Supersymmetric Ad S 6 black holes from matter coupled F(4) gauged supergravity
SSupersymmetric
AdS black holesfrom matter coupled F (4) gauged supergravity Minwoo Suh
Department of Physics, Kyungpook National University, Daegu 41566, Korea [email protected]
Abstract
In matter coupled F (4) gauged supergravity in six dimensions, we study supersymmetric AdS black holes with various horizon geometries. We find new AdS × Σ g × Σ g horizonswith g > g >
1, and present the black hole solution numerically. The full black holeis an interpolating geometry between the asymptotically
AdS boundary and the AdS × Σ g × Σ g horizon. We also find black holes with horizons of K¨ahler four-cycles in Calabi-Yaufourfolds and Cayley four-cycles in Spin (7) manifolds.
October, 2018 a r X i v : . [ h e p - t h ] F e b ontents F (4) gauged supergravity 3 F (4) gauged supergravity coupled to three vector multiplets . . . . . . . . . . . . 5 AdS × Σ g × Σ g horizon 6 U (1) × U (1)-invariant truncation . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 The supersymmetry equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 The AdS solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Numerical black hole solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Spin (7) manifolds . . . . . . . . . . . . . . . . . . . . . . . . 144.2.1 The SU (2)-invariant truncation . . . . . . . . . . . . . . . . . . . . . . . . 144.2.2 The supersymmetry equations . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.3 The AdS solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2.4 Numerical black hole solutions . . . . . . . . . . . . . . . . . . . . . . . . . 18 A The equations of motion 20 In F (4) gauged supergravity in six dimensions [1], there is a unique supersymmetric fixed pointwhich is dual to 5d superconformal U Sp (2 N ) 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 fixedpoint uplifts to AdS × w S near-horizon geometry of the D4-D8 brane system [6, 7]. In thespirit of [8], supergravity solutions of wrapped D4-branes on various supersymmetric cycles werestudied in F (4) gauged supergravity. D4-branes wrapped on two- and three-cycles were studiedin [9]. They found AdS and AdS fixed point solutions. See [10] for more recent results.Recently, by considering D4-branes wrapped on supersymmetric four-cycles, we found super-symmetric AdS black holes of F (4) gauged supergravity in [11]. To be specific, we found thefull black hole solutions which is an interpolating geometry between the asymptotically
AdS boundary and the AdS × H × H horizon. Via the AdS/CFT correspondence, [13], analogous Previously, D4-branes wrapped on supersymmetric four-cycles were studied in [12], but the equations andsolutions were incorrect.
1o the
AdS black hole cases in [14, 15, 16], the Bekenstein-Hawking entropy of the black holesnicely matched with the topologically twisted index of 5d U Sp (2 N ) gauge theory on Σ g × Σ g × S in the large N limit [17, 18]. We also considered black hole horizons of K¨ahler four-cycles inCalabi-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 itis also a consistent truncation of ten-dimensional supergravity, one can couple vector multipletsto pure F (4) gauged supergravity [22]. In this theory, new fixed points and holographic RG flowswere 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 su-persymmetric AdS black holes. We consider F (4) gauged supergravity coupled to three vectormultiplets, and its U (1) × U (1)-invariant truncation first considered in [25]. We consider blackhole solutions with a horizon which is a product of two Riemann surfaces, AdS × Σ g × Σ g . Wederive supersymmetry equations and obtain AdS solutions which was first found in [29]. The AdS horizon exists only for the H × H background, and not for the H × S or S × S backgrounds. We present the full black hole solutions numerically.We also consider black holes with horizons of K¨ahler four-cycles in Calabi-Yau fourfoldsand Cayley four-cycles in Spin (7) manifolds. For Cayley four-cycles in
