Supersymmetric AdS6 black holes from matter coupled F(4) gauged supergravity

In matter coupled F(4) gauged supergravity in six dimensions, we study supersymmetric AdS6 black holes with various horizon geometries. We find new AdS2×Σg1×Σg2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Ad{S}_2\times {\varSigma}_{{\mathfrak{g}}_1}\times {\varSigma}_{{\mathfrak{g}}_2} $$\end{document} horizons with g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{g} $$\end{document}1> 1 and g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{g} $$\end{document}2> 1, and present the black hole solution numerically. The full black hole is an interpolating geometry between the asymptotically AdS6 boundary and the AdS2×Σg1×Σg2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Ad{S}_2\times {\varSigma}_{{\mathfrak{g}}_1}\times {\varSigma}_{{\mathfrak{g}}_2} $$\end{document} horizon. We also find black holes with horizons of Kähler four-cycles in Calabi-Yau fourfolds and Cayley four-cycles in Spin(7) manifolds.


Introduction and conclusions
In F (4) gauged supergravity in six dimensions [1], there is a unique supersymmetric fixed point which is dual to 5d superconformal USp(2N ) gauge theory [2,3]. As it was shown in [4], F (4) gauged supergravity is a consistent truncation of massive type IIA supergravity [5]. The fixed point uplifts to AdS 6 × w S 4 near-horizon geometry of the D4-D8 brane system [6,7]. In the spirit of [8], supergravity solutions of wrapped D4-branes on various supersymmetric cycles were studied in F (4) gauged supergravity. D4-branes wrapped on two-and three-cycles were studied in [9]. They found AdS 4 and AdS 3 fixed point solutions. See [10] for more recent results.
Recently, by considering D4-branes wrapped on supersymmetric four-cycles, we found supersymmetric AdS 6 black holes of F (4) gauged supergravity in [11]. 1 To be specific, we found the full black hole solutions which is an interpolating geometry between the asymptotically AdS 6 boundary and the AdS 2 × H 2 × H 2 horizon. Via the AdS/CFT JHEP02(2019)108 correspondence, [13], analogous to the AdS 4 black hole cases in [14][15][16], the Bekenstein-Hawking entropy of the black holes nicely matched with the topologically twisted index of 5d USp(2N ) gauge theory on Σ g 1 × Σ g 2 × S 1 in the large N limit [17,18]. We also considered black hole horizons of Kähler four-cycles in Calabi-Yau fourfolds and Cayley four-cycles in Spin(7) manifolds.
Pure F (4) gauged supergravity is a consistent truncation of massive type IIA supergravity [4] and type IIB supergravity [19][20][21] on a four-hemisphere. Although it is not known whether it is also a consistent truncation of ten-dimensional supergravity, one can couple vector multiplets to pure F (4) gauged supergravity [22]. In this theory, new fixed points and holographic RG flows were studied in [23][24][25]. See [26][27][28] also for other studies in this theory.
In this paper, in matter coupled F (4) gauged supergravity, we continue our study on supersymmetric AdS 6 black holes. We consider F (4) gauged supergravity coupled to three vector multiplets, and its U(1) × U(1)-invariant truncation first considered in [25]. We consider black hole solutions with a horizon which is a product of two Riemann surfaces, AdS 2 × Σ g 1 × Σ g 2 . We derive supersymmetry equations and obtain new AdS 2 solutions. The AdS 2 horizon exists only for the H 2 × H 2 background, and not for the H 2 × S 2 or S 2 × S 2 backgrounds. We present the full black hole solutions numerically.
We also consider black holes with horizons of Kähler four-cycles in Calabi-Yau fourfolds and Cayley four-cycles in Spin (7) manifolds. For Cayley four-cycles in Spin(7) manifolds, we consider the SU(2) diag -invariant truncation of F (4) gauged supergravity coupled to three vector multiplets. We find new AdS 2 horizons. It will be interesting to have a field theory interpretation of this AdS 2 solution.
In section 2, we review matter coupled F (4) gauged supergravity in six dimensions. In section 3, we consider F (4) gauged supergravity coupled to three vector multiplets, and its U(1) × U(1)-invariant truncation. We consider supersymmetric black hole solutions with a horizon which is a product of two Riemann surfaces. In section 4, we consider supersymmetric black hole solutions with horizons of Kähler four-cycles in Calabi-Yau fourfolds and Cayley four-cycles in Spin(7) manifolds. In appendix A, we present the equations of motion for the U(1) × U(1)-invariant truncation.
Note added. In the final stage of this work, we became aware of [29] which has some overlap with the results presented. The solutions presented in sections 3 and 4.1 are identical to the solutions found in sections 5 and 6 of [29], respectively.

