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Sasaki–Einstein twist of Kerr–AdS black holes
Physics Letters B 600 (2004) 270–274 www.elsevier.com/locate/physletb
Sasaki–Einstein twist of Kerr–AdS black holes Yoshitake Hashimoto a , Makoto Sakaguchi b , Yukinori Yasui c a Department of Mathematics, Osaka City University, Sumiyoshi, Osaka 558-8585, Japan b Osaka City University Advanced Mathematical Institute (OCAMI), Sumiyoshi, Osaka 558-8585, Japan c Department of Physics, Osaka City University, Sumiyoshi, Osaka 558-8585, Japan
Received 23 July 2004; accepted 1 September 2004 Available online 11 September 2004 Editor: M. Cvetiˇc
Abstract We consider Kerr–AdS black holes with equal angular momenta in arbitrary odd spacetime dimensions 5. Twisting the Killing vector fields of the black holes, we reproduce the compact Sasaki–Einstein manifolds constructed by Gauntlett, Martelli, Sparks and Waldram. We also discuss an implication of the twist in string theory and M-theory. 2004 Elsevier B.V. All rights reserved.
Kerr–AdS black holes are characterized by mass, angular momenta and cosmological constant. In spacetime dimension d, the number of angular momenta is equal to the rank of the rotation group SO(d − 1). The fivedimensional Kerr–AdS black holes with two angular momenta were constructed in [1], and recently the general form in arbitrary dimension was found by using the Kerr–Schild ansatz [2]. On the other hand, the Wick rotation of the black holes leads to Riemannian metrics. However, the metrics in general do not extend smoothly to compact manifolds. In [2–4], it was shown that this can be achieved by taking a certain limit (Page limit) which enhances the isometry of the metric. Indeed, the infinite series of Einstein metrics on compact manifolds were explicitly constructed [2,4], and analyzed in detail in [5]. Recently, infinite series of Sasaki–Einstein metrics on compact manifolds were presented in [6,7]. It is expected that these metrics can be related to some Kerr–AdS black holes by a certain limit. Our aim in this Letter is to clarify the relation between them.
E-mail addresses: [email protected] (Y. Hashimoto), [email protected] (M. Sakaguchi), [email protected] (Y. Yasui). 0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.09.002
Y. Hashimoto et al. / Physics Letters B 600 (2004) 270–274
271
We begin with the (2n + 3)-dimensional Kerr–AdS black hole with a negative cosmological constant (2n + 2)λ < 0 (n 1) as follows [1,2]: gˆ = −
2 dr 2 Wˆ (r) 2 ˆ dψ + A + fˆ(r) dt , dt + + r 2 gCP n + b(r) ˆ b(r) Wˆ (r)
(1)
where 2 2M(δ 2 + λJ 2 ) 2MJ 2 ˆ − 2Mδ , + 2n+2 = 1 − λr 2 b(r) Wˆ (r) = 1 − λr 2 − 2n 2n r r r 2 δ 1 2MJ ˆ =1+ 1− . , fˆ(r) = b(r) ˆ J r 2n+2 b(r)
(2)
The metric gCP n is the Fubini–Study metric on CP n with a normalization RicCP n = (2n + 2)gCP n , and the 1-form A is the U(1) connection associated with the Kähler form dA/2 on gCP n . The black hole is parameterized by the mass M, the angular momentum J and a trivial parameter δ. The parameter δ is related to the parameter β introduced in [8] as δ = −λJ 2 β + 1. This metric is a special case that all angular momenta are set to be equal. The metric (1) reduces to the AdS metric at r → ∞ because the metric of the circle bundle over CP n tends to the standard metric