Uplifting supersymmetric $AdS_6$ black holes to type II supergravity

Employing uplift formulae, we uplift supersymmetric $AdS_6$ black holes from $F(4)$ gauged supergravity to massive type IIA and type IIB supergravity. In massive type IIA supergravity, we obtain supersymmetric $AdS_6$ black holes asymptotic to the Brandhuber-Oz solution. In type IIB supergravity, we obtain supersymmetric $AdS_6$ black holes asymptotic to the non-Abelian T-dual of the Brandhuber-Oz solution.


Introduction and conclusions
Recently we are observing progress of the AdS/CFT correspondence, [1], in higher dimensions, e.g., in AdS 7 /CFT 6 and AdS 6 /CFT 5 . For supersymmetric AdS 6 solutions of string/M-theory, the only known solution was the near horizon limit of the D4-D8 brane system, known as the Brandhuber-Oz solution of massive type IIA supergravity [2] which is dual to 5d superconformal field thoeries in [3,4,5,6]. Later, it was shown to be the unique supersymmetric AdS 6 solution of massive type IIA supergravity [7]. After the discovery of the non-Abelian T-dual of the Brandhuber-Oz solution in type IIB supergravity in [8], and derivations of the supersymmetry equations of general AdS 6 solutions of type IIB supergravity in [9,10,11], an infinite family of AdS 6 solutions was discovered in [12,13,14,15]. See also [16]. Although the dual field theory of the non-Abelian T-dual of the Brandhuber-Oz solutions is still unclear [17,18], the infinite family of AdS 6 solutions was shown to be dual of five-brane web theories, e.g., 5d T N theories, [19,20,21,22].
We can also study the AdS 6 /CFT 5 correspondence from six dimensions, i.e., from F (4) gauged supergravity, [23]. Any solution of F (4) gauged supergravity can be uplifted to a solution of massive type IIA, [25], and type IIB supergravity, [26,27,28]. Moreover, more recently, uplift formula for F (4) gauged supergravity coupled to arbitrary number of vector multiplets to type IIB supergravity has been derived in [29]. To the best of our knowledge, it is the first uplift formula for gauged supergravity with arbitary number of matter multiplets. They also showed that uplifts are only possible for F (4) gauged supergravity coupled up to three vector mutiplets. Due to the discovery of the infinite family of supersymmetric AdS 6 solutions of type IIB supergravity, [12,13,14,15], and the developments in exceptional field theory, [30,28,29], these recent progress in uplift formulae was able.
The supersymmetric AdS 6 black hole solutions are asymptotic to the supersymmetric AdS 6 fixed point and have horizon of AdS 2 × Σ g 1 × Σ g 2 , where g 1 > 1 and g 2 > 1 are genus of the Riemann surfaces, [31,41,42]. The supersymmetric AdS 6 fixed point of F (4) gauged supergravity is universal, i.e., it uplifts to i) the Brandhuber-Oz solution of massive type IIA supergravity, [2], ii) the non-Abelian T-dual of the Brandhuber-Oz solution of type IIB supergravity, [8], and iii) the infinite family of AdS 6 solutions of type IIB supergravity, [12,13,14,15]. For the solutions uplifted to massive type IIA supergravity, as the Brandhuber-Oz solution is the unique supersymmetric AdS 6 solution of massive type IIA supergravity, the uplifted solutions are automatically asymptotic to i) the Brandhuber-Oz solution. On the other hand, for the solutions uplifted to type IIB supergravity, solutions can be asymptotic to ii) the non-Abelian T-dual of the Brandhuber-Oz solution or iii) the infinite family of AdS 6 solutions. In the uplift formulae in [28,29], it is determined by choosing holomorphic functions, A ± (z) where z is a complex coordinate on a Riemann surface, [12,13,14,15]. For the solutions asymptotic to the non-Abelian T-dual of the Brandhuber-Oz solutions, the holomorphic functions are given by It was first obtained in [27] and rediscovered in [18]. For the solutions asymptotic to the infinite family of AdS 6 solutions, [12,13,14,15], the holomorphic functions are given by where and r l and s n are the positions of the zeros and poles on the Riemann surfaces, respectively, and A 0 ± and σ are complex constants. However, the infinite family of AdS 6 solutions is not able to be evaluated by elementary functions, and either the uplift of the black holes asymptotic to the infinite family of AdS 6 solutions. Hence, we restrict ourselves to the uplifts of supersymmetric black holes asymptotic to i) the Brandhuber-Oz solutions of massive type IIA supergravity and ii) the non-Abelian T-dual of the Brandhuber-Oz solutions of type IIB supergravity, and not consider iii) the infinite family of supersymmetric AdS 6 solutions of type IIB supergravity. 1 By employing the uplift formula to type IIB supergravity in [29], it will be interesting to uplift the supersymmetric AdS 6 black holes from matter coupled F (4) gauged supergravity in [41,42]. However, there is no consistent truncation to matter coupled F (4) gauged supergravity from solutions asymptotic to the non-Abelian T-dual of the Brandhuber-Oz solution, [29].
In section 2, we review the uplift formula for F (4) gauged supergrvity to massive type IIA supergravity, and employ the uplift formula to obtain supersymmetric AdS 6 black holes asymptotic to the Brandhuber-Oz solutions of massive type IIA supergravity. In section 3, we review the uplift formula for F (4) gauged supergravity to type IIB supergravity, and employ the uplift formula to obtain supersymmetric AdS 6 black holes asymptotic to the non-Abelian T-dual of the Brandhuber-Oz solutions of type IIB supergravity. Some technical details are relegated in appendices.
2 Supersymmetric AdS 6 black holes of massive type IIA supergravity 2.1 Massive type IIA from F (4) gauged supergravity We review the uplift formula for F (4) gauged supergravity to massive type IIA supergravity in [25].
We introduce the bosonic field content of F (4) gauged supergravity, [23], reparametrized to match conventions of the uplift formula in [25]. 2 There are the metric, the real scalar field, φ, an SU (2) gauge field, A I , I = 1, 2, 3, a U (1) gauge field, A, and a two-form gauge potential, B. Their field strengths are respectively defined bỹ There are two parameters, the SU (2) gauge coupling, g, and the mass parameter of the two-form field, m. When g > 0, m > 0 and g = 3m/2, the theory admits a unique supersymmetric AdS 6 fixed point. At the fixed point, all the fields are vanishing except the AdS 6 metric. The uplift formula for F (4) gauged supergravity to massive type IIA supergravity was obtained in [25]. The uplifted metric and the dilaton field are given by, respectively, 3) The couplings and fields in the uplift formula for pure F (4) gauged supergravity to massive type IIA supergravity in [25] are related to the ones of [23] by where the tilded ones are of [25]. and the two-, three-, and four-form fluxes are given by, respectively, 3 We employ the metric and the volume form on the gauged three-sphere by and σ I , I = 1, 2, 3, are the SU (2) left-invariant one-forms which satisfy A choice of the left-invariant one-forms is We also defined quantities, where φ is the scalar field of F (4) gauged supergravity in the conventions of [23].