Matter coupled F (4) gauged supergravity
We review matter coupled F (4) gauged supergravity in six dimensions [22]. The gravity multiplet consists of where they denote the graviton, gravitini, four vector fields, a two-form gauge potential, dilatini, and a real scalar field, respectively. The vector fields, A α µ , α = 0, 1, 2, 3, can be JHEP02(2019)108 used to gauge the SU(2) R × U(1) gauge symmetry. The vector multiplet consists of where they denote a vector field, gaugini, and four real scalar fields, respectively, and I = 1, . . . , n labels the vector multiplets. The fermionic fields are eight-dimensional pseudo-Majorana spinors and transform in the fundamental representation of the SU(2) R ∼ USp(2) R R-symmetry denoted by indices, A, B = 1, 2. We denote the coupling constants of gauge fields from the gravity and vector multiplets by g 1 and g 2 , respectively, and the mass parameter of the two-form gauge potential by m.
When there is no vector multiplet, the theory reduces to pure F (4) gauged supergravity [1]. In pure F (4) gauged supergravity, there are five inequivalent theories: . The N = 4 + theory admits a supersymmetric AdS 6 fixed point when g 1 = 3m. At the supersymmetric AdS 6 fixed point, all the fields are vanishing except the AdS 6 metric.
The bosonic Lagrangian is given by and V = − e 2σ 1 36 where we define

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and The field strengths are collectively defined by 2 The supersymmetry variations of the fermionic fields are given by 3 where we define and The Pauli matrices, σ tA B , satisfy the relations, and σ t AB = σ t (AB) . We also define the chirality matrix by with γ 2 7 = −1 and γ T 7 = −γ 7 . We employ the mostly minus signature, (+ − − − −−).

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2.1 F (4) gauged supergravity coupled to three vector multiplets In F (4) gauged supergravity coupled to three vector multiplets, we have a scalar field from the gravity multiplet and four scalar fields from each vector multiplet: total thirteen scalar fields. The scalar fields from the gravity multiplet and the vector multiplets parametrize each factor of the coset manifold, respectively, .
There are three vector fields from the gravity multiplet and three vector fields from the three vector multiplets. Two sets of three vector fields can be used to gauge SU(2) R × SU(2) gauge group. We denote the coupling constants of two SU(2) factors by g 1 and g 2 . The structure constant in (2.5) splits into (2.20) The generators of SO(4), SU(2) R , SU (2), and the non-compact SO(4, 3) could be represented by, respectively [25], We truncate the theory to the U(1)×U(1)-invariant sector, which was first considered in section 3.1 of [25]. The U(1)×U(1) are generated by J 12 1 and J 12 2 . We find two non-compact SO(4, 3) generators, Y 03 and Y 33 , which are invariant under the action of J 12 1 and J 12 2 . We exponentiate the non-compact generators and obtain the coset representative, We also have two U(1) gauge fields, A 3 and A 6 , and a two-form gauge potential, B µν , in the U(1)×U(1)-invariant truncation. The Lagrangian of the truncation is given by where the scalar potential is JHEP02(2019)108

The supersymmetry equations
In this section, we obtain supersymmetric AdS 6 black holes with a horizon which is a product of two Riemann surfaces. We consider the metric, for the S 2 × S 2 background, and for the H 2 × H 2 background. The only non-vanishing components of the non-Abelian SU(2) gauge field, A Λ µ , Λ = 0, 1, . . . , 6, are given by for the S 2 × S 2 background and for the H 2 × H 2 background, where the magnetic charges, a 1 , a 2 , b 1 , b 2 , are constant. In order to have equal signs for the field strengths, we set opposite signs of the gauge fields for the S 2 × S 2 and H 2 × H 2 backgrounds. We also have a non-trivial two-form gauge potential, B µν , determined by solving the equations of motion, The three-form field strength of the two-form gauge potential, H µνλ , vanishes identically. The supersymmetry equations are obtained by setting the supersymmetry variations of the fermionic fields to zero. From the supersymmetry variations, we obtain where the hatted indices are the flat indices. The t-, θ 1 -, and θ 2 -components of the gravitino variations give (3.9), (3.10), (3.11), the dilatino variation gives (3.12), and the gaugino variation gives (3.13). The φ 1 -, φ 2 -components of the gravitino variations are identical to the θ 1 -, and θ 2 -components beside few more terms, (3.14) We employ the projection conditions, where λ = ±1. Solutions with the projection conditions preserve 1/8 of the supersymmetries. As we check later, in order to have consistent supersymmetry equations with the equations of motion, it is required to have Therefore, from now on, we set ϕ 1 to vanish.