Spin (7) manifolds, weconsider the SU (2) diag -invariant truncation of F (4) gauged supergravity coupled to three vectormultiplets. We find new AdS horizons. It will be interesting to have a field theory interpretationof this AdS solution.In section 2, we review matter coupled F (4) gauged supergravity in six dimensions. In section3, 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 aproduct of two Riemann surfaces. In section 4, we consider supersymmetric black hole solutionswith horizons of K¨ahler 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)-invarianttruncation. Note added:
In the final stage of this work, we became aware of [29] which has some overlapwith the results presented here in section 3 and section 4.1.2
Matter coupled F (4) gauged supergravity We review matter coupled F (4) gauged supergravity in six dimensions [22]. The gravity multipletconsists of (cid:0) e aµ , ψ Aµ , A αµ , B µν , χ A , σ (cid:1) , (2.1)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 αµ , α = 0 , , ,
3, can be used to gaugethe SU (2) R × U (1) gauge symmetry. The vector multiplet consists of( A µ , λ A , ϕ α ) I , (2.2)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-Majoranaspinors and transform in the fundamental representation of the SU (2) R ∼ U Sp (2) R R-symmetrydenoted by indices,
A, B = 1 ,
2. We denote the coupling constants of gauge fields from the gravityand vector multiplets by g and g , respectively, and the mass parameter of the two-form gaugepotential 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 : N = 4 + ( g > m > N = 4 − ( g < m > N = 4 g ( g > m = 0), N = 4 m ( g = 0, m > N = 4 ( g = 0, m = 0). The N = 4 + theory admits a supersymmetric AdS fixed point when g = 3 m . At thesupersymmetric AdS fixed point, all the fields are vanishing except the AdS metric.The scalar fields from the gravity and vector multiplets parametrize each factor of the cosetmanifold, M = SO (1 , × SO (4 , n ) SO (4) × SO ( n ) , (2.3)respectively. The coset representative of the second factor is given by P I α = ( P I , P I r ) = (Ω I , Ω I r ) , (2.4)where we define Ω Λ Σ = ( L − ) Λ Π ∇ L Π Σ , ∇ L Λ Σ = dL Λ Σ − f Γ Λ Π A Γ L Π Σ , (2.5)and Λ = ( α, I ) or Λ = (0 , r, I ) with r = 1 , ,
3. The indices, Λ, Σ, . . . , are raised and loweredby η ΛΣ = diag(1 , , , , − , . . . , − . (2.6)3he bosonic Lagrangian is given by e − L = − R + ∂ µ σ∂ µ σ − P Iαµ P Iαµ − V − e − σ N ΛΣ F Λ µν F Σ µν + 364 e σ H µνλ H µνλ − (cid:15) µνρστκ B µν (cid:18) η ΛΣ F Λ ρσ F Σ τκ + mB ρσ F τκ + 13 m B ρσ B τκ (cid:19) , (2.7)and V = − e σ (cid:18) A + 14 B t B t + 14 C I t C It + D I t D It (cid:19) + m e − σ N − me − σ (cid:18) AL − B t L t (cid:19) , (2.8)where we define N ΛΣ = L ( L − ) + L Λ r ( L − ) r Σ − L Λ I ( L − ) I Σ , (2.9)and A = (cid:15) rst K rst , B t = (cid:15) trs K rs , C I t = (cid:15) trs K rIs , D It = K It . (2.10)The field strengths are collectively defined by F Λ µν = ∂ [ µ A Λ ν ] − mδ Λ0 B µν , H µνλ = ∂ [ µ B νλ ] . (2.11)The supersymmetry variations of the fermionic fields are given by δψ µA = D µ (cid:15) A + i (cid:2) Ae σ + 6 me − σ ( L − ) (cid:3) γ µ (cid:15) A − i (cid:2) B t e σ − me − σ ( L − ) t (cid:3) γ γ µ σ tAB (cid:15) B + 116 e − σ (cid:2) ( L − ) (cid:0) F Λ νλ − mB νλ δ Λ0 (cid:1) γ (cid:15) AB − ( L − ) r Λ F Λ νλ σ rAB (cid:3) (cid:0) γ µ νλ − δ νµ γ λ (cid:1) (cid:15) B + i e σ H νλρ γ γ νλρ γ µ (cid:15) A , (2.12) δχ A = i γ µ ∂ µ σ(cid:15) A + 124 (cid:2) Ae σ − me − σ ( L − ) (cid:3) (cid:15) A + 18 (cid:2) B t e σ + 6 me − σ ( L − ) t (cid:3) γ σ tAB (cid:15) B + i e − σ (cid:2) ( L − ) (cid:0) F Λ µν − mB µν δ Λ0 (cid:1) γ (cid:15) AB + ( L − ) r Λ F Λ µν σ rAB (cid:3) γ µν (cid:15) B + 132 e σ H µνλ γ γ µνλ (cid:15) A , (2.13) δλ IA = iP Iµr γ µ σ rAB (cid:15) B − iP Iµ γ γ µ (cid:15) A − e σ (cid:2) C I t − iγ D I t (cid:3) σ tAB (cid:15) B − me − σ ( L − ) I γ (cid:15) A + i e − σ ( L − ) I Λ F Λ µν γ µν (cid:15) A , (2.14)where we define K rst = g (cid:15) lmn L l r ( L − ) s m L n t + g C IJK L I r ( L − ) s J L K t ,K rs = g (cid:15) lmn L l r ( L − ) s m L n + g C IJK L I r ( L − ) s J L K ,K rIt = g (cid:15) lmn L l r ( L − ) I m L n t + g C LJK L L r ( L − ) I J L K t ,K It = g (cid:15) lmn L l ( L − ) I m L n t + g C LJK L L ( L − ) I J L K t , (2.15) The numerical factors are due to the unusual convention of form fields, e.g. , ω = ω µν dx µ dx ν , in [22]. There is a missing γ in (4.32) and also in the m dependent term in (4.35) of [22]. D µ = ∂ µ (cid:15) A + 14 ω µ ab γ ab (cid:15) A + i (cid:18) g (cid:15) rst Ω µst + iγ Ω µr (cid:19) σ rAB (cid:15) B . (2.16)The Pauli matrices, σ tA B , satisfy the relations, σ tAB = σ tC B (cid:15) CA , (2.17)and σ tAB = σ t ( AB ) . We also define the chirality matrix by γ = iγ γ γ γ γ γ , (2.18)with γ = − γ T = − γ . 