Uplifting Supersymmetric Ads6 Black Holes to Type II Supergravity

Uplifting supersymmetric AdS6 black holes to type II supergravity

Minwoo Suh

Department of Physics, Kyungpook National University, Daegu 41566, Korea

Abstract

Employing uplift formulae, we uplift supersymmetric AdS6 black holes from F (4) gauged supergravity to massive type IIA and type IIB supergravity. In massive type IIA supergrav-

ity, we obtain supersymmetric AdS6 black holes asymptotic to the Brandhuber-Oz solution.

In type IIB supergravity, we obtain supersymmetric AdS6 black holes asymptotic to the non-Abelian T-dual of the Brandhuber-Oz solution. For the uplifted black hole solutions, we calculate the holographic entanglement entropy. In massive type IIA supergravity, it precisely matches the Bekenstein-Hawking entropy of the black hole solutions. arXiv:1908.09846v3 [hep-th] 17 Dec 2020

August, 2019 Contents

1 Introduction and conclusions 1

2 Supersymmetric AdS6 black holes of massive type IIA supergravity 4 2.1 Massive type IIA from F (4) gauged supergravity ...... 4

2.2 Supersymmetric AdS6 black holes ...... 6 2.3 Holographic entanglement entropy ...... 7

3 Supersymmetric AdS6 black holes of type IIB supergravity 9 3.1 Type IIB from F (4) gauged supergravity ...... 9 3.2 Uplifted solutions ...... 11

3.2.1 Supersymmetric AdS6 ﬁxed point ...... 13

3.2.2 Supersymmetric AdS6 black holes ...... 14 3.3 Holographic entanglement entropy ...... 15

A The supersymmetry equations of F (4) gauged supergravity 17

B The equations of motion of type II supergravity 17 B.1 Massive type IIA supergravity ...... 17 B.2 Type IIB supergravity ...... 18

C The non-Abelian T-dual of the Brandhuber-Oz solution 19

1 Introduction and conclusions

Recently we are observing progress of the AdS/CFT correspondence, [1], in higher dimensions, e.g., in AdS7/CFT6 and AdS6/CFT5. For supersymmetric AdS6 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 theories in [3, 4, 5, 6]. Later, it was shown to be the unique supersymmetric AdS6 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 AdS6 solutions of type IIB supergravity in [9, 10, 11], an inﬁnite family of

AdS6 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 AdS6 solutions was shown to be dual of five-brane web theories, e.g., 5d TN theories, [19, 20, 21, 22].

1 We can also study the AdS6/CFT5 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 arbitrary 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 AdS6 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. In this paper, we employ the uplift formulae in [25] and [29] to some solutions of F (4) gauged supergravity, and obtain solutions of massive type IIA and type IIB supergravity. For this purpose, we choose supersymmetric AdS6 black hole solutions from F (4) gauged supergravity in [31]. Analogous to the microscopic entropy counting, [32, 33], of supersymmetric AdS4 black holes, [34, 35, 36], the Bekenstein-Hawking entropy of AdS6 black holes, [31], was successfully counted by the topologically twisted index, [37], of 5d USp(2N) gauge theories, [4, 5, 6]. See also

[38], [39], and [40]. More general AdS6 black holes in matter coupled F (4) gauged supergravity were found in [41] and further studied in [42].

The supersymmetric AdS6 black hole solutions are asymptotic to the supersymmetric AdS6

ﬁxed point and have horizon of AdS2 × Σg1 × Σg2 , where g1 > 1 and g2 > 1 are genus of the

Riemann surfaces, [31, 41, 42]. The supersymmetric AdS6 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 AdS6 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 AdS6 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 AdS6 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 1 i i A = z3∓ z − . (1.1) ± 216 4 108 It was first obtained in [27] and rediscovered in [18]. For the solutions asymptotic to the infinite

2 family of AdS6 solutions, [12, 13, 14, 15], the holomorphic functions are given by

L 0 X l A± = A± + Z± ln(z − rl) , (1.2) l=1 where L−2 L l X X 1 Z± = σ (rl − sn) , (1.3) rl − rk n=1 k6=l and rl and sn are the positions of the zeros and poles on the Riemann surfaces, respectively, and 0 A± and σ are complex constants. However, the infinite family of AdS6 solutions is not able to be evaluated by elementary functions, and either the uplift of the black holes asymptotic to the infinite family of AdS6 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 1 consider iii) the infinite family of supersymmetric AdS6 solutions of type IIB supergravity. By employing the uplift formula to type IIB supergravity in [29], it will be interesting to uplift the supersymmetric AdS6 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]. For the uplifted black hole solutions, we calculate the holographic entanglement entropy, [44, 45]. For the solutions in massive type IIA supergravity, the holographic entanglement entropy precisely matches the Bekenstein-Hawking entropy of the black hole solutions obtained in [31, 39] which is microscopically counted by topologically twisted index of 5d USp(2N) gauge theories, [37]. For the solutions in type IIB supergravity, if the field theory dual of the non-Abelian T- dual of the Brandhuber-Oz solution could be identified, it would be interesting to compare the holographic entanglement entropy we obtain here with the microscopic results. 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 AdS6 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 AdS6 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.

1 Schematic geometry of black holes asymptotic to the inﬁnite family of supersymmetric AdS6 solutions of type IIB supergravity was considered for entropy counting in [43].

3 2 Supersymmetric AdS6 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], in the conventions of [25].2 There are the metric, the real scalar field, φ, an SU(2) gauge field, AI , I = 1, 2, 3, a U(1) gauge field, A, and a two-form gauge potential, B. Their field strengths are respectively defined by 1 F˜I = dAI + gIJK A ∧ A , (2) 2 J K 2 F˜ = dA + gB , (2) 3 ˜ F(3) = dB . (2.2)

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 AdS6

fixed point. At the fixed point, all the fields are vanishing except the AdS6 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,

2 1 cos2 ξ ds2 = X1/8 sin1/12 ξ ∆3/8ds2 + ∆3/8X2dξ2 + ds2 , (2.3) 6 g2 2g2 ∆5/8X S˜3 ∆1/4 eΦ = , (2.4) X5/4 sin5/6 ξ

2The 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

φ φ˜ − √ √ g˜ = 2g , X = e 2 2 = e 2 , 1 1 1 A˜I = AI , A˜ = A , B˜ = B , (2.1) µ 2 µ µ 2 µ µν 2 µν where the tilded ones are of [23].

