 3 × H 2 geometry in the IR.Examples of these solutions for different values of a 6 = 0.70, 0.75, 0.80 are given in figure 2.
Finally, we look for more complicated solutions flowing to AdS 3 × H 2 fixed point 3 with all scalars non-vanishing.Examples of solutions interpolating between N = 4 AdS 5 critical point and this AdS 3 × H 2 fixed point are given in figure 3. It should be noted that at the AdS 3 × H 2 fixed point, only the value of φ 3 is affected by the value of φ * 5 .In addition, the solutions in the figure indicate that apart from the solutions for φ 3 and φ 5 , the entire flow solutions for other fields are not affected by different values of φ * 5 at the AdS 3 × H 2 fixed point.All of these solutions describe black strings with a near horizon geometry of the form AdS 3 × H 2 in asymptotically locally AdS 5 spaces.Holographically, the solutions can also be considered as holographic RG flows from the four-  4 Supersymmetric AdS 5 black strings with SO(2) diag symmetry In this section, we repeat the same analysis for a smaller residual symmetry SO(2) diag that is the diagonal subgroup of SO(2) D × SO(2) R × SO(2) generated by X 0 , X 3 and X 6 .As pointed out in [29], there are nine singlet scalars under SO(2) diag corresponding to the non-compact generators The coset representative is then given by It turns out that the analysis in this case is much more complicated than the previous case of SO (2) diag ×SO(2) symmetry.To proceed further, we will consider a subtruncation of this sector with ϕ 1 = ϕ 2 = ϕ 3 = ϕ 4 = 0.However, consistency among the BPS equations and compatibility between the BPS equations and the field equations further require that φ 4 = φ 5 = 0.The resulting truncation is accordingly the same as that studied in [29] with only φ 1 , φ 2 and φ 3 non-vanishing.
The ansatz for the metric and (A 0 , A 3 , A 6 ) gauge fields are the same as in the previous section.To implement the SO(2) diag symmetry, in this case, we impose the following conditions The twist can be performed in the same way as in the previous section with the twist condition (42) and projectors ( 41) and (43).

Supersymmetric AdS 5 vacua
Since the scalar sector considered here is the same as in [29], the AdS 5 vacua are given by those identified in [29].We will only give the superpotential and supersymmetric AdS 5 critical points but refer to [29] for the explicit form of the scalar potential.The superpotential is given by which admits the following AdS 5 critical points: • I.The trivial critical point, at the origin of SO(5, 5)/SO(5) × SO (5), is given by .
As in the previous section, this critical point preserves the full SO(2) D × SO(3) × SO(3) gauge symmetry and N = 4 supersymmetry.
• II.Unlike the previous case of SO (2) diag ×SO(2) symmetry, there is another N = 4 AdS 5 critical point given by This critical point preserves SO(2) D × SO(3) diag symmetry.
• III.The next critical point is given by .
• IV.There is another N = 2 supersymmetric critical point given by (75) which is invariant under SO(2) diag .
As pointed out in [29], some of these critical points appear in pairs with some sign differences.Since critical points related by these sign changes are physically equivalent, in the above equations, we have chosen a particular sign choice for definiteness.

Supersymmetric AdS 3 × Σ 2 fixed points
With the same anlysis of supersymmetry transformations of fermionic fields, we obtain the BPS equations From these equations, we find the following AdS 3 × Σ 2 fixed points: , G = 1 2 ln 2 ii : iii : , iv : Among these fixed points, only ii leads to AdS 3 × S 2 geometry.All the remaining fixed points correspond to AdS 3 × H 2 solutions.