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 thegravity 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 ofthe coset manifold, respectively, M = SO (1 , × SO (4 , SO (4) × SO (3) . (2.19)There are three vector fields from the gravity multiplet and three vector fields from the threevector multiplets. Two sets of three vector fields can be used to gauge SU (2) R × SU (2) gaugegroup. We denote the coupling constants of two SU (2) factors by g and g . The structureconstant in (2.5) splits into f rst = g (cid:15) rst , f IJK = g C IJK = g (cid:15) IJK . (2.20)The generators of SO (4), SU (2) R , SU (2), and the non-compact SO (4 ,
3) could be representedby, respectively [25], J αβ = e β,α − e α,β , J rs = e s,r − e r,s , J IJ = e J +3 ,I +3 − e I +3 ,J +3 , Y αI = e α,I +3 + e I +3 ,α , (2.21)where we define (cid:0) e ΛΣ (cid:1) ΓΠ = δ ΛΓ δ ΣΠ . (2.22)5 Black holes with
AdS × Σ g × Σ g horizon U (1) × U (1) -invariant truncation We truncate the theory to the U (1) × U (1)-invariant sector, which was first considered in section3.1 of [25]. The U (1) × U (1) are generated by J and J . We find two non-compact SO (4 , Y and Y , which are invariant under the action of J and J . We exponentiatethe non-compact generators and obtain the coset representative, L = e ϕ Y e ϕ Y . (3.1)We also have two U (1) gauge fields, A and A , and a two-form gauge potential, B µν , in the U (1) × U (1)-invariant truncation. The Lagrangian of the truncation is given by e − L = − R + ∂ µ σ∂ µ σ + 14 cosh ϕ ∂ µ ϕ ∂ µ ϕ + 14 ∂ µ ϕ ∂ µ ϕ − V − e − σ cosh(2 ϕ ) F µν F µν − e − σ (cid:0) cosh ϕ cosh(2 ϕ ) + sinh ϕ (cid:1) F µν F µν + 14 e − σ cosh ϕ sinh(2 ϕ ) F µν F µν − m e − σ B µν B µν + 364 e σ H µνρ H µνρ − (cid:15) µνρστκ B µν (cid:18) F ρσ F τκ − F ρσ F τκ + 13 m B ρσ B τκ (cid:19) , (3.2)where the scalar potential is V = − g e σ − g me − σ cosh ϕ cosh ϕ + m e − σ (cid:0) cosh ϕ + sinh ϕ cosh(2 ϕ ) (cid:1) . (3.3) In this section, we obtain supersymmetric
AdS black holes with a horizon which is a product oftwo Riemann surfaces. We consider the metric, ds = e F ( r ) (cid:0) dt − dr (cid:1) − e G ( r ) (cid:0) dθ + sin θ dφ (cid:1) − e G ( r ) (cid:0) dθ + sin θ dφ (cid:1) , (3.4)for the S × S background, and ds = e F ( r ) (cid:0) dt − dr (cid:1) − e G ( r ) (cid:0) dθ + sinh θ dφ (cid:1) − e G ( r ) (cid:0) dθ + sinh θ dφ (cid:1) , (3.5)for the H × H background. The only non-vanishing components of the non-Abelian SU (2)gauge field, A Λ µ , Λ = 0 , , . . . ,
6, are given by A = − a cos θ dφ − a cos θ dφ , A = − b cos θ dφ − b cos θ dφ , (3.6)6or the S × S background and A = 2 a cosh θ dφ + 2 a cosh θ dφ , A = 2 b cosh θ dφ + 2 b cosh θ dφ , (3.7)for the H × H background, where the magnetic charges, a , a , b , b , are constant. In order tohave equal signs for the field strengths, we set opposite signs of the gauge fields for the S × S and H × H backgrounds. We also have a non-trivial two-form gauge potential, B µν , determinedby solving the equations of motion, B tr = − m ( a a − b b ) e σ +2 F − G − G . (3.8)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 thefermionic fields to zero. From the supersymmetry variations, we obtain F (cid:48) e − F γ ˆ r (cid:15) A + i (cid:0) g e σ cosh ϕ + me − σ cosh ϕ (cid:1) (cid:15) A − i me − σ sinh ϕ sinh ϕ γ σ AB (cid:15) B + 14 e − σ sinh ϕ (cid:16) b e − G γ ˆ θ ˆ φ + b e − G γ ˆ θ ˆ φ (cid:17) γ (cid:15) A − e − σ cosh ϕ (cid:16) a e − G γ ˆ θ ˆ φ + a e − G γ ˆ θ ˆ φ (cid:17) σ AB (cid:15) B + 14 e − σ cosh ϕ sinh ϕ (cid:16) b e − G γ ˆ θ ˆ φ + b e − G γ ˆ θ ˆ φ (cid:17) σ AB (cid:15) B + 38 m ( a a − b b ) e σ − G − G γ ˆ t ˆ r γ (cid:15) A = 0 , (3.9) G (cid:48) e − F γ ˆ r (cid:15) A + i (cid:0) g e σ cosh ϕ + me − σ cosh ϕ (cid:1) (cid:15) A − i me − σ sinh ϕ sinh ϕ γ σ AB (cid:15) B − e − σ sinh ϕ (cid:16) b e − G γ ˆ θ ˆ φ − b e − G γ ˆ θ ˆ φ (cid:17) γ (cid:15) A + 14 e − σ cosh ϕ (cid:16) a e − G γ ˆ θ ˆ φ − a e − G γ ˆ θ ˆ φ (cid:17) σ AB (cid:15) B − e − σ cosh ϕ sinh ϕ (cid:16) b e − G γ ˆ θ ˆ φ − b e − G γ ˆ θ ˆ φ (cid:17) σ AB (cid:15) B − m ( a a − b b ) e σ − G − G γ ˆ t ˆ r γ (cid:15) A = 0 , (3.10) G (cid:48) e − F γ ˆ r (cid:15) A + i (cid:0) g e σ cosh ϕ + me − σ cosh ϕ (cid:1) (cid:15) A − i me − σ sinh ϕ sinh ϕ γ σ AB (cid:15) B − e − σ sinh ϕ (cid:16) b e − G γ ˆ θ ˆ φ − b e − G γ ˆ θ ˆ φ (cid:17) γ (cid:15) A + 14 e − σ cosh ϕ (cid:16) a e − G γ ˆ θ ˆ φ − a e − G γ ˆ θ ˆ φ (cid:17) σ AB (cid:15) B − e − σ cosh ϕ sinh ϕ (cid:16) b e − G γ ˆ θ ˆ φ − b e − G γ ˆ θ ˆ φ (cid:17) σ AB (cid:15) B − m ( a a − b b ) e σ − G − G γ ˆ t ˆ r γ (cid:15) A = 0 , (3.11)7 (cid:48) e − F γ ˆ r (cid:15) A − i (cid:0) g e σ cosh ϕ − me − σ cosh ϕ (cid:1) (cid:15) A − i me − σ sinh ϕ sinh ϕ γ σ AB (cid:15) B − e − σ sinh ϕ (cid:16) b e − G γ ˆ θ ˆ φ + b e − G γ ˆ θ ˆ φ (cid:17) γ (cid:15) A + 14 e − σ cosh ϕ (cid:16) a e − G γ ˆ θ ˆ φ + a e − G γ ˆ θ ˆ φ (cid:17) σ AB (cid:15) B − e − σ cosh ϕ sinh ϕ (cid:16) b e − G γ ˆ θ ˆ φ + b e − G γ ˆ θ ˆ φ (cid:17) σ AB (cid:15) B + 18 m ( a a − b b ) e σ − G − G γ ˆ t ˆ r γ (cid:15) A = 0 , (3.12) ϕ (cid:48) e − F γ ˆ r (cid:15) A − ϕ (cid:48) e − F cosh ϕ γ γ ˆ r σ AB (cid:15) B − ig e σ sinh ϕ (cid:15) A − ime − σ sinh ϕ cosh ϕ γ σ AB (cid:15) B − e − σ sinh ϕ (cid:16) a e − G γ ˆ θ ˆ φ + a e − G γ ˆ θ ˆ φ (cid:17) σ AB (cid:15) B + e − σ cosh ϕ cosh ϕ (cid:16) b e − G γ ˆ θ ˆ φ + b e − G γ ˆ θ ˆ φ (cid:17) σ AB (cid:15) B = 0 , (3.13)where the hatted indices are the flat indices. The t -, θ -, and θ -components of the gravitinovariations give (3.9), (3.10), (3.11), the dilatino variation gives (3.12), and the gaugino variationgives (3.13). The φ -, φ -components of the gravitino variations are identical to the θ -, and θ -components beside few more terms, (cid:15) A = − ig a γ ˆ θ ˆ φ σ AB (cid:15) B , (cid:15) A = − ig a γ ˆ θ ˆ φ σ AB (cid:15) B . (3.14)We employ the projection conditions, γ ˆ r (cid:15) A = i(cid:15) A , γ ˆ θ ˆ φ σ AB (cid:15) B = − iλ(cid:15) A , γ ˆ θ ˆ φ σ AB (cid:15) B = − iλ(cid:15) A , (3.15)where λ = ±
1. Solutions with the projection conditions preserve 1 / ϕ = 0 . (3.16)Therefore, from now on, we set ϕ to vanish. 8e present the complete supersymmetry equations, F (cid:48) e − F = − (cid:0) g e σ cosh ϕ + me − σ (cid:1) − m ( a a − b b ) e σ − G − G − λ e − σ cosh ϕ (cid:0) a e − G + a e − G (cid:1) + λ e − σ sinh ϕ (cid:0) b e − G + b e − G (cid:1) ,G (cid:48) e − F = − (cid:0) g e σ cosh ϕ + me − σ (cid:1) + 18 m ( a a − b b ) e σ − G − G + λ e − σ cosh ϕ (cid:0) a e − G − a e − G (cid:1) − λ e − σ sinh ϕ (cid:0) b e − G − b e − G (cid:1) ,G (cid:48) e − F = − (cid:0) g e σ cosh ϕ + me − σ (cid:1) + 18 m ( a a − b b ) e σ − G − G + λ e − σ cosh ϕ (cid:0) a e − G − a e − G (cid:1) − λ e − σ sinh ϕ (cid:0) b e − G − b e − G (cid:1) ,σ (cid:48) e − F = + 12 (cid:0) g e σ cosh ϕ − me − σ (cid:1) − m ( a a − b b ) e σ − G − G + λ e − σ cosh ϕ (cid:0) a e − G + a e − G (cid:1) − λ e − σ sinh ϕ (cid:0) b e − G + b e − G (cid:1) ,ϕ (cid:48) e − F = + 2 g e σ sinh ϕ − λe − σ sinh ϕ (cid:0) a e − G + a e − G (cid:1) + λe − σ cosh ϕ (cid:0) b e − G + b e − G (cid:1) . (3.17)We also obtain twist conditions on the magnetic charges from (3.14), a = − k λg , a = − k λg . (3.18)where k = +1 for the S × S background and k = − H × H background. Thereis no condition on b and b . The supersymmetry equations are consistent with the equations ofmotion. We present the equations of motion in appendix A. AdS solutions In this section, we find
AdS solutions of the supersymmetry equations. The solutions describethe AdS × Σ g × Σ g horizon of six-dimensional black holes.Now we will consider the N = 4 + theory, g > m >
0. When b = b = 0, we find an AdS fixed point solution for the H × H background with k = − e F = 12 / g / m / r , e G = e G = 12 / g / m / , e σ = 2 / m / g / , e ϕ = 1 , (3.19) It is possible to have geometries like S × H for k = +1 and k = −
1, or vice versa. One can easilygeneralize our supersymmetry equations and the twist conditions to that case.
