Supersymmetric AdS4 black holes from matter-coupled N=3,4 gauged supergravities

We study supersymmetric $AdS_4$ black holes in matter-coupled $N=3$ and $N=4$ gauged supergravities in four dimensions. In $N=3$ theory, we consider $N=3$ gauged supergravity coupled to three vector multiplets and $SO(3)\times SO(3)$ gauge group. The resulting gauged supergravity admits two $N=3$ supersymmetric $AdS_4$ vacua with $SO(3)\times SO(3)$ and $SO(3)$ symmetries. We find an $AdS_2\times H^2$ solution with $SO(2)\times SO(2)$ symmetry and an analytic solution interpolating between this geometry and the $SO(3)\times SO(3)$ symmetric $AdS_4$ vacuum. For $N=4$ gauged supergravity coupled to six vector multiplets with $SO(4)\times SO(4)$ gauge group, there exist four supersymmetric $AdS_4$ vacua with $SO(4)\times SO(4)$, $SO(4)\times SO(3)$, $SO(3)\times SO(4)$ and $SO(3)\times SO(3)$ symmetries. We find a number of $AdS_2\times S^2$ and $AdS_2\times H^2$ geometries together with the solutions interpolating between these geometries and all, but the $SO(3)\times SO(3)$, $AdS_4$ vacua. These solutions provide a new class of $AdS_4$ black holes with spherical and hyperbolic horizons dual to holographic RG flows across dimensions from $N=3,4$ SCFTs in three dimensions to superconformal quantum mechanics within the framework of four-dimensional gauged supergravity.


Introduction
String/M-theory has provided a number of insights to various aspects of quantum gravity for many decades. In particular, a resolution for a long-standing problem of black hole entropy has been proposed in [1]. After this pioneering work, many other papers followed and clarified the issues of microscopic entropy of asymptotically flat black holes. For asymptotically AdS 4 black holes, a concrete result on the corresponding microscopic entropy, using AdS/CFT correspondence [2,3,4], has appeared recently in [5,6,7], see also [8,9,10,11,12].
On the gravity side, an important ingredient along this line is AdS 4 black hole solutions interpolating between asymptotic AdS 4 and AdS 2 × Σ 2 spaces with Σ 2 being a Riemann surface. The latter describes the geometry of the black hole horizon with the values of scalars determined by the attractor mechanism. These solutions holographically describe RG flows across dimensions from threedimensional SCFTs, dual to the AdS 4 vacua, to superconformal quantum mechanics, dual to the AdS 2 factor of the horizons. The latter is obtained from twisted compactifications of the former which play an important role in computing Bekenstein-Hawking entropy of the black holes via twisted indices.
In this paper, we are interested in supersymmetric AdS 4 black holes with the horizon geometry AdS 2 × S 2 and AdS 2 × H 2 with S 2 and H 2 being a twosphere and a two-dimensional hyperbolic space, respectively. We will work in matter-coupled N = 3 and N = 4 gauged supergravities. This type of solutions has been extensively studied in N = 2 gauged supergravity for a long time [13,14,15,16,17,18,19], see also [20] for some results in N = 8 gauged supergravity. Similar studies in other gauged supergravities have appeared only recently in [21,22,23,24]. In particular, a study of AdS 2 × Σ 2 solutions in N = 3 with only magnetic charges has been initiated in [21]. We will extend this result by performing a more systematic analysis and including a possible dyonic generalization. We will consider a particular case of N = 3 gauged supergravity coupled to three vector multiplets with a compact SO(3) × SO(3) gauge group. We will see that only one magnetic AdS 2 × H 2 solution with SO(2) × SO (2) symmetry exists. This is very similar to solutions in N = 5 and N = 6 gauged supergravities given in [23] and [24].
For N = 4 case, we will consider N = 4 gauged supergravity coupled to six vector multiplets with SO(4) × SO(4) gauge group. Unlike the N = 3 theory with a purely electric gauging, any N = 4 supergravity that admits supersymmetric AdS 4 vacua must be dyonically gauged [25]. In this case, apart from an AdS 2 × H 2 solution similar to N = 3, 5, 6 gauged supergravities, there exist a number of supersymmetric AdS 2 × S 2 and AdS 2 × H 2 solutions. It should also be pointed out that some AdS 2 × Σ 2 solutions in N = 4 gauged supergravity obtained from a truncation of eleven-dimensional supergravity have also been found in [22]. However, in that case, the gauge group is of non-semisimple form, and the resulting BPS equations are highly complicated. In the present work, we provide a number of much simpler examples of supersymmetric AdS 4 black holes in N = 4 gauged supergravity. In particular, the two-form fields required by the consistency of incorporating magnetic gauge fields can be truncated out in the present case.
The paper is organized as follows. In section 2, we will review the structure of N = 3 gauged supergravity after translating the original construction in group manifold approach to the usual formulae in space-time. This is followed by a general analysis of relevant BPS equations for finding supersymmetric AdS 4 black hole solutions. An AdS 2 × H 2 solution with SO(2) × SO(2) symmetry together with the full flow solution interpolating between this fixed point and the supersymmetric AdS 4 vacuum with SO(3) × SO(3) symmetry are also given. Similar analysis is then performed in section 3 in which we will find a number of AdS 2 × S 2 and AdS 2 × H 2 fixed points and solutions interpolating between them and supersymmetric AdS 4 vacua with various unbroken symmetries in N = 4 gauged supergravity. We end the paper by giving conclusions and comments on the results in section 4.
2 AdS 4 black holes from N = 3 gauged supergravity In this section, we consider matter-coupled N = 3 gauged supergravity and possible supersymmetric AdS 4 black holes. We begin with a review of N = 3 gauged supergravity and the analysis of relevant BPS equations. These are followed by the explicit solutions at the end of the section.