of S 2n+1 .1 A horizon appears for sufficiently small J . If we set δ 2 = −λJ 2 , the Wˆ (r) does not have positive roots so that the curvature singularity at r = 0 is not screened by the horizon, and so is naked. As will be seen below, in the Euclidean picture this solution is shown to be related to the Sasaki–Einstein metrics. The Euclidean Einstein metric with a positive cosmological constant (2n + 2)λ > 0 is extracted from the Kerr– AdS black hole (1) by the substitution t → iτ and J → iJ : g=
2 W (r) 2 dr 2 dτ + + r 2 gCP n + b(r) dψ + A + f (r) dτ , b(r) W (r)
(3)
where 2Mδ 2 2M(δ 2 − λJ 2 ) 2MJ 2 2 b(r) − − = 1 − λr , r 2n r 2n+2 r 2n δ 2MJ 2 1 1− . b(r) = 1 − 2n+2 , f (r) = J b(r) r
W (r) = 1 − λr 2 −
(4)
∂ ∂ The metric has the isometry SU(n + 1) × U(1) × R. The generator of U(1) × R is given by ( ∂ψ , ∂τ ). It is easy to see that under the Page limit and a special choice of the parameters [2–4] this metric reduces to a homogeneous Einstein metric with the isometry SU(n + 1) × SU(2) × U(1) on a circle bundle over CP n × S 2 . Indeed, the metric can be written as 2 k 1 2 2 2 2 , dχ + sin χ dη + r0 gCP n + b0 dψ + A + cos χ dη g0 = (5) W0 2
where 2λ(n + 1)(2(n + 1) − (n + 2)b0 ) , n + 1 − b0 √ 2 (n + 1)b0(1 − b0 ) k=± ∈ Z, b0 (2(n + 1) − (n + 2)b0 )
W0 =
r02 =
n + 1 − b0 , λ(n + 1) (6)
1 If we replace the Fubini–Study CP n by an arbitrary Einstein–Kähler manifold with the same scalar curvature, we obtain another Kerr
black hole with different asymptotic behavior.
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Y. Hashimoto et al. / Physics Letters B 600 (2004) 270–274
and b0 is a constant with 0 < b0 < 1. In the case of n = 1, this reproduces the metric given in Theorem 2 of [4]. Further, for k = 1, it gives the homogeneous Sasaki–Einstein manifold T 1,1 . ˜ 2 (S 2 We now transform the metric to inhomogeneous Sasaki–Einstein metrics on circle bundles over CP n ×S n bundle over CP ) presented in [6,7]. First, we set δ 2 = λJ 2 , then the coefficient of 1/r 2n in W vanishes. Twisting the U(1)×R coordinates as τ˜ = τ + J ψ,
(7)
we obtain g = gK + (J dψ − σ )2 ,
(8)
where the metric gK is a local positive Kähler–Einstein metric in dimension 2n + 2, 2 dr 2 d τ˜ 2 2 + r gCP n + r W (r) +A , gK = W (r) J √ and the Kähler form of gK is given by dσ/2 λ, √ 2 √ λr σ = 1− d τ˜ − λr 2 A. J
(9)
(10)
Thus, as is well known, the metric g in (8) turns out to be locally Sasaki–Einstein. If we write the metric g by the coordinates (τ, τ˜ ), instead of (τ, ψ) or (τ˜ , ψ), we can eliminate the parameter δ after rescaling Mδ 2 → M and J δ −1 → J .2 On the other hand, twisting the coordinates as c ψ˜ = ψ − τ, J we have
(11)
2 g = gC + ω(r) dτ + f (r)(d ψ˜ + A) ,
(12)
where gC =
dr 2 + r 2 gCP n + q(r)(d ψ˜ + A)2 , W (r)
(13)
and the components are given by √ 2 ω(r) = k 2 r 2 W (r) + k λr 2 − 1 ,
f (r) =
√ √ r2 kW (r) + λ k λr 2 − 1 , ω(r)
q(r) =
r 2 W (r) ω(r) (14)
with k = (c + 1)/J . The metric gC is conformally Kähler [7]. The singularities coming from the roots r = ri of W = 0 can be resolved by the restriction of the range of the angle ψ˜ ; putting R 2 = 4(r − ri )/W (ri ) one has in the limit r → ri , dr 2 + q(r) d ψ˜ 2 → dR 2 + Ki2 R 2 d ψ˜ 2 , W (r)2 2 The authors are grateful to Gary Gibbons, Malcolm Perry and Chris Pope for this remark.