4 and the two-, three-, and four-form ﬂuxes are given by, respectively,3

1 2/3 ˜ F(2) = √ sin ξF(2) , (2.5) 2 2/3 ˜ cos ξ ˜ H(3) = sin ξF(3) + F(2) ∧ dξ , (2.6) g sin1/3 ξ √ 2 U sin1/3 ξ cos3 ξ √ sin4/3 ξ cos4 ξ F = − dξ ∧ vol 3 − 2 dX ∧ vol 3 (4) 6 g3∆2 S˜ g3∆2X3 S˜ √ 4 1/3 4/3 X sin ξ cos ξ ˜ 1 sin ξ ˜ − 2 ∗6 F(3) ∧ dξ + √ ∗6 F(2) g 2 X2 1/3 4/3 2 1 sin ξ cos ξ ˜I I 1 sin ξ cos ξ ˜I J K + √ F ∧ h ∧ dξ − √ F ∧ h ∧ h IJK . (2.7) 2 g2 (2) 4 2 g2∆X3 (2) We employ the metric and the volume form on the gauged three-sphere by

3 2 X I I 2 dsS˜3 = σ − gA , I=1

volS˜3 = h1 ∧ h2 ∧ h3 , (2.8) where

hI = σI − gAI , (2.9) and σI , I = 1, 2, 3, are the SU(2) left-invariant one-forms which satisfy 1 dσI = − σJ ∧ σK . (2.10) 2 IJK A choice of the left-invariant one-forms is

1 σ = − sin α2 cos α3dα1 + sin α3dα2 , 2 σ = sin α2 sin α3dα1 + cos α3dα2 , 3 σ = cos α2dα1 + dα3 . (2.11)

We also deﬁned quantities,4

√φ X = e 2 , ∆ = X cos2 ξ + X−3 sin2 ξ , U = X−6 sin2 ξ − 3X2 cos2 ξ + 4X−2 cos2 ξ − 6X−2 . (2.12) 3 ˜ We suspect a typographical error in [25]. We changed the sign of ∗6F(2) term in F(4) from [25]. 4For the scalar ﬁeld, φ, we follow the normalization of [23] than [25].

5 2.2 Supersymmetric AdS6 black holes

We uplift the recently obtained supersymmetric AdS6 black holes of F (4) gauged supergravity in [31] to massive type IIA supergravity.5 We ﬁrst present the six-dimensional solutions of [31] in the conventions of [25]. The solutions are given by the metric,

2 2f(r) 2 2 2g1(r) 2 2 2 2g2(r) 2 2 2 ds6 = e −dt + dr + e dθ1 + sinh θ1dφ1 + e dθ2 + sinh θ1dφ2 , (2.13)

which is asymptotically AdS6 and has a horizon of AdS2 × Σg1 × Σg2 with g1 > 1 and g2 > 1. Only a component of the SU(2) gauge ﬁeld is non-trivial,

3 h i A = 2 a1 cosh θ1dφ1 + a2 cosh θ2dφ2 , (2.14) where the magnetic charges, a1 and a2, are constant. The twist condition on the magnetic charges is 1 k 1 k a = − , a = − , (2.15) 1 2 λg 2 2 λg where k = −1 for g1 > 1 and g2 > 1 and λ = ±1. There is also a two-form field, 2 √ B = 2 − a a e 2φ+2f−2g1−2g2 , (2.16) tr m2 1 2 and the three-form field strength of the two-form field vanishes identically. There is also a √φ non-trivial real scalar field, X(r) = e 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. We ﬁnd the AdS2 horizon for the Σg1 × Σg2 background with g1 > 1 and g2 > 1,

1/4 3/4 φ 1/4 1/4 f 2 1 g g 2 √ 2 m e = , e 1 = e 2 = , e 2 = . (2.17) (2g)3/4m1/4 r (2g)3/4m1/4 (2g)1/4 √ In order to have unit radius for AdS6, L = 1, we set m = 2. 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 detail here. The full geometry is interpolating between the asymptotic AdS6 ﬁxed point, which is the ˜3 1 1 Brandhuber-Oz solution, and the near horizon geometry, AdS2 × Σg1 × Σg2 × S × S , where S is parametrized by the coordinate, 0 ≤ ξ ≤ π/2. 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.

5 See [46] for uplift of another AdS2 solution in F (4) gauged supergravity to massive type IIA supergravity.

6 2.3 Holographic entanglement entropy

˜3 1 The black hole solution interpolates between the boundary, AdS6 × S × S , and the horizon, ˜3 1 AdS2 × Σg1 × Σg2 × S × S . The holographic entanglement entropy, [44, 45], on the boundary was calculated and matched to the five-sphere free energy of the dual field theory, which is 5d USp(2N) gauge theory, [49]. In this section, we calculate the holographic entanglement entropy on the horizon and find that it is precisely the Bekenstein-Hawking entropy of the black hole solutions. We begin by switching the solution to be in string frame,

string Φ/2 Einstein gMN = e gMN , (2.18) and rescale by the conventions of [2, 47, 48], in particular, by (2), (3), and (4) of [48]. The metric and the dilaton are given by6

∆1/8 4L2 1 cos2 ξ ds2 = Ω2 L2∆3/8ds2 + ∆3/8X2dξ2 + ds2 , (2.19) X1/8 sin1/3 ξ 6 9 4 ∆5/8X S˜3

4L2 e−2Φ = Ω−10 sin5/3 ξ , (2.20) 9 where we introduce

3 −1/6 38/3πN 8 − N Ω = m ,L4 = , m = f , (2.21) 2 22/3m1/3 2π and m is the Romans mass. The coupling constant, g, is related to the AdS6 radius, L, by 2 4L2 = . (2.22) g2 9

(10) We also collect some relations of the Newton’s gravitational constant, GN , and the string length, 7 ls, which we take to be ls = 1,

(10) 2 7/2 4 16πGN = 2κ10 , κ10 = 8π ls , (2.23) and, therefore, we obtain 1 2 = . (2.24) (10) (2π)6l8 4GN s 6We suspect that the metric in (4.1) of [49] is missing the factor of Ω2 and the expression of L4 in (4.3) of [49] is not correct. However, they conspire to cancel each other and the ﬁnal answer of entanglement entropy is correct. 7See, for example, footnote 3 in [2].