Matter-coupled N = 3 gauged supergravity
We now give a description of N = 3 gauged supergravity coupled to n vector multiplets. This has been constructed by the geometric group manifold approach in [26], see also [27,28]. However, the final form of the space-time Lagrangian has not been given, and the supersymmetry transformations of fermions have been given in a rather implicit form. We will first collect all these ingredients and specify to the case of n = 3 vector multiplets later on. The interested reader can find a more detailed construction and some discussions on the structure of the scalar manifold and electric-magnetic duality in [26]. We will mostly follow the notations of [26] but in a mostly plus signature for the space-time metric and a slightly different convention for the gauge fields. For N = 3 supersymmetry in four dimensions, there are two types of supermultiplets, the gravity and vector multiplets. The former consists of the following component fields (e a µ , ψ µA , A A µ , χ). e a µ is the graviton, and ψ µA are three gravitini. Space-time and tangent space indices will be denoted by µ, ν, . . . and a, b, . . ., respectively. The gravity multiplet also contains three vector fields A A µ with indices A, B, . . . = 1, 2, 3 denoting the fundamental representation of the SU (3) R part of the full SU (3) R × U (1) R R-symmetry. There is also an SU (3) R singlet spinor field χ. N = 3 supersymmetry allows the gravity multiplet to couple to an arbitrary number of vector multiplets, the only matter fields in this case. The component fields in a vector multiplet are given by the following field content consisting of a vector field A µ , four spinor fields λ and λ A which are respectively singlet and triplet of SU (3) R , and three complex scalars z A in the fundamental of SU (3) R . We will use indices i, j, . . . = 1, . . . , n to label each vector multiplet.
The fermionic fields are subject to the chirality projection conditions These also imply ψ A µ = −γ 5 ψ A µ and λ A = −γ 5 λ A for the corresponding conjugate spinors.
In the matter-coupled supergravity with n vector multiplets, there are 3n complex scalar fields z i A parametrizing the coset space SU (3, n)/SU (3)×SU (n)× U (1). These scalars are conveniently described by the coset representative L Σ Λ . The coset representative transforms under the global G = SU (3, n) and the local H = SU (3) × SU (n) × U (1) symmetries by left and right multiplications, respectively. The SU (3, n) indices Λ, Σ, . . . will take values 1, . . . , n + 3. On the other hand, it is convenient to split the SU (3) × SU (n) × U (1) indices Λ, Σ, . . . as (A, i). We can then write the coset representative as The n + 3 vector fields from both the gravity and vector multiplets are . These are called electric vector fields that appear in the Lagrangian with the usual Yang-Mills (YM) kinetic terms. Accompanied by the corresponding magnetic dual A Λµ , these vector fields transform in the fundamental representation n + 3 of the global symmetry group SU (3, n), also called the duality group.
For the gaugings of the matter-coupled N = 3 supergravity, we will follow the original result of [26] since the complete modern approach using the embedding tensor has not been worked out so far. For general gaugings obtained from the embedding tensor formalism, both electric and magnetic gauge fields can participate in the gaugings. The construction of [26], called electric gaugings, with only electric vector fields becoming the gauge fields results in gauge groups that only account for a smaller class of all possible gaugings. All gauge groups considered in [26] are subgroups of SO(3, n) which is the electric subgroup of the full global symmetry SU (3, n).
After gauging a particular subgroup G 0 of SO(3, n) ⊂ SU (3, n), the corresponding non-abelian gauge field strengths are given by where f ΛΣ Γ denote the structure constants of the gauge group. The gauge gener- Indices Λ, Σ, . . . can be raised and lowered by the SU (3, n) invariant tensor which will become the Killing form of the gauge group G 0 . In order for the gaugings to be consistent with supersymmetry, the structure constants f ΛΣΓ need to satisfy the following constraint which is equivalent to the linear constraint in the embedding tensor formalism. Some examples of possible gauge groups are SO(3) × H n , SO(3, 1) × H n−3 and SO(2, 2) × H n−3 with H n being an n-dimensional compact subgroup of SO(n) ⊂ SU (n). These gaugings together with possible supersymmetric AdS 4 vacua and domain walls have already been studied in [33].
With the fermion mass terms and the scalar potential included as required by supersymmetry, the bosonic Lagrangian of the N = 3 gauged supergravity can be written as This Lagrangian is obtained from translating the first-order Lagrangian in the geometric group manifold approach given in [26] to the usual space-time Lagrangian.
We have also multiplied the whole Lagrangian by a factor of 3 resulting in a factor of 3 in the scalar potential given below as compared to that given in [26]. The self-dual and antiself-dual field strengths are defined by which satisfy the following relations To write down the explicit form of the scalar matrix a ΛΣ in terms of the coset representative, we first identify various components of the coset representative as The symmetric matrix a ΛΣ can be written as in which the matrices f Λ Σ = (L Λ A , (L Λ i ) * ) and h ΛΣ = −(Jf J) ΛΣ are given explicitly by The scalar kinetic terms are written in terms of the vielbein on the SU (3, n)/SU (3) × SU (n) × U (1) obtained from the Maurer-Cartan one-form via the components We also note that the upper and lower indices of SU (3) and SU (n) are related by complex conjugation. Since L Λ Σ is an element of SU (3, n), the inverse (L −1 ) Λ Σ satisfies the following relation The composite connections Q B A , Q j i and Q for the SU (3)×SU (n)×U (1) local symmetry are given by with Q A A = Q i i = 0. The scalar potential is given by with C P = −C M P M . Various components of the fermion-shift matrices are defined in terms of the "boosted" structure constants Finally, the fermionic supersymmetry transformations obtained from the rheonomic parametrization of the fermionic curvatures are given by 1 The covariant derivative for A is defined by The field strengths appearing in the supersymmetry transformations are given by where M ij and M AB are respectively inverse matrices of

BPS equations for supersymmetric AdS 4 black holes
We now look at the BPS equations for supersymmetric AdS 4 black holes with the near horizon geometry given by AdS 2 × Σ 2 . The metric ansatz is taken to be with F (θ) defined by for Σ 2 = S 2 and Σ 2 = H 2 , respectively. The functions f (r) and h(r) together with all other non-vanishing fields only depend on the radial coordinate r. With the following choice of vielbein it is straightforward to compute non-vanishing components of the spin connection 1 We also note an additional factor of 1 2 in the gauge field strengths due to different conventions for differential forms, namely For clarity, we have used the values of flat indices as a, b, . . . , = (t,r,θ,φ).
In the present paper, we are interested in a simple N = 3 gauged supergravity coupled to n = 3 vector multiplets with a compact gauge group SO(3) × SO(3). The non-vanishing components of f ΛΣΓ are given by We also recall that the SO(3) × SO(3) gauge group is electrically gauged with the corresponding gauge fields being the vector fields appearing in the ungauged Lagrangian with YM kinetic terms. To avoid confusion, we will call the first SO(3) factor SO(3) R since this factor is embedded in SU (3) R R-symmetry.