AdS solution first found in [11]. When we consider the S × S background with k = +1, AdS fixed point does not exist. When we consider for non-zero b and b , we obtainthe AdS solutions, first found in [29], e F = e − σ g cosh ϕ r ,e G = λ m e σ ( a cosh ϕ − b sinh ϕ ) ,e G = λ m e σ ( a cosh ϕ − b sinh ϕ ) ,e σ = m a a − b b ) + ( a a + b b ) cosh(2 ϕ ) − ( a b + a b ) sinh(2 ϕ ) g cosh ϕ ( a cosh ϕ − b sinh ϕ )( a cosh ϕ − b sinh ϕ ) ,e ϕ = 1( a − b )( a − b ) (cid:104) ( a a + a b + a b − b b ) − / / − / / (cid:0) a a b + a a b − a a b b − a b b − a b b + 3 b b (cid:1)(cid:21) , (3.21)where we defineΦ = − ( a a − b b ) (cid:0) a b + a b (cid:1) − ( a b + a b ) (cid:0) a a + 10 b b − a a b b (cid:1) − b b (cid:0) a a − b b + a a b b (cid:1) + (cid:113) ( a − b ) ( a − b ) ( a a − b b ) ( a b + a a b − b b ) ( a b + a a b − b b ) . (3.22)All the fields are parametrized by the magnetic charges, ( a , a , b , b ). As ( a , a ) are fixed bythe twist condition in (3.18), there are two free parameters left, ( b , b ).In order to have AdS solutions, we should choose ( b , b ) which makes e F > , e G > , e G > , e σ > , e ϕ > . (3.23)We plot the range of ( b , b ) which satisfies the positivity conditions for H × H , H × S , S × H and S × S , respectively. The positivity ranges are depicted in figure 1. We set m = 1 / g = 3 m to have a unit radius for the AdS boundary. From the plots, weconjecture that only the H × H background gives the AdS solutions. Even in the large regionin the graph for the H × H background, only a small part near origin yields AdS solutions. In order to compare with [11], we have to reparametrize our parameters by σ → √ φ , g , m → √ g, √ m , a , a → √ a , √ a . (3.20) We note that, when G ↔ G , the solutions are invariant under a ↔ a and b ↔ b . However, as a = a - - - - - - - - - - - - - - - Figure 1:
Positivity range of ( b , b ) for the H × H , H × S , S × H and S × S backgrounds,from the top-left and clockwise. In the H × H plot, the black dot at ( b , b ) = (0 , correspondsto the AdS solution in (3.19) . Now we present the full black hole solution numerically. The full black hole solution is aninterpolating geometry between the asymptotically
AdS boundary and the AdS × H × H horizon. We introduce a new radial coordinate, ρ = F + σ . (3.24)This kind of coordinate was introduced in [30]. Employing the supersymmetry equations, weobtain ∂ρ∂r = F (cid:48) + σ (cid:48) = − e F D , (3.25) from (3.18), they are invariant under b ↔ b . When we plot the positivity range for H × S and S × H , thereare small and irregular distributions of points. Most of the points do not respect the invariance under b ↔ b .Even for the points invariant under b ↔ b seem not to give AdS solutions. We presume that the appearanceof these irregular distribution in the positivity range would be due to the complexity of the conditions. D = 2 me − σ + 12 m ( a a − b b ) e σ − G − G . (3.26)Then, the supersymmetry equations are − D ∂F∂ρ = − (cid:0) g e σ cosh ϕ + me − σ (cid:1) − m ( a a − b b ) e σ − G − G − λ e − σ cosh ϕ (cid:0) a e − G + a e − G (cid:1) + λ e − σ sinh ϕ (cid:0) b e − G + b e − G (cid:1) , − D ∂G ∂ρ = − (cid:0) g e σ cosh ϕ + me − σ (cid:1) + 18 m ( a a − b b ) e σ − G − G + λ e − σ cosh ϕ (cid:0) a e − G − a e − G (cid:1) − λ e − σ sinh ϕ (cid:0) b e − G − b e − G (cid:1) , − D ∂G ∂ρ = − (cid:0) g e σ cosh ϕ + me − σ (cid:1) + 18 m ( a a − b b ) e σ − G − G + λ e − σ cosh ϕ (cid:0) a e − G − a e − G (cid:1) − λ e − σ sinh ϕ (cid:0) b e − G − b e − G (cid:1) , − D ∂σ∂ρ = + 12 (cid:0) g e σ cosh ϕ − me − σ (cid:1) − m ( a a − b b ) e σ − G − G + λ e − σ cosh ϕ (cid:0) a e − G + a e − G (cid:1) − λ e − σ sinh ϕ (cid:0) b e − G + b e − G (cid:1) , − D ∂ϕ ∂ρ = + 2 g e σ sinh ϕ − λe − σ sinh ϕ (cid:0) a e − G + a e − G (cid:1) + λe − σ cosh ϕ (cid:0) b e − G + b e − G (cid:1) . (3.27)In the r -coordinate, the UV or asymptotically AdS boundary is at r = 0, and the IR or AdS × H × H horizon is at r = ∞ . In this ρ -coordinate, the UV is at ρ = + ∞ , and the IRis at ρ = −∞ . We present some representative plots of the full black hole solutions in figure 2. In this section, we obtain more black hole solutions with other horizon geometries by consideringD4-branes wrapped on K¨ahler 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 D4-branes can wrap in F (4) gauged supergravity. D4-branes on two Riemann surfaces in the previous section fall into aspecial case of D4-branes on K¨ahler four-cycles in Calabi-Yau fourfolds. The analogous solutionsof M5-branes wrapped on supersymmetric four-cycles were studied in [31, 32].12 -