7 Now we readily calculate the holographic entanglement entropy on the horizon of the black holes. For the gravitational theory with a dilaton ﬁeld, the holographic entanglement entropy is given by, [50], Z 1 8 −2Φ√ SEE = (10) d xe γ , (2.25) 4GN where γ is determinant of the induced metric of co-dimension two entangling surface in string frame. For the metric in (2.19), the determinant of the induced metric is

√ 4π 4/3 16L4 2g1+2g2 −5/2 1/2 3 −4/3 γ = e X ∆ cos ξ sin ξ sinh θ1 sinh θ2 sin α2 . (2.26) 3(8 − Nf ) 81

˜3 1 We compute the holographic entanglement entropy on the horizon of AdS2 ×Σg1 ×Σg2 ×S ×S .

Unlike computing entanglement entropy on the asymptotically AdS6 boundary in [49], the minimal surface is just a point for AdS2, [51]. In this case, the holographic entanglement entropy computes the entanglement between CFTs on two boundaries of AdS2 space. Then, the holographic entanglement entropy is obtained to be

3N 5/2 Z π/2 2g1 2g2 3 1/3 SEE = e volΣ e volΣ volS3 dξ cos ξ sin ξ , 3p g1 g2 π 2(8 − N ) 0 √ f 5/2 8 2π(g1 − 1)(g2 − 1)N = p , (2.27) 5 8 − Nf where we use e2g1+2g2 = 2/33 from (2.17) and that the areas are given by

2 3 volΣg6=1 = 4π(g − 1) , volS = 2π . (2.28)

The holographic entanglement entropy in (2.27) precisely matches the Bekenstein-Hawking entropy of the black hole calculated in [31, 39]. It is microscopically counted by topologically twisted index of 5d USp(2N) gauge theories, [37]. See also [38].

8 3 Supersymmetric AdS6 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 AdS6 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, 1 2 A± = if ±f , (3.1) where f α = −pα + ikα , (3.2) and pα and kα are real functions. The SL(2, R) indices, α, β = 1, 2, are raised and lowered by 12 12 = = 1, α αβ β f = fβ , fα = f βα . (3.3) We also define quantities,8

α dγ = − pαdk , (3.4) 1 |dk| = ∂ k ∂αkβ , (3.5) 2 α β 2 4 γ σ δ ∆ = 3γ|dk| + X |dk|pγpδ∂σk ∂ p , (3.6) where X is the real scalar ﬁeld of F (4) gauged supergravity,

√φ X = e 2 . (3.7)

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, AA, A = 1, 2, 3, a U(1) gauge field, A4, and a two-form gauge potential, B. Their field strengths are respectively defined by

˜A A 3 ABC F = dA + √ AB ∧ AC , (2) 2 2R √ 2 F˜4 = dA4 + B, (2) R ˜ F(3) = dB . (3.8)

8The function, γ, was denoted by r in [28, 29].

9 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 AdS6

fixed point. At the fixed point, all the fields are vanishing except the AdS6 metric. The uplifted geometry is going to be of the form,

˜2 M1,5 × S × Σ , (3.9) where M1,5 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, yA, A = 1, 2, 3, their derivatives,

A A 3 ABC Dy = dy + √ AByC , (3.10) 2R and their Hodge dual coordinates,

˜ B C θA = ∗S˜2 DyA = ABC y Dy . (3.11)

The metric and the volume form of the gauged two-sphere are, respectively, given by

A B dsS˜2 = δABDy ⊗ Dy ,

1 A B C vol 2 = y Dy ∧ Dy . (3.12) S˜ 2 ABC 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

1/4 1/4 1/4 2 5/4 3/2 2 2 1/4 2 4 3 c6γ ∆ 2 4c6R γ |dk| X 2 4c6R ∆ α ds = 1/2 ds6 + 3/4 dsS˜2 + 3/4 3/4 1/2 2 dk ⊗dpα , |dk| 33/4∆ 3 γ |dk| X 4 1 X |dk| √ γ Hαβ = √ √ pαpβ + 3 γ∂γkα∂ pβ , 3∆ γ 2 4 α 4c6R α X γ|dk| γ β α C = − k + pγ∂βk ∂ p vol ˜2 (2) 3 ∆ S √ √ A α α 4 α α α A B C + 2 2c6RA ∧ (yAdk + k DyA) + 2 2c6RA ∧ dp + 4c6Bp − 3c6k ABC y A ∧ A , (3.13) where R and c6 are constants. The ﬁve-form ﬂux is obtained from

F(5) = F(2,3) + F(3,2) + F(4,1) , (3.14)

10 where √ 2 3 4 8 2c6R |dk| Ã ˜ X γ|dk| Ã α ˜4 α F(2,3) = F ∧ θA ∧ volΣ + yAF ∧ pαdp − F ∧ pαdk ∧ vol ˜2 , 3 (2) ∆ (2) (2) S 2 2 2 2 γ |dk| ˜ |dk| ˜ F(3,2) = 16c R F(3) ∧ vol ˜2 + ∗6 F(3) ∧ volΣ , 6 ∆ S X4 √ 2 2 Ã ˜4 α Ã α F(4,1) = 8 2c6RX −γ ∗6 F(2) ∧ DyA + ∗6F(2) ∧ pαdp + yA ∗6 F(2) ∧ pαdk , (3.15) and F(4,1) = ∗F(2,3).