To preserve some amount of supersymmetry, we implement a topological twist by turning an In addition, we can also turn on an SO(2) ⊂ SO(3) gauge field from the second SO(3) factor. We will choose these gauge fields to be A 3 µ and A 6 µ with the following ansatz for F (θ) = dF (θ) dθ . The corresponding field strengths are given by Throughout the paper, we will use to denote a derivative with respect to the radial coordinate r with an exception for F (θ) = dF (θ) dθ . In this equation, we have also introduced a parameter κ via the relation F (θ) = −κF (θ) with κ = 1 and κ = −1 for Σ 2 = S 2 and Σ 2 = H 2 , respectively. Imposing the Bianchi's identity DF Λ = 0 implies p Λ = 0, so p Λ are constant and will be identified with magnetic charges.
It is useful to recall the definition of electric and magnetic charges given by with G Λ = δS δF Λ . To further fix the ansatz for the gauge fields, we consider the Lagrangian for the gauge fields in which we have rewritten the relevant terms in the Lagrangian (7) in differential form language. We have also used the following definition R ΛΣ = Re a ΛΣ and I ΛΣ = Im a ΛΣ .
From the above Lagrangian, we find which, together with the above definition of (q Λ , p Λ ) and F Λ θφ = κp Λ F (θ), leads to We have written the inverse of R ΛΣ as R ΛΣ . For later convenience, we also note the Maxwell equations obtained from the Lagrangian (7) This can be rewritten in form language as with P Ai = P Ai µ dx µ . It should also be emphasized that the left-hand side is related to a radial derivative of electric charges via the definition in (35). Therefore, in general, electric charges are not conserved if the YM currents are non-vanishing as also pointed out in [17].
We are now in a position to perform the analysis of BPS equations. The analysis is closely parallel to that in N = 2 gauged supergravity given in [14] and [17]. We will work in Majorana representation with all γ a real but γ 5 = iγtγrγθγφ purely imaginary. In this representation, the two chiral components A and A of the Killing spinors are related to each other by complex conjugation. In addition, in all of the solutions considered in this work, we assume that the Killing spinors depend only on the radial coordinate r. We are only interested in solutions with SO(2) × SO(2) and SO(2) diag symmetries, but in this section, we will consider the general structure of the BPS equations.
We begin with the BPS equation from the variation δψφ A given by The matrix S AB is symmetric and can be diagonalized. The corresponding eigenvalues will lead to the superpotential W in terms of which the scalar potential can be written. We then write, without summation on A, in which W A denote eigenvalues of S AB . It is also useful to define the central charge matrix as We now impose the following projector and rewrite equation (42) as We have used the explicit form of Ωφ which is valid for both cases we are interested in. We now notice that only the terms in the second bracket of equation (46) depend on θ. Therefore, these terms must cancel against each other, and using the gauge field ansatz (33), we find that Since only p 3 is non-vanishing, we find that the supersymmetry corresponding to 3 must be broken. Imposing the twist condition and writting AB3 = ÂB forÂ,B = 1, 2 and 12 = 1, we obtain the following projector In this analysis, we have written A = ( Â , 3 ). We also remark that indiceŝ A,B, . . . of ÂB and ÂB are simply raised and lowered by the Kronecker delta δÂB and δÂB.
In general, WÂ for a particular value ofÂ gives the superpotential corresponding to the eigenvalue of SÂB along the directions of the Killing spinors Â . We will simply denote this eigenvalue by W. Moreover, it turns out that in the cases we will consider, only G 3 µν is non-vanishing. We then find that in which we have defined a complex number Z sometimes called the "central charge" as With all these, we finally obtain the BPS equation from δψφ A h e iΛ = W + Z which implies Using all of the results previously obtained, we can perform a similar analysis for δψθ A . This results, as expected, in the same BPS equations given in (54). We now move to the variation δψtÂ of the form with We then impose another projector It should be noted that this is not an independent projector since it is implied by the γr and γθφ projectors given in (45) and (49) by the relation We note here that the central charge matrix can also be written as With all the previous results, we can write equation (55) as and The second equation fixes the form of A 3 t . Finally, we consider the variation δψr A which gives In all the cases we will consider, it turns out that Q r = 0 and Q rÂB = 0. Using δψtÂ = 0 equation, we can rewrite this equation as which gives with (0) A being r-independent spinors subject to the projectors Consistency with the projector (45) leads to a flow equation for the phase Λ Since all scalars depend only on the radial coordinate r, the BPS equations obtained from δχ, δλ i and δλ A i only involve γr. By using the projector (45) and phase factor in (54) in these variations, we eventually obtain flow equations for scalars. Before giving the solutions, we end this section with the conditions for the near horizon geometry meaning that the function h and all scalars are constant, and f is linear in r in this limit. We will also choose an upper sign choice in (54) for definiteness.

Solutions with SO(2) × SO(2) symmetry
We now consider supersymmetric solutions to the BPS equations with the general structure given in the previous section. We begin with explicit parametrization of the SU With the structure constants given in (32), the SO(3) R × SO(3) gauge generators are given by The residual SO(2) × SO(2) symmetry is generated by T (1) 3 and T 3 . There are two singlet scalars corresponding to the following SU (3, 3) non-compact generatorsŶ The coset representative can be written as In this case, the YM currents vanish, so the electric charges are constant. The scalar potential is given by This potential admits a unique N = 3 supersymmetric vacuum at φ 1 = φ 2 = 0 with the cosmological constant V 0 = − 3 2 g 2 1 . The AdS 4 radius is given by the relation in which we have taken g 1 > 0 for convenience. We also note that truncating all vector multiplets out gives rise to pure N = 3 gauged supergravity with SO(3) R gauge group and cosmological constant − 3 2 g 2 1 constructed in [29] and [30], see also a more recent result [31] in which pure N = 3 gauged supergravity is embedded in massive type IIA theory.
The matrix S AB is given by in which W 1 and W 2 are given by It turns out that only W 2 gives the superpotential in terms of which the scalar potential (72) can be written as, see more detail in [33], In this case, the supersymmetry associated with 1,2 , which are relevant to the present work, is broken. For φ 2 = 0, W 1 can give rise to the superpotential leading to unbroken supersymmetry along 1,2 , and in this case, W 1 and W 2 are equal. We then set φ 2 = 0 in the following analysis. We will also write W = W 1 = W 2 and φ = φ 1 . In addition, it is worth noting that setting pseudo-scalars, corresponding to imaginary parts of the complex scalars z A i , to zero always gives I ΛΣ = 0. This implies that the components F Λ tr are given only in terms of electric charges and vanish for purely magnetic solutions.