Supersymmetric black string solutions
We now look for supersymmetric black string solutions interpolating between AdS 5 and AdS 3 × Σ 2 geometries.We begin with solutions flowing to AdS  We now move to solutions involving AdS 3 × Σ 2 fixed point ii.Unlike other cases, the solutions in this case only exist for κ = 1.Some examples of solutions for φ 3 = 0 are given in figure 5.There is a solution flowing directly from N = 4 AdS 5 vacuum I to AdS 3 × S 2 fixed point ii (red line).There are also solutions that flow to AdS 3 × S 2 fixed point ii and approach arbitrarily close to N = 4 AdS 5 critical point II shown by green, purple and blue lines.Similarly, setting φ 2 = 0, we find solutions interpolating between AdS 5 critical point I and AdS 3 × H 2 fixed point iii with some examples of these solutions shown in figure 6.
Finally, we consider solutions that flow to AdS 3 × H 2 critical point iv.We first note that the values of scalar fields are the same as those for N = 2 AdS 5 critical point IV.By setting all scalar fields to the values at these critical points, we find a solution involving only F (r) and G(r) given by For more general solutions with scalars depending on r, we can find only numerical solutions.Examples of these solutions can be found in figure 7.In this case, the solutions are more interesting than those of the previous cases.There is a solution interpolating between AdS 5 critical point I to AdS 3 × H 2 fixed point iv shown by the purple line.On the other hand, there are solutions interpolating among AdS 5 critical points I, II and IV and AdS 3 × H 2 fixed point iv.Some of these solutions connects all these critical points within a single flow (pink line).
There are also solutions interpolating between two AdS 5 vacua and AdS 3 × H 2 fixed point iv as shown by the orange and cyan lines.The former begins at critical point I and flows to critical points IV and iv while the latter flows from critical point I to critical points II and iv.
As in the previous section, these solutions should also holographically describe various possible RG flows from N = 2 and N = 1 SCFTs in four dimensions to N = (2, 0) two-dimensional conformal fixed points in the IR.

Conclusions and discussions
In this paper, we have constructed a large number of supersymmetric AdS     superconformal symmetry in two dimensions.Some of the solutions can even be analytically obtained.Although most of the solutions describe black strings with AdS 3 ×H 2 near horizon geometry, we have also found one AdS 3 ×S 2 solution with SO(2) diag symmetry.The solutions could be of interest in microscopic counting of black string entropy along the line of [11,12,13,14].Holographically, the solutions also describe various RG flows from four-dimensional N = 1 and N = 2 SCFTs to two-dimensional N = (2, 0) SCFTs in the IR via twisted compactifications on Σ 2 .The solutions given here are expected to be useful in holographic study of strongly coupled N = 1, 2 SCFTs in four dimensions with topological twists as well.
It should be pointed out that the N = 2 AdS 5 vacuum with SO(2) diag × SO(3) symmetry, obtained from SO(2) diag × SO(2) sector, together with AdS 3 × H 2 fixed point 2 and related flow solutions can be embedded in N = 4 gauged supergravity with n = 2, 3 vector multiplets.The latter can be obtained from the gauged supergravity considered here by truncating out φ 1 and the gauge field A 6 µ resulting in N = 4 gauged supergravity with SO(2) D × SO(3) R gauge group.It has been shown in [37] that this gauged supergravity can be embedded in eleven dimensions.Therefore, within the above truncation, the solutions given here can be uplifted to M-theory.It is then of particular interest to construct the truncation ansatz from the result of [37] and uplift the black string solutions found here to M-theory.This would lead to a new holographic dual of N = (2, 0) SCFTs in two dimensions within string/M-theory context.
Moreover, it could be interesting to identify the N = 1 and N = 2 SCFTs dual to the AdS 5 vacua and the two-dimensional conformal fixed points in the IR as well as the associated RG flows in the field theory context.Partial results along this direction have been given in [41] in which the possible dual N = 1 and N = 2 SCFTs have been identified.It would be useful to extend these results to the case with topological twists.Furthermore, constructing similar solutions in the form of AdS 3 × Σ with Σ being a spindle or a half-spindle along the line of recent results in [42,43,44,45] is also worth considering.Finally, finding similar solutions within N = 4 gauged supergravity with gauge groups identified in [37] as embeddable in eleven dimensions would lead to new holographic solutions in string/M-theory framework.We leave all these and related issues for future work.