3.2 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], 1 1 3 i i A± = z ∓ z − , (3.16) c6 216 4 108 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., 1 1 f 1 = − z3 − 2i , f 2 = − z , (3.17) 216ic6 4ic6 1 1 p1 = − z3 − z¯3 − 4i , p2 = − (z − z¯) , (3.18) 432ic6 8ic6 1 1 k1 = − z3 +z ¯3 , k2 = − (z +z ¯) , (3.19) 432c6 8c6 and 1 1 4 4 1 3 3 γ = 2 z − z¯ + zz¯ − zz¯ + z +z ¯ , (3.20) 864c6 8i 4i

1 2 2 |dk| = 2 z − z¯ . (3.21) 576ic6 For the gauged two-sphere, we introduce angular coordinates,

y1 = sin θ sin φ , y2 = sin θ cos φ , y3 = − cos θ . (3.22)

In terms of the angular coordinates, a Hodge dual coordinate and the volume form are

˜ 2 θ3 = − sin θDφ , (3.23)

volS˜2 = sin θdθDφ . (3.24)

11 For the Riemann surface, we also introduce angular coordinates, e.g., (4.3) of [27],

z = x1 + ix2 , (3.25) where 2˜g2 x1 = ρ , x2 = sin2/3 ξ . (3.26) 3 The metric and the volume form are given, respectively, by

4˜g4 cos2 ξ ds2 = dzdz¯ = dρ2 + dξ2 , (3.27) Σ 9 g˜4 sin2/3 ξ 2 1 α β 4˜g cos ξ volΣ = αβdx ∧ dx = dρ ∧ dξ . (3.28) 2 9 sin1/3 ξ For the later use we also give explicit expressions of pα and kα in terms of the angular coordinates on the Riemann surface,

1 1 kα = − 4˜g6ρ3 − 27˜g2ρ sin4/3 ξ , − g˜2ρ , 2916 6 1 1 1 dkα = − 4˜g6ρ2 − 9˜g2 sin4/3 ξ dρ + g˜2ρ cos ξ sin1/3 ξdξ, − g˜2dρ , 972 81 6 1 1 pα = 6 + 3 sin2 ξ − 4˜g4ρ2 sin2/3 ξ , − sin2/3 ξ , 648 4 1 1 4˜g4ρ2 1 cos ξ dpα = − g˜4ρ sin2/3 ξdρ + cos ξ 9 sin ξ − dξ, − dξ . (3.29) 81 972 sin1/3 ξ 6 sin1/3 ξ

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], by

(3g3m)1/4 g˜ = . (3.30) 2 For the solutions we consider, we will choose the parameters to be √ g = 3m = 2 2. (3.31)

Therefore, we have √ g˜ = 2 . (3.32) Accordingly, we also specify the constants, 3 c = 1 ,R = . (3.33) 6 2

12 3.2.1 Supersymmetric AdS6 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, 1/4 3/2 4 2 2 1/2 1/4 1/2 2 1 F cos ξ 2 2 4˜g X ρ tan ξ 2 F tan ξ 2 ds = √ ds + R ds 2 + ds , (3.34) 3 3 sin1/6 ξ AdS6 9 F 3/4 S X2 Σ where F is given by

F = 4˜g4ρ2 sec2 ξ sin2/3 ξ + X4 1 + 4˜g4ρ2 sec4 ξ sin8/3 ξ . (3.35)

Employing an SL(2, R) rotation,   0 1 U =   , (3.36) −1 0 to the dilaton-axion ﬁeld matrix, Hαβ, we obtain   2 −1 Φ |τ| C(0) U HU = e   , (3.37) C(0) 1 where −Φ τ = C(0) + ie . (3.38) The dilaton and axion ﬁelds are obtained to be, respectively, 27 cos2 ξ + X4 sin2 ξ eΦ = , F 1/2 sin1/3 ξ cos3 ξ 2˜g4 sin4/3 ξ 3 cos2 ξ + X4 2 + sin2 ξ C = ρ2 − . (3.39) (0) 81 54 cos2 ξ + X4 sin2 ξ The RR and NSNS two-form gauge potentials are, respectively, given by

1 C(2) = C(2) 2 2 g˜ R ρh 4 2 4/3 4 2 4 2 2 2 4/3 i = F 4˜g ρ − 27 sin ξ − 3X sec ξ 4˜g ρ (1 + 5 sin ξ) − 9 cos ξ sin ξ vol 2 , 37F S 2 2 4 2 2˜g R (F − X )ρ B = C = vol 2 . (3.40) (2) (2) 9F S 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.

13 3.2.2 Supersymmetric AdS6 black holes

We uplift the recently obtained supersymmetric AdS6 black holes of F (4) gauged supergravity in [31] to type IIB supergravity.9 We ﬁrst present the six-dimensional solutions of [31] in the conventions of [28, 29]. The solutions are given by the metric,

2 2f(r) 2 2 2g1(r) 2 2 2 2g2(r) 2 2 2 ds6 = e −dt + dr + e dθ1 + sinh θ1dφ1 + e dθ2 + sinh θ1dφ2 , (3.41)

which is asymptotically AdS6 and has a horizon of AdS2 × Σg1 × Σg2 with g1 > 1 and g2 > 1. Only a component of the SU(2) gauge ﬁeld is non-trivial,

3 h i A = 2 a1 cosh θ1dφ1 + a2 cosh θ2dφ2 , (3.42) where the magnetic charges, a1 and a2, are constant. The twist condition on the magnetic charges is k k a = − , a = − , (3.43) 1 λg 2 λg where k = −1 for g1 > 1 and g2 > 1 and λ ± 1. There is also a two-form field, 2 √ B = 2 − a a e 2φ+2f−2g1−2g2 , (3.44) tr m2 1 2 and the three-form field strength of the two-form field vanishes identically. There is also a √φ non-trivial real scalar field, X(r) = e 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 AdS6 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,