With 3 = 0, we find that δχ = 0 and δλ i = 0 identically. By using the coset representative (71) with φ 2 = 0, we find a consistent BPS equation for φ from δλ A i provided that one of these two conditions is satified q 3 = q 6 = 0 or The first one corresponds to a purely magnetic case while the second one is a dyonic case with only q 3 and p 3 non-vanishing. Setting q 3 = q 6 = 0 and using the BPS equations given in the previous section, we find the following set of BPS equations We note that both W and Z are real giving rise to e iΛ = ±1. The existence of AdS 2 × Σ 2 fixed points requires p 6 = 0. In this case, we find the fixed point given by for a constant φ 0 . For real h and 2g 1 p 3 = 1 > 0, we need to take κ = −1, so this is an AdS 2 × H 2 fixed point. For p 6 = q 6 = 0, we find leading to the BPS equations together withq which fixes the time component of the gauge field ansatz. We also note that upon using the BPS equations for f , h and φ , we find in agreement with the gauge field ansatz given in (39). The existence of AdS 2 × Σ 2 fixed points requires q 3 = 0. This can be clearly seen from the condition h = 0. With q 3 = 0, the AdS 2 × Σ 2 fixed point is just the AdS 2 × H 2 vacuum given in (81). We then find that all supersymmetric black hole solutions will be magnetically charged without any dyonic generalization.
We now look for a solution interpolating between the supersymmetric AdS 4 vacuum and this AdS 2 × H 2 critical point. To find the relevant solution, we can further set p 6 = 0 and q 3 = 0 in the two sets of the BPS equations. In this case, the two sets lead to the same BPS equations which we repeat here for convenience These equations are very similar to those given in N = 5 and N = 6 gauged supergravities studied in [23] and [24]. By a similar analysis, we can obtain an analytic solution with the new radial coordinate ρ defined by dρ dr = e −φ . As φ ∼ 0, we find that the solution becomes which is an asymptotically locally AdS 4 preserving the full N = 3 supersymmetry. On the other hand, for φ ∼ φ 0 with the solution approaches the AdS 2 × H 2 fixed point with 4κp 3 −g 1 . We end this section by a comment on the solution of pure N = 3 gauged supergravity in which we set φ = 0 for the entire solution. This simply gives the following solution for a constant r 0 . This solution can be embedded in massive type IIA theory via S 6 truncation given in [31]. Alternatively, this solution can also be embedded in eleven dimensions using a consistent truncation on a trisasakian manifold given in [32].

Solutions with SO(2) diag symmetry
We now consider solutions with SO(2) diag symmetry generated by T 3 . There are six singlet scalars corresponding to the following non-compact genera- The coset representative is given by In this case, the scalar potential turns out to be highly complicated. We refrain from giving its explicit form here, but it is useful to note that there are two supersymmetric AdS 4 vacua, see more detail in [33]. The first one is the N = 3 supersymmetric AdS 4 vacuum with all scalars vanishing and the full SO (3)×SO (3) gauge group unbroken. This is the same as the AdS 4 critical point mentioned in the previous section. The second one is another N = 3 AdS 4 critical point with with all other scalars vanishing. We now repeat the same analysis as in the previous case with an additional condition g 2 A 6 = g 1 A 3 implementing the SO(2) diag subgroup. This condition results in the same component Qφ A B as in the SO(2) × SO(2) case, so the twist can be performed by the same procedure. We will not repeat all the details here to avoid repetition.
As in the previous case, it turns out that all pseudo-scalars must be truncated out in order to preserve supersymmetry along 1 and 2 . Therefore, we need to set φ 2 = φ 4 = φ 6 = 0. Consistency for the scalar equations also requires all electric charges to vanish resulting in a real phase e iΛ = ±1. We will accordingly set q Λ = 0 and obtain the following BPS equations We also note that these equations can be written more compactly as For AdS 2 × Σ 2 fixed points to exist, we immediately see from φ 3 and φ 5 equations that there are two possibilities; φ 3 = φ 5 = 0 or φ 1 = 1 2 ln g 2 −g 1 g 2 +g 1 . However, both of these choices do not lead to any AdS 2 × Σ 2 fixed point, so there are no supersymmetric AdS 4 black holes with SO(2) diag symmetry.
At this point, it should be noted that similar BPS equations have been considered in [21] with more vector multiplets (n = 8), and a number of AdS 2 ×Σ 2 fixed points have been given. A truncation of that results to three vector multiplets can be performed resulting in the BPS equations given above. It is worth pointing out here that there is a sign error in the BPS equations considered in [21] regarding to the contribution of the gauge fields to the supersymmetry transformations. The corresponding equations from the present analysis are correct and compatible with the second-order field equations. Therefore, the AdS 2 × Σ 2 fixed points with SO(2) diag × SO(2) symmetry found in [21] do not exist.
3 AdS 4 black holes from N = 4 gauged supergravity In this section, we repeat the same analysis as in the previous section for mattercoupled N = 4 gauged supergravity. Unlike the N = 3 gauged supergravity considered in the previous section, gaugings of N = 4 supergravity that can give rise to supersymmetric AdS 4 vacua need to be dyonic, involving both electric and magnetic vector fields. However, there always exists a symplectic frame in which the resulting gaugings are purely electric. As in the previous section, we will begin with a review of N = 4 gauged supergravity coupled to n vector multiplets.

Matter-coupled N = 4 gauged supergravity
Unlike the N = 3 gauged supergravity, N = 4 gauged supergravity has completely been constructed in the embedding tensor formalism in [34]. We will mainly follow the construction and notation used in [34]. Similar to the N = 3 theory, N = 4 supersymmetry in four dimensions only allows for the graviton and vector multiplets. Unlike N = 3 supersymmetry, the graviton multiplet in N = 4 supersymmetry does contain scalars with the full field content given by The component fields 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 parametrizing the SL(2, R)/SO(2) coset. In this case, indices m, n = 1, . . . , 6 and i, j = 1, 2, 3, 4 respectively describe the vector and chiral spinor representations of the SO(6) R ∼ SU (4) R R-symmetry. The former is equivalent to a second-rank anti-symmetric tensor representation of SU (4) R . Furthermore, in this section, we denote flat space-time indices byμ,ν, . . . to avoid confusion with indices labeling the vector multiplets to be introduced later.