10 10 20 ρ - -
10 10 20 ρ - -
10 10 20 ρ - - - - - σ - -
10 10 20 ρ - - - - - ϕ Figure 2:
Numerical black hole solutions with m = 1 / and g = 3 m . For the magnetic charges, ( b , b ) , we have ( − . , − . , purple, ( − . , − . , red, and ( − . , − . , orange. We consider the U (1) × U (1)-invariant truncation presented in section 3.1. We consider the metric, ds = e F ( r ) (cid:0) dt − dr (cid:1) − e G ( r ) ds M , (4.1)where M is a K¨ahler four-cycle in Calabi-Yau fourfolds. The curved coordinates on the K¨ahlerfour-cycles will be denoted by { x , x , x , x } , and the hatted ones are the flat coordinates. ForK¨ahler four-cycles in Calabi-Yau fourfolds, there are four directions transverse to D4-branes inthe fourfolds. The normal bundle of the four-cycle has U (2) ⊂ SO (4) structure group. Weidentify 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) gaugefield, A Λ µ , Λ = 0 , , . . . ,
6, are given by F x ˆ x = a e − G , F x ˆ x = a e − G ,F x ˆ x = b e − G , F x ˆ x = b e − G , (4.2)where the magnetic charges, a , a , b , b , are constant. The only non-vanishing component ofthe two-form gauge potential is B tr = − m ( a a − b b ) e σ +2 F − G . (4.3)We employ the projection conditions, γ ˆ r (cid:15) A = i(cid:15) A , γ ˆ x ˆ x σ AB (cid:15) B = − iλ(cid:15) A , γ ˆ x ˆ x σ AB (cid:15) B = − iλ(cid:15) A , (4.4)13here λ = ±
1. Solutions with the projection conditions preserve 1 / F (cid:48) e − F = − (cid:0) g e σ cosh ϕ + me − σ (cid:1) − λ e − σ − G ( a cosh ϕ − b sinh ϕ ) − m (cid:0) a − b (cid:1) e σ − G G (cid:48) e − F = − (cid:0) g e σ cosh ϕ + me − σ (cid:1) + λ e − σ − G ( a cosh ϕ − b sinh ϕ ) + 18 m (cid:0) a − b (cid:1) e σ − G σ (cid:48) e − F = + 12 (cid:0) g e σ cosh ϕ − me − σ (cid:1) + λ e − σ − G ( a cosh ϕ − b sinh ϕ ) − m (cid:0) a − b (cid:1) e σ − G ϕ (cid:48) e − F = + 2 g e σ sinh ϕ − λe − σ − G ( a sinh ϕ − b cosh ϕ ) . (4.5)with the twist conditions, a ≡ a = a = − k λg , b ≡ b = b . (4.6)where k determines the curvature of the K¨ahler four-cycles in Calabi-Yau fourfolds. There is nocondition on b and b .The product of two Riemann surfaces considered in the previous section is a special caseof K¨ahler four-cycles in Calabi-Yau fourfolds. When we identify G ≡ G = G in the su-persymmetry equations for D4-branes wrapped on two Riemann surfaces, (3.17), we obtain thesupersymmetry equations here, (4.5). By solving the supersymmetry equations, we find the AdS fixed point solutions which are identical to the ones obtained in the previous section. Spin (7) manifolds SU (2) -invariant truncation We considered matter coupled F (4) gauged supergravity coupled to three vector multiplets whichhas SU (2) R × SU (2) gauge symmetry. In this section we truncate the theory to the SU (2) diag ⊂ SU (2) R × SU (2) invariant sector, which was first considered in [23] and again in section 4.1of [25]. There is one singlet under SU (2) diag which corresponds to Y + Y + Y by the non-compact generators defined in (2.21). We exponentiate the non-compact generators and obtainthe coset representative, L = e ϕ ( Y + Y + Y ) . (4.7)14e also have a non-Abelian SU (2) gauge field and a two-form gauge potential in the SU (2) diag -invariant truncation. The Lagrangian of the truncation is given by e − L = − R + ∂ µ σ∂ µ σ + 34 ∂ µ ϕ∂ µ ϕ − V + 18 sinh (2 ϕ ) (cid:2) ( g A µ − g A µ ) + ( g A µ − g A µ ) + ( g A µ − g A µ ) (cid:3) − e − σ cosh(2 ϕ ) (cid:0) F µν F µν + F µν F µν + F µν F µν + F µν F µν + F µν F µν + F µν F µν (cid:1) + 14 e − σ sinh(2 ϕ ) (cid:0) F µν F µν + F µν F µν + F µν F µν (cid:1) − m e − σ B µν B µν + 364 e σ H µνρ H µνρ − (cid:15) µνρστκ B µν (cid:18) F ρσ F τκ + F ρσ F τκ + F ρσ F τκ − F ρσ F τκ − F ρσ F τκ − F ρσ F τκ + 13 m B ρσ B τκ (cid:19) , (4.8)where the scalar potential is V = 116 g e σ (cosh(6 ϕ ) − ϕ ) −