1/4 3/2 4 2 2 1/2 1/4 1/2 2 1 F cos ξ 2 2 4˜g X ρ tan ξ 2 F tan ξ 2 ds = √ ds + R ds ˜2 + ds , (3.45) 3 3 sin1/6 ξ 6 9 F 3/4 S X2 Σ where F is given in (3.35). The dilaton and axion ﬁelds are, respectively,

27 cos2 ξ + X4 sin2 ξ eΦ = , F 1/2 sin1/3 ξ cos3 ξ 2˜g4 sin4/3 ξ 3 cos2 ξ + X4 2 + sin2 ξ C = ρ2 − . (3.46) (0) 81 54 cos2 ξ + X4 sin2 ξ

9 See [52] for more AdS2 solutions of type IIB supergravity.

14 The RR and NSNS two-form gauge potentials are, respectively, given by

1 C(2) = C(2) 2 2 g˜ c6R ρh 4 2 4/3 4 2 4 2 2 2 4/3 i = 7 F 4˜g ρ − 27 sin ξ − 3X sec ξ 2˜g ρ (1 + 5 sin ξ) − 9 cos ξ sin ξ volS˜2 3 F √ 3 1 3 1 1 + 2 2c6R A ∧ y3dk + A ∧ k Dy3 + 4c6Bp , 2 2 4 √ 2 2˜g c6R (F − X )ρ 3 2 3 2 2 B = C = vol 2 + 2 2c R A ∧ y dk + A ∧ k Dy + 4c Bp . (2) (2) 9F S˜ 6 3 3 6 (3.47)

There is also a non-trivial self-dual ﬁve-form ﬂux given by

F(5) = F(2,3) + F(3,2) + F(4,1) , (3.48) where √ 8 2c2R3 g˜2 F = 6 ρ sin2/3 ξ dA3 ∧ yBDyC ∧ vol (2,3) 3 216 3BC Σ √ 8 6 4 ! ! 2 3 X ρ 3 α 2 α + y dA ∧ p dp − B ∧ p dk ∧ vol 2 , F cos4 ξ 3 α R α S˜ √ ! √ 2 F = 8 2c2RX2 −γ ∗ dA3 ∧ Dy + ∗ B ∧ p dpα + y ∗ dA3 ∧ p dkα , (4,1) 6 6 3 R 6 α 3 6 α

F(3,2) = 0 . (3.49)

The full geometry is interpolating between the asymptotic AdS6 ﬁxed point, which is the non-

Abelian T-dual of the Brandhuber-Oz solution, and the near horizon geometry, AdS2×Σg1 ×Σg2 × 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.

3.3 Holographic entanglement entropy

˜2 The black hole solution interpolates between the boundary, AdS6 × S × Σ, and the horizon, ˜2 AdS2 × Σg1 × Σg2 × S × Σ. The holographic entanglement entropy, [44, 45], on the boundary was calculated in [17]. In this section, we calculate the holographic entanglement entropy on the horizon.

15 For the gravitational theory with a dilaton ﬁeld, the holographic entanglement entropy is given by, [50], Z 1 8 −2Φ√ SEE = (10) d xe γ , (3.50) 4GN where γ is determinant of the induced metric of co-dimension two entangling surface in string frame. We compute the holographic entanglement entropy on the horizon of AdS2 × Σg1 × Σg2 × ˜2 S × Σ. Unlike computing entanglement entropy on the asymptotically AdS6 boundary in [49], the minimal surface is just a point for AdS2, [51]. In this case, the holographic entanglement entropy computes the entanglement between CFTs on two boundaries of AdS2 space. Then, the holographic entanglement entropy is obtained to be

1 Z 24 S = d8x c4R4g˜6e2g1+2g2 ρ2 cos2 ξ sin1/3 ξ sinh θ sinh θ sin θ EE (10) 310 6 1 2 4GN 1 24 Z R Z π/2 4 4 6 2g1 2g2 2 2 1/3 = c R g˜ e volΣ e volΣ vol ˜2 dρρ dξ cos ξ sin ξ (10) 310 6 g1 g2 S 4GN 0 0 24 = c4R7g˜6(g − 1)(g − 1) , (3.51) 3125π3 6 1 2 where we use e2g1+2g2 = 2/33 from (2.17). If the ﬁeld theory dual of the non-Abelian T-dual of the Brandhuber-Oz solution could be identiﬁed, it would be interesting to compare the holographic entanglement entropy we obtain here with the microscopic results.

Acknowledgements We would like to thank Emanuel Malek for explaining his work, and for very important comments on a preprint which corrected an error and improved the preprint. We also thank Tatsuma Nishioka for helpful communications. This research was supported by the National Research Foundation of Korea under the grant NRF-2019R1I1A1A01060811.

16 A The supersymmetry equations of F (4) gauged supergravity

We present the supersymmetry equations of AdS6 black holes from F (4) gauged supergravity, [31], in the conventions of [23, 31],

1 √φ √3φ λ √φ 3 √φ 0 −f − − −2g1 −2g2 −2g1−2g2 f e = − √ ge 2 + me 2 − √ e 2 a1e + a2e − √ a1a2e 2 , 4 2 2 2 2m

1 √φ √3φ λ √φ 1 √φ 0 −f − − −2g1 −2g2 −2g1−2g2 g e = − √ ge 2 + me 2 + √ e 2 3a1e − a2e + √ a1a2e 2 , 1 4 2 2 2 2m

1 √φ √3φ λ √φ 1 √φ 0 −f − − −2g2 −2g1 −2g1−2g2 g e = − √ ge 2 + me 2 + √ e 2 3a2e − a1e + √ a1a2e 2 , 2 4 2 2 2 2m