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We present the complete supersymmetry equations, We also obtain twist conditions on the magnetic charges from (3.14), where k = +1 for the S 2 × S 2 background and k = −1 for the H 2 × H 2 background. 4 There is no condition on b 1 and b 2 . The supersymmetry equations are consistent with the equations of motion. We present the equations of motion in appendix A.

The AdS 2 solutions
In this section, we find AdS 2 solutions of the supersymmetry equations. The solutions describe the AdS 2 × Σ g 1 × Σ g 2 horizon of six-dimensional black holes. Now we will consider the N = 4 + theory, g 1 > 0, m > 0. When b 1 = b 2 = 0, we find an 19) which is the AdS 2 solution first found in [11]. 5 When we consider the S 2 × S 2 background with k = +1, AdS 2 fixed point does not exist. 4 It is possible to have geometries like S 2 × H 2 for k1 = +1 and k2 = −1, or vice versa. One can easily generalize our supersymmetry equations and the twist conditions to that case. 5 In order to compare with [11], we have to reparametrize our parameters by

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When we consider for non-zero b 1 and b 2 , we obtain new AdS 2 solutions, where we define All the fields are parametrized by the magnetic charges, (a 1 , a 2 , b 1 , b 2 ). As (a 1 , a 2 ) are fixed by the twist condition in (3.18), there are two free parameters left, (b 1 , b 2 ).
In order to have AdS 2 solutions, we should choose (b 1 , b 2 ) which makes We plot the range of (b 1 , b 2 ) which satisfies the positivity conditions for H 2 × H 2 , H 2 × S 2 , S 2 × H 2 and S 2 × S 2 , respectively. The positivity ranges are depicted in figure 1. We set m = 1/2 and g 1 = 3m to have a unit radius for the AdS 6 boundary. From the plots, we conjecture that only the H 2 × H 2 background gives the AdS 2 solutions. 6 Even in the large region in the graph for the H 2 × H 2 background, only a small part near origin yields AdS 2 solutions.

Numerical black hole solutions
Now we present the full black hole solution numerically. The full black hole solution is an interpolating geometry between the asymptotically AdS 6 boundary and the AdS 2 × H 2 × H 2 horizon. We introduce a new radial coordinate, (3.24) 6 We note that, when G1 ↔ G2, the solutions are invariant under a1 ↔ a2 and b1 ↔ b2. However, as This kind of coordinate was introduced in [30]. Employing the supersymmetry equations, we obtain ∂ρ ∂r where we define Then, the supersymmetry equations are

−D
In the r-coordinate, the UV or asymptotically AdS 6 boundary is at r = 0, and the IR or AdS 2 × H 2 × H 2 horizon is at r = ∞. In this ρ-coordinate, the UV is at ρ = +∞, and the IR is at ρ = −∞. We present some representative plots of the full black hole solutions in figure 2.

Black holes with other horizons
In this section, we obtain more black hole solutions with other horizon geometries by considering D4-branes wrapped on Kähler four-cycles in Calabi-Yau fourfolds and on Cayley four-cycles in Spin (7) manifolds. We believe these are all possible four-cycles on which D4branes can wrap in F (4) gauged supergravity. D4-branes on two Riemann surfaces in the previous section fall into a special case of D4-branes on Kähler four-cycles in Calabi-Yau fourfolds. The analogous solutions of M5-branes wrapped on supersymmetric four-cycles were studied in [31,32].

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where M 4 is a Kähler four-cycle in Calabi-Yau fourfolds. The curved coordinates on the Kähler four-cycles will be denoted by {x 1 , x 2 , x 3 , x 4 }, and the hatted ones are the flat coordinates. For Kähler four-cycles in Calabi-Yau fourfolds, there are four directions transverse to D4-branes in the fourfolds. The normal bundle of the four-cycle has U(2) ⊂ SO(4) structure group. We identify U(1) part of the structure group with U(1) gauge field from the non-Abelian SU(2) gauge group, [31,33]. The only non-vanishing components of the field strength of SU(2) gauge field, A Λ µ , Λ = 0, 1, . . . , 6, are given by where the magnetic charges, a 1 , a 2 , b 1 , b 2 , are constant. The only non-vanishing component of the two-form gauge potential is We employ the projection conditions, where λ = ±1. Solutions with the projection conditions preserve 1/8 of the supersymmetries. By employing the projection conditions, we obtain the complete supersymmetry equations, with the twist conditions, where k determines the curvature of the Kähler four-cycles in Calabi-Yau fourfolds. There is no condition on b 1 and b 2 . The product of two Riemann surfaces considered in the previous section is a special case of Kähler four-cycles in Calabi-Yau fourfolds. When we identify G ≡ G 1 = G 2 in the supersymmetry equations for D4-branes wrapped on two Riemann surfaces, (3.17), we obtain the supersymmetry equations here, (4.5). By solving the supersymmetry equations, we find the AdS 2 fixed point solutions which are identical to the ones obtained in the previous section.