8) + 116 g e σ (cosh(6 ϕ ) − ϕ ) + 8) − g g e σ sinh (2 ϕ ) − g me − σ cosh ϕ + 4 g me − σ sinh ϕ + m e − σ . (4.9) We consider the metric, ds = e F ( r ) (cid:0) dt − dr (cid:1) − e G ( r ) ds M , (4.10)where M is a Cayley four-cycle in manifolds with Spin (7) holonomy. The curved coordinateson the Cayley four-cycles will be denoted by { x , x , x , x } , and the hatted ones are the flatcoordinates. 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, SO (4) → SU (2) + × SU (2) − , (4.11)with the non-Abelian SU (2) gauge group, [31, 33]. The self-duality is defined by γ µν = ± (cid:15) µνρσ γ ρσ , (4.12)and we denoted the self-duality and anti-self-duality by + and − , respectively. For the self-dualpart, components are identified by γ ˆ x ˆ x = γ ˆ x ˆ x , γ ˆ x ˆ x = γ ˆ x ˆ x , γ ˆ x ˆ x = γ ˆ x ˆ x . (4.13)15he only non-vanishing components of the field strength of the SU (2) gauge field, A Λ µ , Λ =0 , , . . . ,
6, are given by F x ˆ x = F x ˆ x = a e − G , F x ˆ x = F x ˆ x = b e − G ,F x ˆ x = F x ˆ x = a e − G , F x ˆ x = F x ˆ x = b e − G ,F x ˆ x = F x ˆ x = a e − G , F x ˆ x = F x ˆ x = b e − G , (4.14)where the magnetic charges, a , a , a , b , b , b , are constant. As we have one SU (2) diag ⊂ SU (2) R × SU (2) non-Abelian gauge field, they are related by g a = g b , g a = g b , g a = g b , (4.15)where g and g are the gauge coupling constants of SU (2) R and SU (2), respectively. The onlynon-vanishing component of the two-form gauge potential is B tr = − m (cid:0) a + a + a − b − b − b (cid:1) e σ +2 F − G . (4.16)We employ the projection conditions, γ ˆ r (cid:15) A = i(cid:15) A ,γ ˆ x ˆ x σ AB (cid:15) B = γ ˆ x ˆ x σ AB (cid:15) B = − iλ(cid:15) A ,γ ˆ x ˆ x σ AB (cid:15) B = γ ˆ x ˆ x σ AB (cid:15) B = − iλ(cid:15) A ,γ ˆ x ˆ x σ AB (cid:15) B = γ ˆ x ˆ x σ AB (cid:15) B = − iλ(cid:15) A , (4.17)where λ = ±
1. Solutions with the projection conditions preserve 1 /
16 of the supersymmetries.Employing the projection conditions, from δλ IA = 0, we obtain for I = 1 , ,
3, respectively, ϕ (cid:48) e − F = + ( g e σ cosh ϕ − g e σ sinh ϕ ) sinh(2 ϕ ) − λe − σ − G ( a sinh ϕ − b cosh ϕ ) ,ϕ (cid:48) e − F = + ( g e σ cosh ϕ − g e σ sinh ϕ ) sinh(2 ϕ ) − λe − σ − G ( a sinh ϕ − b cosh ϕ ) ,ϕ (cid:48) e − F = + ( g e σ cosh ϕ − g e σ sinh ϕ ) sinh(2 ϕ ) − λe − σ − G ( a sinh ϕ − b cosh ϕ ) . (4.18)Therefore, we conclude that the magnetic charges are a ≡ a = a = a , b ≡ b = b = b . (4.19)16e present the complete supersymmetry equations, F (cid:48) e − F = − (cid:0) g e σ cosh ϕ − g e σ sinh ϕ + me − σ (cid:1) − λ e − σ − G ( a cosh ϕ − b sinh ϕ ) − m (cid:0) a − b (cid:1) e σ − G ,G (cid:48) e − F = − (cid:0) g e σ cosh ϕ − g e σ sinh ϕ + me − σ (cid:1) + 3 λ e − σ − G ( a cosh ϕ − b sinh ϕ )+ 38 m (cid:0) a − b (cid:1) e σ − G ,σ (cid:48) e − F = + 12 (cid:0) g e σ cosh ϕ − g e σ sinh ϕ − me − σ (cid:1) + 3 λ e − σ − G ( a cosh ϕ − b sinh ϕ ) − m (cid:0) a − b (cid:1) e σ − G ,ϕ (cid:48) e − F = + ( g e σ cosh ϕ − g e σ sinh ϕ ) sinh(2 ϕ ) − λe − σ − G ( a sinh ϕ − b cosh ϕ ) . (4.20)We also obtain twist conditions on the magnetic charges, a = − k λg , b = − k λg , (4.21)where k determines the curvature of the Cayley four-cycles in Spin (7) manifolds. The twistcondition on b comes from the SU (2) diag condition, (4.15). AdS solutions Now we will consider the N = 4 + theory, g > m >
0. When b = 0, we find an AdS fixedpoint solution for k = − e F = 3 / / g / m / r , e G = 12 / / g / m / , e σ = 2 / m / / g / , e ϕ = 1 . (4.22)After the reparametrization of the parameters given in (3.20), this is the AdS solution firstfound in [11].When we consider for non-zero b , we find new AdS solutions in terms of the scalar field, ϕ , e F = e − σ g cosh ϕ − g sinh ϕ ) 1 r ,e G = 3 λ m e σ ( a cosh ϕ − b sinh ϕ ) ,e σ = m a − b ) + 3( a + b ) cosh(2 ϕ ) − ab sinh(2 ϕ )( g cosh ϕ − g sinh ϕ )( a cosh ϕ − b sinh ϕ ) . (4.23)Then, the scalar field, ϕ , should be expressed in terms of the magnetic charges, a and b , but theexpression is very unwieldy. Alternatively, we present the