1 1 √φ √3φ λ √φ 1 √φ 0 −f − − −2g1 −2g2 −2g1−2g2 √ φ e = √ ge 2 − 3me 2 + √ e 2 a1e + a2e − √ a1a2e 2 . 2 4 2 2 2 2m (A.1)

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 string Φ/2 Einstein gMN = e gMN . (B.1) We also deﬁne 1 1 |F |2 = F F M1···Mp , |F |2 = F F M1···Mp−1 . (B.2) (p) p! (p)M1···Mp (p) (p) MN (p − 1)! (p)MM1···Mp−1 (p)N

B.1 Massive type IIA supergravity

The ﬁeld content of massive type IIA supergravity is the metric, gMN , 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, [53]. We follow the conventions of [25]. The ﬂuxes are deﬁned by m F = dC + mB ,H = dB ,F = dC + C ∧ dB + B ∧ B , (B.1) (2) (1) (2) (3) (2) (4) (3) (1) (2) 2 (2) (2) 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 √ 2 m = g . (B.2) 3

17 The equations of motion are given by 1 1 2 5 Φ 1 3 Φ 2 1 2 R − ∂ Φ∂ Φ − m e 2 g − e 2 |F | − |F | g MN 2 M N 16 MN 2 (2) MN 8 (2) MN 1 −Φ 2 1 2 1 1 Φ 2 3 2 − e |H | − |H | g − e 2 |F | − |F | g = 0 , (B.3) 2 (3) MN 4 (3) MN 2 (4) MN 8 (4) MN

1 √ MN 5 2 5 Φ 3 3 Φ 2 1 −Φ 2 1 1 Φ 2 √ ∂ −gg ∂ Φ − m e 2 − e 2 |F | + e |H | − e 2 |F | = 0 , (B.4) −g M N 4 4 (2) 2 (3) 4 (4) 1 d e−Φ ∗ H + F ∧ F + me3Φ/2 ∗ F + eΦ/2 ∗ F ∧ F = 0 (3) 2 (4) (4) (2) (4) (2) 3Φ/2 Φ/2 d e ∗ F(2) + e ∗ F(4) ∧ H(3) , Φ/2 d e ∗ F(4) + H(3) ∧ F(4) = 0 , (B.5) and the Bianchi identities are

dF(2) = mH(3) , dH(3) = 0 , dF(4) = F(2) ∧ H(3) . (B.6)

B.2 Type IIB supergravity

The ﬁeld content of type IIB supergravity is the metric, gMN , 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, [54, 55]. We follow the conventions of [56]. The ﬂuxes are

F(1) = dC(0) ,H(3) = dB(2) ,F(3) = dC(2) − C(0)H(3) F(5) = F(5) + ∗10F(5) , (B.1) where 1 F = dC − C ∧ H − B ∧ dC . (B.2) (5) (4) 2 (2) (3) (2) (2) The equations of motion are given by 1 1 1 1 R − ∂ Φ∂ Φ − e2Φ|F |2 − eΦ |F |2 − |F |2g MN 2 M N 2 (1) MN 2 (3) MN 4 (3) MN 1 1 1 1 − e−Φ |H |2 − |H |2g − e−Φ |F |2 − |F |2g = 0 , (B.3) 2 (3) MN 4 (3) MN 4 (5) MN 2 (5) MN 1 √ 1 1 √ ∂ −ggMN ∂ Φ − e2Φ|F |2 + e−Φ|H |2 − eΦ|F |2 = 0 , (B.4) −g M N (1) 2 (3) 2 (3)

−Φ Φ d e ∗ H(3) − F(3) ∧ F(5) − e F(1) ∧ ∗F(3) = 0 , 2Φ Φ d e ∗ F(1) + e H(3) ∧ ∗F(3) = 0 , Φ d e ∗ F(3) + H(3) ∧ F(5) = 0 , (B.5) and the Bianchi identities are

dF(1) = 0 , dH(3) = 0 , dF(3) − H(3) ∧ F(1) = 0 dF(5) − H(3) ∧ F(3) = 0 . (B.6)

18 C The non-Abelian T-dual of the Brandhuber-Oz solution

We present the non-Abelian T-dual of the Brandhuber-Oz solution, [8], as it appeared in [26].10 The solution is consist of the metric, 2 2A 2 −Φ/2 1/2 −1/3 2 ρ e 2 −2A 2 2 −2 1/2 −1/3 2 ds = e ∆˜ sin ξds + ds 2 + e dρ + X ∆˜ sin ξdξ , AdS6 ρ2 + e4A S g˜2 (C.1) the dilaton and axion ﬁelds, respectively, X∆˜ 1/4 eΦ = , eApρ2 + e4A sin5/6 ξ

F(1) = − G1 +mρdρ ˜ , (C.2) and the NSNS two-form gauge potential and the RR three-form flux are, respectively, ρ3 B = − vol 2 , (2) ρ2 + e4A S 2 ρ 4A F = ρG +me ˜ dρ ∧ vol 2 , (C.3) (3) ρ2 + e4A 1 S where we define ∆˜ = cos2 ξ + X4 sin2 ξ , X cos ξ eA = √ , 2˜g∆˜ 1/4 sin1/6 ξ 1 1 G = − UX2∆˜ −2 sin1/3 ξ cos3 ξdξ − X3∆˜ −2 sin4/3 ξ cos4 ξdX , 1 12 4 U = X6 sin2 ξ − 3X−2 cos2 ξ + 4X2 cos2 ξ − 6X2 . (C.4) The five-form flux vanishes identically. In order to match the non-Abelian T-dual of the Brandhuber-Oz solution with the uplifted

AdS6 fixed point in section 3.2.1, the uplifted solution should be rescaled by 1 1 g˜ = g , eΦ˜ =m ˜ 3eΦ , C˜ = C , µν m˜ 3/2 µν (0) m˜ 3 (0) 1 1 C˜ = C , B˜ = B , F˜ = F (C.5) (2) m˜ 3 (2) (2) (2) (5) m˜ 3 (5) and take, e.g., in [26], √ 2 2 m˜ = g˜ = , (C.6) 3 3 where the tilded ones are the rescaled fields. 10The scalar field in [26] is related to the scalar field in this work by Xthere = 1/Xhere. We use our X in this appendix. We also rescaled the metric in string frame in [26] to the metric in Einstein frame.