The supersymmetry equations
We consider the metric, where M 4 is a Cayley four-cycle in manifolds with Spin (7) holonomy. The curved coordinates on the Cayley four-cycles will be denoted by {x 1 , x 2 , x 3 , x 4 }, and the hatted ones are the flat coordinates. In order to preserve supersymmetry for D4-branes wrapped on Cayley four-cycles in Spin(7) manifolds, we identify self-dual SU(2) + subgroup of the SO(4) isometry of the four-cycle,

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with the non-Abelian SU(2) gauge group, [31,33]. The self-duality is defined by (4.12) and we denoted the self-duality and anti-self-duality by + and −, respectively. For the self-dual part, components are identified by The only non-vanishing components of the field strength of the SU(2) gauge field, A Λ µ , Λ = 0, 1, . . . , 6, are given by where the magnetic charges, a 1 , a 2 , a 3 , b 1 , b 2 , b 3 , are constant. As we have one SU(2) diag ⊂ SU(2) R × SU(2) non-Abelian gauge field, they are related by where g 1 and g 2 are the gauge coupling constants of SU(2) R and SU (2), respectively. The only non-vanishing component of the two-form gauge potential is We employ the projection conditions, where λ = ±1. Solutions with the projection conditions preserve 1/16 of the supersymmetries.
Employing the projection conditions, from δλ I A = 0, we obtain for I = 1, 2, 3, respectively, (4.18) Therefore, we conclude that the magnetic charges are

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We present the complete supersymmetry equations, We also obtain twist conditions on the magnetic charges, where k determines the curvature of the Cayley four-cycles in Spin (7) manifolds. The twist condition on b comes from the SU(2) diag condition, (4.15).

The AdS 2 solutions
Now we will consider the N = 4 + theory, g 1 > 0, m > 0. When b = 0, we find an After the reparametrization of the parameters given in (3.20), this is the AdS 2 solution first found in [11]. When we consider for non-zero b, we find new AdS 2 solutions in terms of the scalar field, ϕ, Then, the scalar field, ϕ, should be expressed in terms of the magnetic charges, a and b, but the expression is very unwieldy. Alternatively, we present the magnetic charge, b, in terms of the scalar field, ϕ, b = − a g 1 1 + e 2ϕ 3 1 − 6e 2ϕ + e 4ϕ + g 2 1 − e 2ϕ 3 1 + 6e 2ϕ + e 4ϕ + 4e 2ϕ g 2 1 (1+e 2ϕ ) 4 (5−6e 2ϕ +5e 4ϕ )+g 2 2 (1−e 2ϕ ) 4 (5+6e 2ϕ +5e 4ϕ )+10g 1 g 2 (1−e 4ϕ ) 3 The solutions are parametrized by two magnetic charges, a and b. It will be interesting to have a field theory interpretation of this AdS 2 fixed point solution.
Unlike the black holes with a horizon of two Riemann surfaces, for this case, we could not device a way to determine the positivity range for AdS 2 solutions. However, as we see in the next subsection, we obtained a number of AdS 2 solutions with negative curvature horizon, k = −1, numerically. On the other hand, we could not find any solutions with positive curvature horizon, k = +1. Thus, we will concentrate on solutions with negative curvature horizon, k = −1.

Numerical black hole solutions
Now we present the full black hole solution numerically. The full black hole solution is an interpolating geometry between the asymptotically AdS 6 boundary and the AdS 2 × Cayley 4 horizon. As we explained at the end of the last subsection, we will concentrate on solutions with negative curvature horizon, k = −1. We introduce a new radial coordinate, ρ = F + σ . (4.25) This kind of coordinate was introduced in [30]. Employing the supersymmetry equations, we obtain ∂ρ ∂r where we define D = 2me −3σ + 3 2m a 2 − b 2 e σ−4G .  In the r-coordinate, the UV or asymptotically AdS 6 boundary is at r = 0, and the IR or AdS 2 × Cayley 4 horizon is at r = ∞. In this ρ-coordinate, the UV is at ρ = +∞, and the IR is at ρ = −∞. We set m = 1/2 and g 1 = 3m to have a unit radius for the AdS 6 boundary. Then, a is determined by (4.21), and there is one free parameter left, b. Employing (4.23) and (4.24) to determine boundary conditions, we solve the supersymmetry Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.