(15)
Y. Hashimoto et al. / Physics Letters B 600 (2004) 270–274
273
where (n + 2)λri2 − (n + 1) √ . k λri2 − 1 √ If we set λ(n + 2)/(n + 1) = k λ, that is, n + 2√ c= λJ − 1, n+1 Ki =
(16)
(17)
then Ki is independent of ri . Under a suitable condition on the parameter MJ 2 , the corresponding metric g has an SU(n + 1) × U(1) × U(1) symmetry, and it reproduces a Sasaki–Einstein metric on a compact manifold given by Gauntlett et al. in [6,7]. We shall comment on the implication of our method in the higher-dimensional context. As explained above, the higher-dimensional backgrounds are related to each other as follows: AdSp × S q Wick rot.
H p × dSq
q
AdSp × MSE Wick rot. and δ 2 =λJ 2 q
H p × MdS
cosmo.
cosmo.
S p × AdSq
S p × MAdS
q
where (p, q) = (5, 5), (4, 7), and a p-form flux is associated with them. The left-hand side shows that the maximally supersymmetric backgrounds are related to each other. Under the Wick rotation and a sign change of the cosmological constant, the AdS5 ×S 5 solution in the type-IIB string theory is mapped to itself, while AdS4 ×S 7 beq q comes S 4 × AdS7 . In the right-hand side, we have generalized S q to MSE , where MSE stands for the q-dimensional q q Sasaki–Einstein manifold specified by (12). This shows the relation between AdSp × MSE and S p × MAdS , where q MAdS means the q-dimensional Kerr–AdS black hole. It is known that the former solution admits supersymmetry q q due to the Sasaki–Einstein structure of MSE , and that string/M-theory on AdSp × MSE is dual to supersymmetric q Yang–Mills theory in (p − 1)-dimensions. Though the condition δ 2 = λJ 2 implies a naked singularity for MAdS q q and cannot be imposed consistently on MdS , where MdS means the Kerr–dS black hole, it may be interesting to q examine string/M-theory on S p × MAdS and the dual Yang–Mills theory. Acknowledgements M.S. gave a talk at the YITP Workshop on “Quantum Field Theory 2004” (13–16 July) at the Yukawa Institute for Theoretical Physics. The authors thank the organizers for giving the chance and participants for useful comments. This Letter is supported by the 21 COE program “Constitution of wide-angle mathematical basis focused on knots”. Research of Y.H. is supported in part by the Grant-in Aid for scientific Research (Nos. 14540088, 15540040 and 15540090) from Japan Ministry of Education. Research of Y.Y. is supported in part by the Grant-in Aid for Scientific Research (Nos. 14540073 and 14540275) from Japan Ministry of Education. References [1] S.W. Hawking, C.J. Hunter, M.M. Taylor-Robinson, Phys. Rev. D 59 (1999) 064005, hep-th/9811056. [2] G.W. Gibbons, H. Lü, D.N. Page, C.N. Pope, hep-th/0404008.
274
[3] [4] [5] [6] [7] [8]
Y. Hashimoto et al. / Physics Letters B 600 (2004) 270–274
D.N. Page, Phys. Lett. B 79 (1978) 235. Y. Hashimoto, M. Sakaguchi, Y. Yasui, hep-th/0402199, Commun. Math. Phys., in press. G.W. Gibbons, S.A. Hartnoll, Y. Yasui, hep-th/0407030. J.P. Gauntlett, D. Martelli, J. Sparks, D. Waldram, hep-th/0403002. J.P. Gauntlett, D. Martelli, J.F. Sparks, D. Waldram, hep-th/0403038. M. Cvetiˇc, H. Lü, C.N. Pope, hep-th/0406196.