magnetic charge, b , in terms of the17calar field, ϕ , b = − a (cid:104) g (cid:0) e ϕ (cid:1) (cid:0) − e ϕ + e ϕ (cid:1) + g (cid:0) − e ϕ (cid:1) (cid:0) e ϕ + e ϕ (cid:1) +4 e ϕ (cid:113) g (1 + e ϕ ) (5 − e ϕ + 5 e ϕ ) + g (1 − e ϕ ) (5 + 6 e ϕ + 5 e ϕ ) + 10 g g (1 − e ϕ ) (cid:105) / (cid:2)(cid:0) − e ϕ (cid:1) (cid:0) g (cid:0) − e ϕ − e ϕ + e ϕ (cid:1) + g (cid:0) − e ϕ + 7 e ϕ − e ϕ (cid:1)(cid:1)(cid:3) . (4.24)The solutions are parametrized by two magnetic charges, a and b . It will be interesting to havea field theory interpretation of this AdS fixed point solution.Unlike the black holes with a horizon of two Riemann surfaces, for this case, we could notdevice a way to determine the positivity range for AdS solutions. However, as we see in the nextsubsection, we obtained a number of AdS solutions with negative curvature horizon, k = − k = +1. Thus, we will concentrate on solutions with negative curvature horizon, k = − Now we present the full black hole solution numerically. The full black hole solution is aninterpolating geometry between the asymptotically
AdS boundary and the AdS × Cayley horizon. As we explained at the end of the last subsection, we will concentrate on solutions withnegative 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, weobtain ∂ρ∂r = F (cid:48) + σ (cid:48) = − e F D , (4.26)where we define D = 2 me − σ + 32 m (cid:0) a − b (cid:1) e σ − G . (4.27)18hen, the supersymmetry equations are − D ∂F∂ρ = − (cid:0) g e σ cosh ϕ − g e σ sinh ϕ + me − σ (cid:1) − λ e − σ − G ( a cosh ϕ − b sinh ϕ ) − m (cid:0) a − b (cid:1) e σ − G , − D ∂G∂ρ = − (cid:0) g e σ cosh ϕ − g e σ sinh ϕ + me − σ (cid:1) + 3 λ e − σ − G ( a cosh ϕ − b sinh ϕ )+ 38 m (cid:0) a − b (cid:1) e σ − G , − D ∂σ∂ρ = + 12 (cid:0) g e σ cosh ϕ − g e σ sinh ϕ − me − σ (cid:1) + 3 λ e − σ − G ( a cosh ϕ − b sinh ϕ ) − m (cid:0) a − b (cid:1) e σ − G , − D ∂ϕ ∂ρ = + ( g e σ cosh ϕ − g e σ sinh ϕ ) sinh(2 ϕ ) − λe − σ − G ( a sinh ϕ − b cosh ϕ ) . (4.28)In the r -coordinate, the UV or asymptotically AdS boundary is at r = 0, and the IR or AdS × Cayley horizon is at r = ∞ . In this ρ -coordinate, the UV is at ρ = + ∞ , and the IR isat ρ = −∞ . We set m = 1 / g = 3 m to have a unit radius for the AdS boundary. Then, a is determined by (4.21), and there is one free parameter left, b . Employing (4.23) and (4.24) todetermine boundary conditions, we solve the supersymmetry equations numerically. With somechoices of b , we present representative plots of the full black hole solutions in figure 3. - -
10 10 20 ρ - -
10 10 20 ρ - -
10 10 20 ρ - - - σ - -
10 10 20 ρ ϕ Figure 3:
Numerical black hole solutions with m = 1 / and g = 3 m . We have b = − . forpurple, b = − . for red, and b = − . for orange. cknowledgements We would like to thank Parinya Karndumri for helpful discussions. This research was supportedby the National Research Foundation of Korea under the grant NRF-2017R1D1A1B03034576.
A The equations of motion
In this appendix, we present the equations of motion for the U (1) × U (1)-invariant truncation in(4.8) with ϕ = 0 as in (3.16), R µν − Rg µν = 2 (cid:20) V g µν + 2 (cid:18) ∂ µ σ∂ ν σ − g µν g ρσ ∂ ρ σ∂ σ σ (cid:19) + 12 (cid:18) ∂ µ ϕ ∂ ν ϕ − g µν g ρσ ∂ ρ ϕ ∂ σ ϕ (cid:19) − e − σ cosh(2 ϕ ) (cid:18) g ρσ F µρ F νσ − g µν F ρσ F ρσ (cid:19) − e − σ cosh(2 ϕ ) (cid:18) g ρσ F µρ F νσ − g µν F ρσ F ρσ (cid:19) + e − σ sinh(2 ϕ ) (cid:18) g ρσ F µρ F νσ − g µν F ρσ F ρσ (cid:19) − m e − σ (cid:18) g ρσ B µρ B νσ − g µν B ρσ B ρσ (cid:19)(cid:21) , (A.1)1 √− g ∂ µ (cid:0) √− gg µν ∂ ν σ (cid:1) + 12 ∂V∂σ − e − σ cosh(2 ϕ ) F µν F µν − e − σ cosh(2 ϕ ) F µν F µν + 14 e − σ cosh(2 ϕ ) F µν F µν − m e − σ B µν B µν = 0 , (A.2)1 √− g ∂ µ (cid:0) √− gg µν ∂ ν ϕ (cid:1) + 2 ∂V∂ϕ + 12 e − σ cosh(2 ϕ ) F µν F µν + 12 e − σ cosh(2 ϕ ) F µν F µν − e − σ cosh(2 ϕ ) F µν F µν = 0 . (A.3) D ν (cid:0) e − σ F Λ νµ (cid:1) = 124 e(cid:15) µνρστκ F Λ νρ H στκ , (A.4) D ρ (cid:0) e σ H ρµν (cid:1) = − e(cid:15) µνρστκ η ΛΣ F Λ ρσ F Σ τκ − me − σ δ F Λ µν . (A.5)20 eferences [1] L. J. Romans, The F(4) Gauged Supergravity in Six-dimensions,
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