Supersymmetric AdS 6 black holes
We uplift the recently obtained supersymmetric AdS 6 black holes of F (4) gauged supergravity in [31] to massive type IIA supergravity. 4 We first present the six-dimensional solutions of [31]. The solutions are given by the metric, which is asymptotically AdS 6 and has a horizon of AdS 2 × Σ g 1 × Σ g 2 with g 1 > 1 and g 2 > 1.
Only a component of the SU (2) gauge field is non-trivial, where the magnetic charges, a 1 and a 2 , are constant. The twist condition on the magnetic charges is where k = −1 for g 1 > 1 and g 2 > 1 and λ = ±1. There is also a two-form field, 16) and the three-form field strength of the two-form field vanishes identically. There is also a non-trivial real scalar field, X(r) = e φ √ 2 , but the U (1) gauge field is vanishing. To match the conventions of [25], there are additional overall factors of 2 to the solutions of [31] in (2.14), (2.16) and 1/2 in (2.15). Also we set m = 2g/3.
As the uplifted solution is simply the uplift formula in (2.3), (2.4), (2.5), (2.6), and (2.7) with the six-dimensional fields in (2.13), (2.14), and (2.16), it is unnecessary to present them in detial here. The full geometry is interpolating between the asymptotic AdS 6 fixed point, which is the Brandhuber-Oz solution, and the near horizon geometry, By employing the supersymmetry equations of F (4) gauged supergravity, we explicitly checked that the uplifted solution solves the equations of motion of massive type IIA supergravity. We present the supersymmetry equations of F (4) gauged supergravity in appendix A and the equations of motion of massive type IIA supergravity in appendix B.1.
3 Supersymmetric AdS 6 black holes of type IIB supergravity 3.1 Type IIB from F (4) gauged supergravity We review the uplift formula for F (4) gauged supergravity to type IIB supergravity in [29].
The general supersymmetric AdS 6 solutions of [12,13,14,15] are specified by two holomorphic functions, A ± . For the uplift formula in [28,29], the functions, A ± , were redefined by two holomorphic functions, where and p α and k α are real functions. The SL(2, R) indices, α, β = 1, 2, are raised and lowered by We also define quantities, 5 where X is the real scalar field of F (4) gauged supergravity, We introduce the bosonic field content of F (4) gauged supergravity, [23], reparametrized to match conventions of the uplift formula in [28,29]. There are the metric, the real scalar field, φ, an SU (2) gauge field, A A , A = 1, 2, 3, a U (1) gauge field, A 4 , and a two-form gauge potential, B. Their field strengths are respectively defined bỹ There are two parameters, the SU (2) gauge coupling, g, and the mass parameter of the two-form field, m. When g > 0, m > 0 and g = 3m, the theory admits a unique supersymmetric AdS 6 fixed point. At the fixed point, all the fields are vanishing except the AdS 6 metric. The uplifted geometry is going to be of the form, where M 1,5 is given by the metric of F (4) gauged supergravity, and Σ is the Riemann surface parametrized by a complex coordinate, z. For the gauged two-sphere,S 2 , we introduce real coordinates, y A , A = 1, 2, 3, their derivatives, and their Hodge dual coordinates, The metric and the volume form of the gauged two-sphere are, respectively, given by Now we readily present the uplift formula for F (4) gauged supergravity to type IIB supergravity in [29]. The metric, the dilaton-axion fields, and the NSNS and RR two-form gauge potentials are respectively obtained from ds 2 = 4 3 1/4 c 6 γ 1/4 ∆ where R and c 6 are constants. The five-form flux is obtained from where and F (4,1) = * F (2,3) .