As in the N = 3 theory, the supergravity multiplet can couple to an arbitrary number n of vector multiplets. Each vector multiplet will be labeled by indices a, b = 1, . . . , n and contain the following field content corresponding to vector fields A a µ , gaugini λ ia and scalars φ ma . The 6n scalar fields can be described by SO(6, n)/SO(6) × SO(n) coset. We also note the wellknown fact that the field contents of the vector multiplet in N = 3 and N = 4 supersymmetries are the same.
All fermionic fields and supersymmetry parameters that transform in the fundamental representation of SU (4) R R-symmetry are subject to the chirality projections Similarly, the conjugate fields transforming in the anti-fundamental representation of SU (4) R satisfy The most general gaugings of the matter-coupled N = 4 supergravity can be efficiently described by the embedding tensor Θ. There are two components of the embedding tensor ξ αM and f αM N P with α = (+, −) and M, N = (m, a) = 1, . . . , n + 6 denoting respectively fundamental representations of SL(2, R) × SO(6, n) global symmetry. The electric vector fields A M + = (A m µ , A a µ ) together with their magnetic dual A M − , collectively denoted by A M α , form a doublet of SL(2, R). The existence of AdS 4 vacua requires ξ αM = 0 [25], so we will consider gaugings with only f αM N P non-vanishing and set ξ αM to zero from now on.
The embedding tensor implements the minimal coupling to various fields via the covariant derivative where ∇ µ is the space-time covariant derivative including (possibly) the spin connections. t M N denote SO(6, n) generators which can be chosen as with η M N = diag(−1, −1, −1, −1, −, 1−, 1, 1, 1, . . . , 1) being the SO(6, n) invariant tensor. The gauge coupling constant g can also be absorbed in the definition of the embedding tensor f αM N P . In addition to ξ αM = 0, the existence of AdS 4 vacua requires the gaugings to be dyonic involving both electric and magnetic vector fields. In this case, both A M + and A M − enter the Lagrangian, and f αM N P with α = ± are nonvanishing. Consistency requires the presence of two-form fields when magnetic vector fields are included. In the case of ξ αM = 0, the two-forms transform as an anti-symmetric tensor under SO(6, n) and will be denoted by The two-forms are also needed to define covariant gauge field strengths given by In particular, for non-vanishing f −M N P the electric field strengths H M + acquire a contribution from the two-form fields.
The scalar coset manifold SL(2, R)/SO(2) in the graviton multiplet can be described by a coset representative or equivalently by a symmetric matrix We also note the relation Im(V α V * β ) = αβ . The complex scalar τ can in turn be written in terms of the dilaton φ and the axion χ as For the SO(6, n)/SO(6) × SO(n) coset from vector multiplets, we introduce the coset representative V A M transforming by left and right multiplications under SO(6, n) and SO(6) × SO(n), respectively. The SO(6) × SO(n) index will be split as A = (m, a) according to which the coset representative can be written as Being an element of SO(6, n), the matrix V A M satisfies the relation The SO(6, n)/SO(6) × SO(n) coset can also be parametrized in terms of a symmetric matrix defined by with a manifest SO(6) × SO(n) invariance. The bosonic Lagrangian of the N = 4 gauged supergravity for ξ αM = 0 is given by where e is the vielbein determinant. The scalar potential is given by where M M N is the inverse of M M N , and M M N P QRS is defined by with indices raised by η M N . The covariant derivative of M M N is defined by The magnetic vectors and two-form fields do not have kinetic terms. They are auxiliary fields and enter the Lagrangian through topological terms. The corresponding field equations give rise to the duality relation between two-forms and scalars and the electric-magnetic duality between A M + and A M − , respectively. The field equations resulting from varying the Lagrangian with respect to A M ± µ and B M N µν are given by written in differential form language for computational convenience. By substituting H M − from (126) in (124), we obtain the usual Yang-Mills equations for H M + while equation (125) simply gives the relation between the Hodge dual of the three-form field strengths and the scalars due to the usual Bianchi identity of the gauge field strengths defined by The supersymmetry transformations of fermionic fields are given by with the fermion shift matrices defined by and similarly for its inverse We note that G ij m satisfy the relations We will choose the explicit form of these matrices as follows The covariant derivative of i is given by Finally, it should be noted that the scalar potential can be written in terms of A 1 and A 2 tensors as which is usually referred to as supersymmetric Ward's identity. We also recall that upper and lower i, j, . . . indices are related by complex conjugation. We end this section by giving some relations which are very useful in deriving the BPS equations in subsequent analysis. With the explicit form of V α given in (114) and equation (126), it is straightforward to derive the following identities in which we have used the following relations for the SO(6, n) coset representative [35] It should be noted that these relations are slightly different from those given in [34] due to a different convention on V α in terms of the scalar τ namely V α used in this paper satisfies V + /V − = τ while that used in [34] gives V + /V − = τ * .

Solutions with SO(2)×SO(2)×SO(2)×SO(2) symmetry
In this paper, we are interested in N = 4 gauged supergravity with n = 6 vector multiplets and SO(4 The corresponding embedding tensor takes the following form [36] f +mnp = g 1 mnp , f +âbĉ =g 1 âbĉ , f −mñp = g 2 mñp , f −ãbc =g 2 ãbc . We We now consider solutions preserving SO(2) × SO(2) × SO(2) × SO(2) symmetry. To proceed further, we first give an explicit parametrization of the SO(6, 6)/SO(6) × SO(6) coset. The scalar sector of SO(2) × SO(2) × SO(2) × SO(2) singlets have already been studied recently in [37]. We will mostly take various results from [37] in which more details can be found. By using SO(6, 6) generators in the fundamental representation of the form given in (112), we can identify the SO(6, 6) non-compact generators as There are four SO(2)×SO(2)×SO(2)×SO(2) singlet scalars from the SO(6, 6)/SO(6)× SO(6) coset. With the SO(2) × SO(2) × SO(2) × SO(2) generators chosen to be X +3 , X −6 , X +9 and X −12 , the non-compact generators corresponding to these singlets are given by 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, there are six scalars in the SO(2) × SO(2) × SO(2) × SO(2) sector. The scalar potential for these singlet scalars is given by which admits a unique AdS 4 critical point at with the cosmological constant and AdS 4 radius given by This AdS 4 vacuum preserves N = 4 supersymmetry and the full SO(4) × SO(4) symmetry. We can also choose g 2 = g 1 = g, by shifting the dilaton, to make the dilaton vanish at this critical point. Holographic RG flows and Janus solutions in this sector have been extensively studied in [37]. In the present work, we look for supersymmetric AdS 4 black holes with the horizons of AdS 2 × Σ 2 geometry. The analysis is parallel to the N = 3 case considered in the previous section with some modifications to incorporate the magnetic gauge fields. Similar analyses can be found in [18,19,38] and [22] in the contexts of N = 2 and N = 4 gauged supergravities, respectively. We will closely follow the procedure in [22].
