Supersymmetric $AdS_6$ black holes from matter coupled $F(4)$ gauged supergravity

In matter coupled $F(4)$ gauged supergravity in six dimensions, we study supersymmetric $AdS_6$ black holes with various horizon geometries. We find new $AdS_2\,\times\,\Sigma_{\mathfrak{g}_1}\times\Sigma_{\mathfrak{g}_2}$ horizons with $\mathfrak{g}_1>1$ and $\mathfrak{g}_2>1$, and present the black hole solution numerically. The full black hole is an interpolating geometry between the asymptotically $AdS_6$ boundary and the $AdS_2\,\times\,\Sigma_{\mathfrak{g}_1}\times\Sigma_{\mathfrak{g}_2}$ horizon. We also find black holes with horizons of K\"ahler 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 U Sp(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, we studied D4-branes wrapped on supersymmetric four-cycles in [11]. 1 To be specific, we considered Kähler four-cycles in Calabi-Yau fourfolds and Cayley four-cycles in Spin(7) manifolds, and new AdS 2 fixed point solutions were found. When the four-cycle is a product of two Riemann surfaces, an AdS 2 × Σ g 1 × Σ g 2 solution was found. By employing this solution, entropy of asymptotically AdS 6 black holes with AdS 2 × Σ g 1 × Σ g 2 horizon was calculated. Via the AdS/CFT correspondence [13], analogous to the AdS 4 black hole entropy [14,15], this entropy nicely matched with topologically twisted index of 5d U Sp(2N ) gauge theory on Σ g 1 × Σ g 2 × S 1 in the large N limit [16,17].
Pure F (4) gauged supergravity is a consistent truncation of massive type IIA supergravity [4] and type IIB supergravity [18,19,20]. Although it is not known that it is also a consistent truncation of ten-dimensional supergravity, one can couple vector multiplets to pure F (4) gauged supergravity [21]. In this theory, new fixed points and holographic RG flows were studied in [22,23,24]. See [25,26,27] also for other studies in this theory.
In this paper, we continue our study on D4-branes wrapped on supersymmetric four-cycles in matter coupled F (4) supergravity. We consider F (4) gauged supergravity coupled to three vector multiplets, and its U (1) × U (1)-invariant truncation first studied in [24]. We specify to D4-branes wrapped on a product of two Riemann surfaces, and derive supersymmetry equations. We obtain the AdS 2 solutions which were first found and used to calculate entropy of asymptotically AdS 6 black holes with AdS 2 × Σ g 1 × Σ g 2 horizon in [11] and also in [28].
We also study 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 a new AdS 2 solution. 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 section 3, we consider F (4) gauged supergravity coupled to three vector multiplets, and its U (1) × U (1)-invariant truncation. We derive the supersymmetry equations for D4-branes wrapped on a product of two Riemann surfaces, and obtain AdS 2 solutions. In section 4, we consider Kähler four-cycles in Calabi-Yau fourfolds and Cayley four-cycles in Spin(7) manifolds, and obtain a new AdS 2 solution. 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 [28] which has some overlap with the results presented here in section 3.

Matter coupled F (4) gauged supergravity
We review matter coupled F (4) gauged supergravity in six dimensions [21]. The gravity multiplet consists of e a µ , ψ A µ , A α µ , B µν , χ A , σ , 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 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 ∼ U Sp(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 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.
The scalar fields from the gravity and vector multiplets parametrize each factor of the coset manifold, respectively. The coset representative of the second factor is given by where we define The bosonic Lagrangian is given by and V = −e 2σ 1 36 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, (+ − − − −−).

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 multiplets: 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 [24], 3 D4-branes wrapped on two Riemann surfaces

The U (1)×U (1)-invariant truncation
We truncate the theory to the U (1)×U (1)-invariant sector, which was first considered in section 3.1 of [24]. 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

The supersymmetry equations
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 S 2 × S 2 background and for 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 field strengths, we set opposite signs of the gauge fields for 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 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. In order to obtain consistent equations, it is required to have Therefore, from now on, we set ϕ 1 to vanish.
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 satisfy the equations of motion. We present the equations of motion in appendix A.

The AdS 2 solutions
Now we will consider the N = 4 + theory, g 1 > 0, m > 0. When b 1 = b 2 = 0, we find an   3.19) which is the AdS 2 solution first found in [11]. 5 4 It is possible to have geometries like S 2 × H 2 for k 1 = +1 and k 2 = −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 When we consider for non-zero b 1 and b 2 , we obtain the AdS 2 solution in terms of the scalar field, ϕ 2 , first found in [28], for the H 2 × H 2 background with k = −1. When we consider the S 2 × S 2 background with k = +1, AdS 2 fixed point does not exist.

D4-branes wrapped on supersymmetric four-cycles 4.1 D4-branes wrapped on Kähler four-cycles in Calabi-Yau fourfolds
We consider the U (1)×U (1)-invariant truncation presented in section 3.1. We consider the metric, 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, [29,30]. 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 . 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.

D4-branes wrapped on Cayley four-cycles in Spin(7) manifolds 4.2.1 The SU (2)-invariant truncation
We considered matter coupled F (4) gauged supergravity coupled to three vector multiplets which has SU (2) R ×SU (2) gauge symmetry. In this section we truncate the theory to SU (2) diag ⊂ SU (2) R × SU (2) invariant sector, which was first considered in [22] and again in section 4.1 of [24]. There is one singlet under SU (2) diag which corresponds to Y 11 + Y 22 + Y 33 by the noncompact generators defined in (2.21). We exponentiate the non-compact generators and obtain the coset representative, L = e ϕ(Y 11 +Y 22 +Y 33 ) . (4.7) We also have a non-Abelian SU (2) gauge field and a two-form gauge potential in the SU (2) diaginvariant truncation. The Lagrangian of the truncation is given by where the scalar potential is (4.9)

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 and the hatted ones are the flat coordinates. In order to preserve supersymmetry for D4-branes wrapped on Cayley fourcycles in Spin(7) manifolds, we identify self-dual SU (2) + subgroup of the SO(4) isometry of the four-cycle, with the non-Abelian SU (2) gauge group, [29,30]. The self-duality is defined by 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 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, ϕ e −F = + (g 1 e σ cosh ϕ − g 2 e σ sinh ϕ) sinh(2ϕ) − 2λe −σ−2G (a 1 sinh ϕ − b 1 cosh ϕ) , Therefore, we conclude that the magnetic charges are 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).