19 References

[1] J. M. Maldacena, The large N limit of superconformal ﬁeld theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep- th/9711200].

[2] A. Brandhuber and Y. Oz, The D-4 - D-8 brane system and ﬁve-dimensional ﬁxed points, Phys. Lett. B 460, 307 (1999) [hep-th/9905148].

[3] S. Ferrara, A. Kehagias, H. Partouche and A. Zaﬀaroni, AdS(6) interpretation of 5-D superconformal ﬁeld theories, Phys. Lett. B 431, 57 (1998) [hep-th/9804006].

[4] N. Seiberg, Five-dimensional SUSY ﬁeld theories, nontrivial ﬁxed points and string dynam- ics, Phys. Lett. B 388, 753 (1996) [hep-th/9608111].

[5] D. R. Morrison and N. Seiberg, Extremal transitions and ﬁve-dimensional supersymmetric ﬁeld theories, Nucl. Phys. B 483, 229 (1997) [hep-th/9609070].

[6] K. A. Intriligator, D. R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497, 56 (1997) [hep- th/9702198].

[7] A. Passias, A note on supersymmetric AdS6 solutions of massive type IIA supergravity, JHEP 1301, 113 (2013) [arXiv:1209.3267 [hep-th]].

[8] Y. Lozano, E. O Colgain, D. Rodriguez-Gomez and K. Sfetsos, Supersymmetric AdS6 via T Duality, Phys. Rev. Lett. 110, no. 23, 231601 (2013) [arXiv:1212.1043 [hep-th]].

[9] F. Apruzzi, M. Fazzi, A. Passias, D. Rosa and A. Tomasiello, AdS6 solutions of type II supergravity, JHEP 1411, 099 (2014) Erratum: [JHEP 1505, 012 (2015)] [arXiv:1406.0852 [hep-th]].

[10] H. Kim, N. Kim and M. Suh, Supersymmetric AdS6 Solutions of Type IIB Supergravity, Eur. Phys. J. C 75, no. 10, 484 (2015) [arXiv:1506.05480 [hep-th]].

[11] H. Kim and N. Kim, Comments on the symmetry of AdS6 solutions in string/M-theory and Killing spinor equations, Phys. Lett. B 760, 780 (2016) [arXiv:1604.07987 [hep-th]].

2 [12] E. D’Hoker, M. Gutperle, A. Karch and C. F. Uhlemann, Warped AdS6 × S in Type IIB supergravity I: Local solutions, JHEP 1608, 046 (2016) [arXiv:1606.01254 [hep-th]].

20 [13] E. D’Hoker, M. Gutperle and C. F. Uhlemann, Holographic duals for ﬁve-dimensional superconformal quantum ﬁeld theories, Phys. Rev. Lett. 118, no. 10, 101601 (2017) [arXiv:1611.09411 [hep-th]].

2 [14] E. D’Hoker, M. Gutperle and C. F. Uhlemann, Warped AdS6 × S in Type IIB supergravity II: Global solutions and ﬁve-brane webs, JHEP 1705, 131 (2017) [arXiv:1703.08186 [hep-th]].

2 [15] E. D’Hoker, M. Gutperle and C. F. Uhlemann, Warped AdS6 × S in Type IIB supergravity III: Global solutions with seven-branes, JHEP 1711, 200 (2017) [arXiv:1706.00433 [hep-th]].

[16] F. Apruzzi, J. C. Geipel, A. Legramandi, N. T. Macpherson and M. Zagermann, 2 Minkowski4 × S solutions of IIB supergravity, Fortsch. Phys. 66, no. 3, 1800006 (2018) [arXiv:1801.00800 [hep-th]].

[17] Y. Lozano, E. O Colgain and D. Rodriguez-Gomez, Hints of 5d Fixed Point Theories from Non-Abelian T-duality, JHEP 1405, 009 (2014) [arXiv:1311.4842 [hep-th]].

2 [18] Y. Lozano, N. T. Macpherson and J. Montero, AdS6 T-duals and type IIB AdS6× S geome- tries with 7-branes, JHEP 1901, 116 (2019) [arXiv:1810.08093 [hep-th]].

[19] M. Gutperle, C. Marasinou, A. Trivella and C. F. Uhlemann, Entanglement entropy vs. free energy in IIB supergravity duals for 5d SCFTs, JHEP 1709, 125 (2017) [arXiv:1705.01561 [hep-th]].

[20] O. Bergman, D. Rodriguez-Gomez and C. F. Uhlemann, Testing AdS6/CFT5 in Type IIB with stringy operators, JHEP 1808, 127 (2018) [arXiv:1806.07898 [hep-th]].

[21] M. Fluder and C. F. Uhlemann, Precision Test of AdS6/CFT5 in Type IIB String Theory, Phys. Rev. Lett. 121, no. 17, 171603 (2018) [arXiv:1806.08374 [hep-th]].

[22] C. F. Uhlemann, Exact results for 5d SCFTs of long quiver type, JHEP 11, 072 (2019) [arXiv:1909.01369 [hep-th]].

[23] L. J. Romans, The F(4) Gauged Supergravity in Six-dimensions, Nucl. Phys. B 269, 691 (1986).

[24] L. Andrianopoli, R. D’Auria and S. Vaula, Matter coupled F(4) gauged supergravity La- grangian, JHEP 0105, 065 (2001) [hep-th/0104155].

[25] M. Cvetic, H. Lu and C. N. Pope, Gauged six-dimensional supergravity from massive type IIA, Phys. Rev. Lett. 83, 5226 (1999) [hep-th/9906221].

21 [26] J. Jeong, O. Kelekci and E. O Colgain, An alternative IIB embedding of F(4) gauged supergravity, JHEP 1305, 079 (2013) [arXiv:1302.2105 [hep-th]].