Sasaki–Einstein twist of Kerr–AdS black holes Yoshitake Hashimoto a , Makoto Sakaguchi b , Yukinori Yasui c a Department of Mathematics, Osaka City University, Sumiyoshi, Osaka 558-8585, Japan b Osaka City University Advanced Mathematical Institute (OCAMI), Sumiyoshi, Osaka 558-8585, Japan c Department of Physics, Osaka City University, Sumiyoshi, Osaka 558-8585, Japan
Received 23 July 2004; accepted 1 September 2004 Available online 11 September 2004 Editor: M. Cvetiˇc
Abstract We consider Kerr–AdS black holes with equal angular momenta in arbitrary odd spacetime dimensions 5. Twisting the Killing vector fields of the black holes, we reproduce the compact Sasaki–Einstein manifolds constructed by Gauntlett, Martelli, Sparks and Waldram. We also discuss an implication of the twist in string theory and M-theory. 2004 Elsevier B.V. All rights reserved.
Kerr–AdS black holes are characterized by mass, angular momenta and cosmological constant. In spacetime dimension d, the number of angular momenta is equal to the rank of the rotation group SO(d − 1). The fivedimensional Kerr–AdS black holes with two angular momenta were constructed in [1], and recently the general form in arbitrary dimension was found by using the Kerr–Schild ansatz [2]. On the other hand, the Wick rotation of the black holes leads to Riemannian metrics. However, the metrics in general do not extend smoothly to compact manifolds. In [2–4], it was shown that this can be achieved by taking a certain limit (Page limit) which enhances the isometry of the metric. Indeed, the infinite series of Einstein metrics on compact manifolds were explicitly constructed [2,4], and analyzed in detail in [5]. Recently, infinite series of Sasaki–Einstein metrics on compact manifolds were presented in [6,7]. It is expected that these metrics can be related to some Kerr–AdS black holes by a certain limit. Our aim in this Letter is to clarify the relation between them.
E-mail addresses: [email protected] (Y. Hashimoto), [email protected] (M. Sakaguchi), [email protected] (Y. Yasui). 0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.09.002
Y. Hashimoto et al. / Physics Letters B 600 (2004) 270–274
271
We begin with the (2n + 3)-dimensional Kerr–AdS black hole with a negative cosmological constant (2n + 2)λ < 0 (n 1) as follows [1,2]: gˆ = −
2 dr 2 Wˆ (r) 2 ˆ dψ + A + fˆ(r) dt , dt + + r 2 gCP n + b(r) ˆ b(r) Wˆ (r)
(1)
where 2 2M(δ 2 + λJ 2 ) 2MJ 2 ˆ − 2Mδ , + 2n+2 = 1 − λr 2 b(r) Wˆ (r) = 1 − λr 2 − 2n 2n r r r 2 δ 1 2MJ ˆ =1+ 1− . , fˆ(r) = b(r) ˆ J r 2n+2 b(r)
(2)
The metric gCP n is the Fubini–Study metric on CP n with a normalization RicCP n = (2n + 2)gCP n , and the 1-form A is the U(1) connection associated with the Kähler form dA/2 on gCP n . The black hole is parameterized by the mass M, the angular momentum J and a trivial parameter δ. The parameter δ is related to the parameter β introduced in [8] as δ = −λJ 2 β + 1. This metric is a special case that all angular momenta are set to be equal. The metric (1) reduces to the AdS metric at r → ∞ because the metric of the circle bundle over CP n tends to the standard metric of S 2n+1 .1 A horizon appears for sufficiently small J . If we set δ 2 = −λJ 2 , the Wˆ (r) does not have positive roots so that the curvature singularity at r = 0 is not screened by the horizon, and so is naked. As will be seen below, in the Euclidean picture this solution is shown to be related to the Sasaki–Einstein metrics. The Euclidean Einstein metric with a positive cosmological constant (2n + 2)λ > 0 is extracted from the Kerr– AdS black hole (1) by the substitution t → iτ and J → iJ : g=