Uplifted solutions
For the solutions asymptotic to the non-Abelian T-dual of the Brandhuber-Oz solution, the form of the functions, A ± , was first obtained in [27] and rediscovered in [18], where z is a complex coordinate on the Riemann surface. We can obtain explicit expressions for the quantities defined in the previous subsection, e.g., and γ = 1 864c 2 For the gauged two-sphere, we introduce angular coordinates, y 1 = sin θ sin φ , y 2 = sin θ cos φ , y 3 = − cos θ . For the Riemann surface, we also introduce angular coordinates, e.g., (4.3) of [27], The metric and the volume form are given, respectively, by For the later use we also give explicit expressions of p α and k α in terms of the angular coordinates on the Riemann surface, In addition to the SU (2) gauge coupling, g, and the mass parameter of the two-form field, m, of F (4) gauged supergravity, we introduced a parameter,g, e.g., [26], bỹ For the solutions we consider, we will choose the parameters to be Accordingly, we also specify the constants, (3.33)

Supersymmetric AdS 6 fixed point
As the simplest exercise, we uplift the supersymmetric fixed point of F (4) gauged supergravity to the non-Abelian T-dual of the Brandhuber-Oz solution in type IIB supergravity, [8]. This uplift was previously done in [26,27].
Employing the uplift formula with A ± in (3.16), we obtain the metric, where F is given by Employing an SL(2, R) rotation, to the dilaton-axion field matrix, H αβ , we obtain The dilaton and axion fields are obtained to be, respectively, e Φ = 27 cos 2 ξ + X 4 sin 2 ξ F 1/2 sin 1/3 ξ cos 3 ξ , C (0) = 2g 4 81 ρ 2 − sin 4/3 ξ 3 cos 2 ξ + X 4 2 + sin 2 ξ 54 cos 2 ξ + X 4 sin 2 ξ . (3.39) The RR and NSNS two-form gauge potentials are, respectively, given by The five-form flux vanishes identically. We checked that the uplifted solution satisfies the equations of motion of type IIB supergravity. Up to some overall factors, the uplifted solution precisely coincides with the non-Abelian T-dual of the Brandhuber-Oz solution in [8] which we present in appendix C.

Supersymmetric AdS 6 black holes
We uplift the recently obtained supersymmetric AdS 6 black holes of F (4) gauged supergravity in [31] to type IIB supergravity. 6 We first present the six-dimensional solutions of [31]. The solutions are given by the metric, which is asymptotically AdS 6 and has a horizon of AdS 2 × Σ g 1 × Σ g 2 with g 1 > 1 and g 2 > 1.
Only a component of the SU (2) gauge field is non-trivial, where the magnetic charges, a 1 and a 2 , are constant. The twist condition on the magnetic charges is where k = −1 for g 1 > 1 and g 2 > 1 and λ ± 1. There is also a two-form field, 44) and the three-form field strength of the two-form field vanishes identically. There is also a non-trivial real scalar field, X(r) = e φ √ 2 , but the U (1) gauge field is vanishing. To match the conventions of [28,29], there are additional overall factors of 2 to the solutions of [31] in (3.42) and (3.44). Now we perform uplifting of the supersymmetric AdS 6 black holes to type IIB supergravity. As it was done for the previous example, employing the uplift formula with A ± in (3.16), we obtain the uplifted metric, where F is given in (3.35). The dilaton and axion fields are, respectively, e Φ = 27 cos 2 ξ + X 4 sin 2 ξ F 1/2 sin 1/3 ξ cos 3 ξ , C (0) = 2g 4 81 ρ 2 − sin 4/3 ξ 3 cos 2 ξ + X 4 2 + sin 2 ξ 54 cos 2 ξ + X 4 sin 2 ξ . (3.46) The RR and NSNS two-form gauge potentials are, respectively, given by There is also a non-trivial self-dual five-form flux given by The full geometry is interpolating between the asymptotic AdS 6 fixed point, which is the non-Abelian T-dual of the Brandhuber-Oz solution, and the near horizon geometry, AdS 2 ×Σ g 1 ×Σ g 2 × S 2 × Σ. By employing the supersymmetry equations of F (4) gauged supergravity, we explicitly checked that the uplifted solution solves the equations of motion. For the Einstein equations, due to the complexity of the solution, we checked them for numerous specific numerical values of the coordinates on the Riemann surface, (ρ, ξ). We present the supersymmetry equations of F (4) gauged supergravity in appendix A and the equations of motion of type IIB supergravity in appendix B.2.

A The supersymmetry equations of F (4) gauged supergravity
We present the supersymmetry equations of AdS 6 black holes from F (4) gauged supergravity in [31],

B The equations of motion of type II supergravity
We present the equations of motion of type II supergravity in Einstein frame. The metric in string frame is obtained by g string We also define

B.1 Massive type IIA supergravity
The field content of massive type IIA supergravity is the metric, g M N , the dilaton field, Φ, the NSNS two-form gauge potential, B (2) , and the RR one-and three-form gauge potentials, C (1) and C (3) , respectively, [46]. We follow the conventions of [25]. The fluxes are defined by where m is the mass parameter of B (2) , known as the Romans' mass, and is related to the SU (2) coupling of F (4) gauged supergravity by The equations of motion are given by

B.2 Type IIB supergravity
The field content of type IIB supergravity is the metric, g M N , the dilaton and the axion fields, Φ and C (0) , the NSNS and RR two-form gauge potential, B (2) and C (2) , and the RR four-form gauge potential, C (4) , respectively, [47,48]. We follow the conventions of [49]. The fluxes are defined by