We first consider the ansatz for SO(2) × SO(2) × SO(2) × SO(2) gauge fields of the form We also note that the gauge fields participating in the SO(4) × SO(4) gauging are given by A 3+ , A 6− , A 9+ and A 12− while the above ansatz includes all of their electric-magnetic duals. The ansatz for relevant two-form fields is given by The metric ansatz is still given by (28). In addition, to avoid some confusion and make various expressions less cumbersome, we will denote the magnetic charges with a subscript, p M = (p 3 , p 6 , p 9 , p 12 ).
With the embedding tensor (142), it is straightforward to compute the covariant gauge field strengths In this SO(2) × SO(2) × SO(2) × SO(2) sector, it turns out that all components of YM current are zero Equations (124) and (125) then imply that DH M ± = 0. Therefore, we find that all the fields b i (r) and electric charges e i are constant.
As pointed out in [37], supersymmetric solutions with SO(2) × SO(2) × SO(2) × SO(2) symmetry can arise from two possibilities, χ = φ 2 = φ 3 = 0 or χ = φ 1 = φ 4 = 0. For definiteness, we will choose the first possibility. Choosing the second one results in relabeling the scalars. With Re τ = χ = 0, equations (126) gives All these relations fix the ansatz for the H M α 0r components of the field strengths in terms of scalars and various charges.
We now consider topological twists along Σ 2 . The scalar coset representative (144) gives the composite connection of the form with σ a , a = 1, 2, 3, are usual Pauli matrices. To perform a twist, we consider relevant terms in the variation δψ iφ of the form There are a few possibilities to satisfy this condition. These are given by the following two main categories: • N = 4 twists: By setting either p 3 = 0 or e 6 = 0, all four i can be nonvanishing. These two choices lead to the following twist conditions and projectors e 6 = 0; We will refer to these two cases as N = 4 twists which have a similar structure to the N = 3 theory.
• N = 2 twists: By using the relation we can rewrite the condition (155) as This can be solved by imposing the following conditions The last projector simply sets 2 = 3 = 0 reducing half of the original supersymmetry. Accordingly, we will call this case N = 2 twists.
We also note that the situation is very similar to AdS 5 black strings in fivedimensional N = 4 gauged supergravity considered in [39]. In addition, the two possibilities of N = 4 twists correspond to the H-twist and C-twist of the dual N = 4 SCFT in three dimensions considered in [40]. By a similar analysis performed in the N = 3 theory, we find a general structure of the BPS equations given by together with an algebraic constraint In these equations, W is the superpotential obtained from the eigenvalue of the A ij 1 tensor along the Killing spinors, and Z is the central charge as in the previous section. We have also imposed the following projector Using this projector in the supersymmetry transformations δχ i and δλ i a leads to the BPS equations for scalars in the gravity and vector multiplets, respectively.

Solutions with N = 4 twists
We begin with the case of N = 4 twist by A 3+ . In addition to setting e 6 = 0, unbroken N = 4 supersymmetry also requires b 6 = b 12 = e 12 = p 6 = p 12 = 0 . (164) Moreover, consistency of the scalar equations imposes further conditions of the form All these lead to the following set of consistent BPS equations However, there do not exist any AdS 2 × Σ 2 fixed points in these equations. We then look at the case of N = 4 twist by A 6− in which consistency similarly requires the following conditions b 3 = b 9 = e 3 = e 9 = p 9 = b 6 = b 12 = p 6 = p 12 = 0 . (171) The BPS equations are given by g 1 cosh φ 1 + e φ g 2 cosh φ 4 + κe −2h (e 6 cosh φ 4 + e 12 sinh φ 4 ) , (173) which do not admit any AdS 2 × Σ 2 fixed points as in the case of A 3+ twist.

Solutions with N = 2 twists
We now move to a more interesting and more complicated case of N = 2 twists by both A 3+ and A 6− . The resulting BPS conditions are much more involved than those in the previous case. However, we are able to find a number of solutions for special values of electric and magnetic charges.
• Solutions from pure N = 4 gauged supergravity We will begin with a simple case of pure N = 4 gauged supergravity with φ 1 = φ 4 = 0 and A 9+ = A 12− = 0. In this case, the constraint (162) requires e 3 = p 6 = 0, and we find We then find the following BPS equations [e 2h (g 1 + e φ g 2 ) + κe 6 + κp 3 e φ ] 2 + (e 2h g 2 + κp 3 ) 2 χ 2 . (183) From these equations, we find an AdS 2 × H 2 fixed point given by for constants φ = φ 0 and χ = χ 0 provided that g 2 e 6 = g 1 p 3 . We note that for χ = 0, the above BPS equations and the AdS 2 × H 2 fixed point are the same as those considered in [41] with an appropriate change of symplectic frame to purely electric SO(4) gauge group. We have slightly generalized the equations in [41] by including a non-vanishing axion. We now give the flow solutions interpolating between the AdS 4 vacuum and the AdS 2 × H 2 geometry. Before giving explicit solutions, we first simplify the expressions by setting g 2 = g 1 according to which the twist condition gives p 3 = e 6 = 1 2g 1 . For χ = 0 and κ = −1, we find a much simpler set of BPS equations These equations take a very similar form to those of N = 5, 6 gauged supergravities and N = 3 gauged supergravity given in the previous section. We then expect that the resulting solutions are related to each other by truncations of N = 6 gauged supergravity to gauged supergravities with lower amounts of supersymmetry. The solution is given by This solution flows to the AdS 2 × H 2 fixed point (184) for φ 0 given by For χ = 0, we have the BPS equations with the solution given by for a constant C 0 . However, we are not able to find an analytic solution for χ(r).