2 W (r) 2 dr 2 dτ + + r 2 gCP n + b(r) dψ + A + f (r) dτ , b(r) W (r)
(3)
where 2Mδ 2 2M(δ 2 − λJ 2 ) 2MJ 2 2 b(r) − − = 1 − λr , r 2n r 2n+2 r 2n δ 2MJ 2 1 1− . b(r) = 1 − 2n+2 , f (r) = J b(r) r
W (r) = 1 − λr 2 −
(4)
∂ ∂ The metric has the isometry SU(n + 1) × U(1) × R. The generator of U(1) × R is given by ( ∂ψ , ∂τ ). It is easy to see that under the Page limit and a special choice of the parameters [2–4] this metric reduces to a homogeneous Einstein metric with the isometry SU(n + 1) × SU(2) × U(1) on a circle bundle over CP n × S 2 . Indeed, the metric can be written as 2 k 1 2 2 2 2 , dχ + sin χ dη + r0 gCP n + b0 dψ + A + cos χ dη g0 = (5) W0 2
where 2λ(n + 1)(2(n + 1) − (n + 2)b0 ) , n + 1 − b0 √ 2 (n + 1)b0(1 − b0 ) k=± ∈ Z, b0 (2(n + 1) − (n + 2)b0 )
W0 =
r02 =
n + 1 − b0 , λ(n + 1) (6)
1 If we replace the Fubini–Study CP n by an arbitrary Einstein–Kähler manifold with the same scalar curvature, we obtain another Kerr
black hole with different asymptotic behavior.
272
Y. Hashimoto et al. / Physics Letters B 600 (2004) 270–274
and b0 is a constant with 0 < b0 < 1. In the case of n = 1, this reproduces the metric given in Theorem 2 of [4]. Further, for k = 1, it gives the homogeneous Sasaki–Einstein manifold T 1,1 . ˜ 2 (S 2 We now transform the metric to inhomogeneous Sasaki–Einstein metrics on circle bundles over CP n ×S n bundle over CP ) presented in [6,7]. First, we set δ 2 = λJ 2 , then the coefficient of 1/r 2n in W vanishes. Twisting the U(1)×R coordinates as τ˜ = τ + J ψ,
(7)
we obtain g = gK + (J dψ − σ )2 ,
(8)
where the metric gK is a local positive Kähler–Einstein metric in dimension 2n + 2, 2 dr 2 d τ˜ 2 2 + r gCP n + r W (r) +A , gK = W (r) J √ and the Kähler form of gK is given by dσ/2 λ, √ 2 √ λr σ = 1− d τ˜ − λr 2 A. J
(9)
(10)
Thus, as is well known, the metric g in (8) turns out to be locally Sasaki–Einstein. If we write the metric g by the coordinates (τ, τ˜ ), instead of (τ, ψ) or (τ˜ , ψ), we can eliminate the parameter δ after rescaling Mδ 2 → M and J δ −1 → J .2 On the other hand, twisting the coordinates as c ψ˜ = ψ − τ, J we have
(11)
2 g = gC + ω(r) dτ + f (r)(d ψ˜ + A) ,
(12)
where gC =
dr 2 + r 2 gCP n + q(r)(d ψ˜ + A)2 , W (r)
(13)
and the components are given by √ 2 ω(r) = k 2 r 2 W (r) + k λr 2 − 1 ,
f (r) =
√ √ r2 kW (r) + λ k λr 2 − 1 , ω(r)
q(r) =
r 2 W (r) ω(r) (14)
with k = (c + 1)/J . The metric gC is conformally Kähler [7]. The singularities coming from the roots r = ri of W = 0 can be resolved by the restriction of the range of the angle ψ˜ ; putting R 2 = 4(r − ri )/W (ri ) one has in the limit r → ri , dr 2 + q(r) d ψ˜ 2 → dR 2 + Ki2 R 2 d ψ˜ 2 , W (r)2 2 The authors are grateful to Gary Gibbons, Malcolm Perry and Chris Pope for this remark.