[27] J. Hong, J. T. Liu and D. R. Mayerson, Gauged Six-Dimensional Supergravity from Warped IIB Reductions, JHEP 1809, 140 (2018) [arXiv:1808.04301 [hep-th]].

[28] E. Malek, H. Samtleben and V. Vall Camell, Supersymmetric AdS7 and AdS6 vacua and their minimal consistent truncations from exceptional ﬁeld theory, Phys. Lett. B 786, 171 (2018) [arXiv:1808.05597 [hep-th]].

[29] E. Malek, H. Samtleben and V. Vall Camell, Supersymmetric AdS7 and AdS6 vacua and their consistent truncations with vector multiplets, JHEP 1904, 088 (2019) [arXiv:1901.11039 [hep-th]].

[30] E. Malek, Half-Maximal Supersymmetry from Exceptional Field Theory, Fortsch. Phys. 65, no. 10-11, 1700061 (2017) [arXiv:1707.00714 [hep-th]].

[31] M. Suh, Supersymmetric AdS6 black holes from F(4) gauged supergravity, JHEP 1901, 035 (2019) [arXiv:1809.03517 [hep-th]].

[32] F. Benini and A. Zaﬀaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 1507, 127 (2015) [arXiv:1504.03698 [hep-th]].

[33] F. Benini, K. Hristov and A. Zaﬀaroni, Black hole microstates in AdS4 from supersymmetric localization, JHEP 1605, 054 (2016) [arXiv:1511.04085 [hep-th]].

[34] S. L. Cacciatori and D. Klemm, Supersymmetric AdS(4) black holes and attractors, JHEP 1001, 085 (2010) [arXiv:0911.4926 [hep-th]].

[35] G. Dall’Agata and A. Gnecchi, Flow equations and attractors for black holes in N = 2 U(1) gauged supergravity, JHEP 1103, 037 (2011) [arXiv:1012.3756 [hep-th]].

[36] K. Hristov and S. Vandoren, Static supersymmetric black holes in AdS4 with spherical symmetry, JHEP 1104, 047 (2011) [arXiv:1012.4314 [hep-th]].

[37] S. M. Hosseini, I. Yaakov and A. Zaﬀaroni, Topologically twisted indices in ﬁve dimensions and holography, JHEP 1811, 119 (2018) [arXiv:1808.06626 [hep-th]].

[38] P. M. Crichigno, D. Jain and B. Willett, 5d Partition Functions with A Twist, JHEP 1811, 058 (2018) [arXiv:1808.06744 [hep-th]].

[39] M. Suh, On-shell action and the Bekenstein-Hawking entropy of supersymmetric black holes

in AdS6, arXiv:1812.10491 [hep-th].

22 [40] N. Kim and M. Shim, Wrapped Brane Solutions in Romans F (4) Gauged Supergravity, Nucl. Phys. B 951, 114882 (2020) [arXiv:1909.01534 [hep-th]].

[41] S. M. Hosseini, K. Hristov, A. Passias and A. Zaﬀaroni, 6D attractors and black hole microstates, JHEP 1812, 001 (2018) [arXiv:1809.10685 [hep-th]].

[42] M. Suh, Supersymmetric AdS6 black holes from matter coupled F (4) gauged supergravity, JHEP 1902, 108 (2019) [arXiv:1810.00675 [hep-th]].

[43] M. Fluder, S. M. Hosseini and C. F. Uhlemann, Black hole microstate counting in Type IIB from 5d SCFTs, JHEP 1905, 134 (2019) [arXiv:1902.05074 [hep-th]].

[44] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006) [arXiv:hep-th/0603001 [hep-th]].

[45] S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08, 045 (2006) [arXiv:hep-th/0605073 [hep-th]].

[46] G. Dibitetto and N. Petri, AdS2 solutions and their massive IIA origin, JHEP 1905, 107 (2019) [arXiv:1811.11572 [hep-th]].

[47] O. Bergman and D. Rodriguez-Gomez, 5d quivers and their AdS(6) duals, JHEP 07, 171 (2012) [arXiv:1206.3503 [hep-th]].

[48] O. Bergman and D. Rodriguez-Gomez, Probing the Higgs branch of 5d ﬁxed point theories with dual giant gravitons in AdS(6), JHEP 12, 047 (2012) [arXiv:1210.0589 [hep-th]].

[49] D. L. Jafferis and S. S. Pufu, Exact results for five-dimensional superconformal field theories with gravity duals, JHEP 05, 032 (2014) [arXiv:1207.4359 [hep-th]].

[50] I. R. Klebanov, D. Kutasov and A. Murugan, Entanglement as a probe of conﬁnement, Nucl. Phys. B 796, 274-293 (2008) [arXiv:0709.2140 [hep-th]].

[51] T. Azeyanagi, T. Nishioka and T. Takayanagi, Near Extremal Black Hole Entropy as En- tanglement Entropy via AdS(2)/CFT(1), Phys. Rev. D 77, 064005 (2008) [arXiv:0710.2956 [hep-th]].

6 [52] D. Corbino, E. D’Hoker, J. Kaidi and C. F. Uhlemann, Global half-BPS AdS2 ×S solutions in Type IIB, JHEP 1903, 039 (2019) [arXiv:1812.10206 [hep-th]].

[53] L. J. Romans, Massive N=2a Supergravity in Ten-Dimensions, Phys. Lett. B 169, 374 (1986).

23 [54] J. H. Schwarz, Covariant Field Equations of Chiral N=2 D=10 Supergravity, Nucl. Phys. B 226, 269 (1983).

[55] P. S. Howe and P. C. West, The Complete N=2, D=10 Supergravity, Nucl. Phys. B 238, 181 (1984).

[56] N. Bobev, F. F. Gautason and J. Van Muiden, Precision Holography for N = 2∗ on S4 from type IIB Supergravity, JHEP 1804, 148 (2018) [arXiv:1802.09539 [hep-th]].