It can be verified that for appropriate values of the parameters, this critical point is valid for both κ = 1 and κ = −1 resulting in a class of AdS 2 ×S 2 and AdS 2 ×H 2 geometries. Since p 12 = 0 in this case, the solutions carry only electric charges of A 12− . Examples of solutions interpolating between AdS 4 and AdS 2 × H 2 vacua with g 2 = g 1 = 1, and e 12 = 1, 2, 3 are shown in figure 1. We also note that the value of e 6 is fixed by the twist condition g 1 (p 3 + e 6 ) = 1.  A number of interpolating solutions between AdS 4 and AdS 2 × S 2 critical points are shown in figure 2 with the following numerical values and e 12 = 4, 6, 8.
Case ii: In this case, the solutions carry magnetic charges of A 12− , and the resulting BPS equations are given by  From these equations, we find a family of AdS 2 × Σ 2 fixed points given by .
Similar to the previous case, both AdS 2 × S 2 and AdS 2 × H 2 geometries are possible depending on the values of various parameters. Examples of flow solutions from the AdS 4 vacuum to AdS 2 × H 2 fixed points with and p 12 = 1, 2, 3 are given in figure 3. For flow solutions to AdS 2 ×S 2 fixed points, we give some representative solutions for p 12 = 3, 6, 9 and in figure 4.

Solutions with SO(2) diag × SO(2) diag symmetry
In this section, we repeat the same analysis for a smaller residual symmetry SO(2) diag ×SO(2) diag . As we will see, a new feature is the appearance of a number of non-trivial supersymmetric AdS 4 vacua. All of these vacua are not new but have recently been found in [42] to which we refer for more details. Since the analysis of SO(2) diag × SO(2) diag singlet scalars has not previously appeared, we will give more detail than the SO(2) × SO(2) × SO(2) × SO(2) sector considered in the previous section. We begin with the scalars from SO(6, 6)/SO(6) × SO(6) coset which contains six singlets corresponding to the following non-compact generators The coset representative can be then written as With this coset representative, scalar kinetic terms are given by The tensor A ij 1 is proportional to the identity matrix of which the four-fold degenerate eigenvalue gives the superpotential only for χ = 0. Since the complete expressions are much more complicated and will not play any important role in subsequent analysis, we will only give the potential and superpotential for the case of χ = 0. These are given respectively by and g 2 e φ cosh φ 4 (1 + cosh 2φ 5 cosh 2φ 6 ) +g 2 e φ sinh φ 4 (1 − cosh 2φ 5 cosh 2φ 6 ) .
All of these vacua have already been found in [42], but we repeat them here for later convenience. We also note the unbroken gauge symmetries for these solutions which are given respectively by SO (4)  To find supersymmetric AdS 4 black hole solutions, we now turn to the analysis of Yang-Mills equations. To implement the SO(2) diag × SO(2) diag symmetry, we impose the following conditions on the gauge fields and which lead to the same composite connection given in (154). Therefore, the twist conditions and relevant projectors are the same. Unlike the SO(2) × SO(2) × SO(2) × SO(2) case, the YM currents are non-vanishing in this case. From equation (124), we find which, from the ansatz of the gauge fields, imply that b 3 and b 9 are constant and Similarly, equation (125) gives which lead to constant b 6 and b 12 together with We also note that the radial component of the composite connection is given by which identically vanishes whenever φ 2 = 0 or φ 3 = 0 and φ 5 = 0 or φ 6 = 0. In order to find solutions interpolating between supersymmetric AdS 4 vacua identified above, we will choose a definite choice We then consider equation (126). Equations for H 3− and H 9− givẽ together with Forg 1 = g 1 which is needed for the existence of non-trivial AdS 4 vacua, the last equation implies e 3 = e 9 = b 3 = b 9 = 0 (237) which in turn gives Similarly, equations for H 6− and H 12− give together with With χ = φ 3 = φ 6 = 0, we find that both W and Z are real and given by It can be readily verified that critical points I, II, III, and IV are critical points of W as expected for supersymmetric vacua. As in the previous case, there are two possible topological twists, N = 4 and N = 2 twists. The N = 4 twists do not give rise to any AdS 2 × Σ 2 fixed points, so we will only give the results on N = 2 twists. Since both W and Z are real, we find the phase e iΛ = ±1, and the BPS equations are given by From φ 2 and φ 5 equations, we immediately see that there are four possibilities for AdS 2 × Σ 2 fixed points to exist: ii : φ 2 = 0 and φ 4 = 1 2 ln g 2 + g 2 g 2 − g 2 , iii : φ 5 = 0 and φ 1 = 1 2 ln g 1 + g 1 g 1 − g 1 iv : • ii: In this case, we have φ 2 = 0 and with Y = g 1 p 3 (g 2 1 + g 2 1 ) cosh 4φ 1 − 4g 1g1 sinh 2φ 1 (e 6 g 2 + g 1 p 3 cosh 2φ 1 ) +g 3 1 p 3 −g 2 1 (4e 6 g 2 + g 1 p 3 ) + 4e 6 g 2g • iii: For this final possibility, we have φ 5 = 0 and g 2g2 (e 2φ 4 − 1) , φ = ln e 6 g 2 (e 6 g 3 2 + 2g 1g 2 2 p 3 − 2g 1g 2 2 p 3 cosh 2φ 4 + 2g 1 g 2g2 p 3 sinh 2φ 4 ) + e 6 g 2 2 + ln g 1 e φ 4 (coth φ 4 − 1) 2g 2g2 p 3 g 2 1 − g 2 1 , φ 4 = 1 2 ln g 2 (e 6 g 2 + 2g 1 p 3 ) + e 2 6 g 4 2 + 4e 6 g 1 g 2g 2 2 p 3 + 4g 2 1g 2 2 p 2 3 e 6 g 2 (g 2 −g 2 ) .
In each case, we have not explicitly given the expressions for L AdS 2 due to their complexity. These can be obtained from f equation by using the values of the other fields at the fixed points. We have verified that all the above three cases indeed lead to valid AdS 2 × Σ 2 fixed points in each case. This will also be clearly seen later in numerical analyses. For critical point ii, we have found only AdS 2 × H 2 solutions as in critical point i. An example of the solutions interpolating between supersymmetric AdS 4 critical points I and II and an AdS 2 × H 2 geometry with g 2 = g 1 = 1,g 1 = 2g 1 , g 2 = 3g 2 and p 3 = −3 is shown in figure 6. We have set φ 2 = 0 along the entire solution. We also note that the solution indeed exhibits an intermediate Unlike the previous two cases, in critical point iii, we only find AdS 2 × S 2 solutions. An example of flow solutions is shown in figure 7 with g 2 = g 1 = 1, g 1 = 2g 1 ,g 2 = 3g 2 and p 3 = 3. Along the entire flow, we have set φ 5 = 0. As in the flow solution to AdS 2 × H 2 critical point ii, the solution exhibits an intermediate AdS 4 critical point III with φ = 0.143841, so the solution interpolates between AdS 4 critical points I and II and AdS 2 × S 2 geometry in the IR. The solutions in this case and the flow to critical point ii are similar to solutions describing RG flows across dimensions in half-maximal gauged supergravities in five, six and seven dimensions [39,43,44,45]. Moreover, there also exist solutions that flow directly from AdS 4 critical point I to these AdS 2 ×S 2 and AdS 2 ×H 2 fixed points. We will not give these solutions here since they are similar to the solutions in SO(2) × SO(2) × SO(2) × SO(2) case without non-trivial AdS 4 vacua.