(15)
Y. Hashimoto et al. / Physics Letters B 600 (2004) 270–274
273
where (n + 2)λri2 − (n + 1) √ . k λri2 − 1 √ If we set λ(n + 2)/(n + 1) = k λ, that is, n + 2√ c= λJ − 1, n+1 Ki =
(16)
(17)
then Ki is independent of ri . Under a suitable condition on the parameter MJ 2 , the corresponding metric g has an SU(n + 1) × U(1) × U(1) symmetry, and it reproduces a Sasaki–Einstein metric on a compact manifold given by Gauntlett et al. in [6,7]. We shall comment on the implication of our method in the higher-dimensional context. As explained above, the higher-dimensional backgrounds are related to each other as follows: AdSp × S q Wick rot.
H p × dSq
q
AdSp × MSE Wick rot. and δ 2 =λJ 2 q
H p × MdS
cosmo.
cosmo.
S p × AdSq
S p × MAdS
q
where (p, q) = (5, 5), (4, 7), and a p-form flux is associated with them. The left-hand side shows that the maximally supersymmetric backgrounds are related to each other. Under the Wick rotation and a sign change of the cosmological constant, the AdS5 ×S 5 solution in the type-IIB string theory is mapped to itself, while AdS4 ×S 7 beq q comes S 4 × AdS7 . In the right-hand side, we have generalized S q to MSE , where MSE stands for the q-dimensional q q Sasaki–Einstein manifold specified by (12). This shows the relation between AdSp × MSE and S p × MAdS , where q MAdS means the q-dimensional Kerr–AdS black hole. It is known that the former solution admits supersymmetry q q due to the Sasaki–Einstein structure of MSE , and that string/M-theory on AdSp × MSE is dual to supersymmetric q Yang–Mills theory in (p − 1)-dimensions. Though the condition δ 2 = λJ 2 implies a naked singularity for MAdS q q and cannot be imposed consistently on MdS , where MdS means the Kerr–dS black hole, it may be interesting to q examine string/M-theory on S p × MAdS and the dual Yang–Mills theory. Acknowledgements M.S. gave a talk at the YITP Workshop on “Quantum Field Theory 2004” (13–16 July) at the Yukawa Institute for Theoretical Physics. The authors thank the organizers for giving the chance and participants for useful comments. This Letter is supported by the 21 COE program “Constitution of wide-angle mathematical basis focused on knots”. Research of Y.H. is supported in part by the Grant-in Aid for scientific Research (Nos. 14540088, 15540040 and 15540090) from Japan Ministry of Education. Research of Y.Y. is supported in part by the Grant-in Aid for Scientific Research (Nos. 14540073 and 14540275) from Japan Ministry of Education. References [1] S.W. Hawking, C.J. Hunter, M.M. Taylor-Robinson, Phys. Rev. D 59 (1999) 064005, hep-th/9811056. [2] G.W. Gibbons, H. Lü, D.N. Page, C.N. Pope, hep-th/0404008.
274
[3] [4] [5] [6] [7] [8]
Y. Hashimoto et al. / Physics Letters B 600 (2004) 270–274
D.N. Page, Phys. Lett. B 79 (1978) 235. Y. Hashimoto, M. Sakaguchi, Y. Yasui, hep-th/0402199, Commun. Math. Phys., in press. G.W. Gibbons, S.A. Hartnoll, Y. Yasui, hep-th/0407030. J.P. Gauntlett, D. Martelli, J. Sparks, D. Waldram, hep-th/0403002. J.P. Gauntlett, D. Martelli, J.F. Sparks, D. Waldram, hep-th/0403038. M. Cvetiˇc, H. Lü, C.N. Pope, hep-th/0406196.