We end this section by noting that there do not exist any AdS 2 × Σ 2 fixed points for case iv discussed above. Therefore, there are no flow solutions from the supersymmetric AdS 4 vacuum IV to AdS 2 × Σ 2 geometries in the IR. This is in line with the N = 3 gauged supergravity studied in the previous section in which no AdS 2 × Σ 2 fixed points exist for RG flows involving the non-trivial N = 3 AdS 4 critical point with SO(3) symmetry. On the other hand, as we have seen above, AdS 2 × Σ 2 critical points ii and iii do exist and are connected to non-trivial AdS 4 critical points II and III. However, the latter do not have an analogue in the case of N = 3 gauged supergravity.   (f) f (r) solution Figure 6: A supersymmetric AdS 4 black hole with AdS 2 × H 2 horizon (ii) for g 2 = g 1 = 1,g 1 = 2g 1 ,g 2 = 3g 2 and p 3 = −3. (f) f (r) solution Figure 7: A supersymmetric AdS 4 black hole with AdS 2 × S 2 horizon (iii) for g 2 = g 1 = 1,g 1 = 2g 1 ,g 2 = 3g 2 and p 3 = 3.

Conclusions and discussions
We have studied a number of supersymmetric black hole solutions in asymptotically AdS 4 space from matter-coupled N = 3 and N = 4 gauged supergravities. In N = 3 theory, we have found an AdS 2 × H 2 solution with SO(2) × SO(2) symmetry. We have also given a complete solution interpolating between SO(3)×SO (3) symmetric AdS 4 vacuum and this AdS 2 × H 2 geometry with a non-vanishing scalar. The resulting solution has a very similar structure to those given in N = 5, 6 gauged supergravities. The solution with vanishing scalars is a solution of pure N = 3 gauged supergravity and can be embedded in massive type IIA theory using the result of [31]. We have also shown that there are no AdS 4 black hole solutions with SO(2) diag symmetry. Therefore, in N = 3 gauged supergravity under consideration here, it is clear that there are no other solutions. Although we have considered only a particular case of three vector multiplets, it has been shown in [46] that the SO(3) R ⊂ SU (3) R symmetry must be gauged in order for the gaugings to admit a supersymmetric AdS 4 vacuum. This is also an essential part in performing topological twists since the gravitini and Killing spinors are charged exclusively under this symmetry or a diagonal subgroup with parts of the symmetry of vector multiplets. Therefore, even with extra vector multiplets and possibly larger gauge groups, the structure of the topological twists should be the same and eventually leads to a similar conclusion.
In pure N = 4 gauged supergravity, we have recovered an AdS 2 ×H 2 solution studied in [41]. However, we have included a non-vanishing axion and given the interpolating solutions between this geometry and the supersymmetric AdS 4 vacuum. For matter-coupled N = 4 gauged supergravity, we have found a number of AdS 2 ×S 2 and AdS 2 ×H 2 solutions with SO(2)×SO(2)×SO(3)×SO(2) symmetry. We have also given various examples of numerical solutions interpolating between these geometries and the AdS 4 vacuum with SO(4) × SO(4) symmetry. The BPS equations are very complicated, and we are not able to completely carry out the analysis. However, we have given a number of possible AdS 4 black hole solutions with both spherical and hyperbolic horizons. We note that unlike N = 5 and N = 6 gauged supergravities, there exist matter multiplets in N = 4 theory, and the two SO(2) factors involving in the twists are not necessarily equal though related, see the twist condition in (160). This gives a weaker constraint on the charges and leaves more freedom to find AdS 2 × Σ 2 solutions. This is also supported by the fact that, when restricted to the case of pure N = 4 gauged supergravity, the charges of A 3+ and A 6− must be equal, and only one AdS 2 × H 2 solution which is an analogue of similar solutions in N = 5, 6 theories exists.
We have also found AdS 2 × S 2 and AdS 2 × H 2 solutions with SO(2) diag × SO(2) diag symmetry. Similar to the N = 3 theory, in this case, we have performed a complete analysis and classified all possible supersymmetric AdS 2 × Σ 2 solutions with the aforementioned residual symmetry at least for the case of six vector multiplets. In this case, apart from the trivial AdS 4 critical point with the full SO(4) × SO(4) symmetry, there exist additional three supersymmetric AdS 4 vacua with SO(4)×SO(3), SO(3)×SO(4) and SO(3)×SO(3) symmetries. Except for the last critical point, we have found black hole solutions interpolating between these vacua and AdS 2 ×S 2 and AdS 2 ×H 2 geometries. We hope all these solutions could be useful in black hole physics and holographic studies of twisted compactifications of N = 3 and N = 4 SCFTs in three dimensions on a Riemann surface.
It is interesting to look for more general solutions in the SO(2) × SO(2) × SO(2) × SO(2) case in particular solutions carrying both electric and magnetic charges of the same gauge fields. In this paper, we have given only some representative examples of the possible solutions which carry either electric or magnetic charges of a given gauge field. Another direction is to find an embedding of the solutions given here in string/M-theory. Solutions in pure N = 3 and N = 4 gauged supergravities can be embedded in ten and eleven dimensions using consistent truncations given respectively in [31,32] and [47]. It would be useful to find similar embedding for the solutions in matter-coupled gauged supergravities. It could also be of particular interest to study the dual three-dimensional N = 3, 4 SCFTs with topological twists and compute microscopic entropy of the black holes. Finally, it would be interesting to study similar solutions in other gauged supergravities such as ω-deformed N = 8 gauged supergravity and N = 4 truncation of massive type IIA on S 6 given in [48] and [49], respectively.