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THE NEAR-HORIZON GEOMETRY OF DILATON-AXION BLACK HOLES G. CLE´MENT LAPTH (CNRS), B.P.110, F-74941 Annecy–le–Vieux cedex, France E-mail: [email protected] 1 D. GAL’TSOVa 0 Department of Theoretical Physics, MSU, 119899, Moscow, Russia 0 E-mail: [email protected] 2 n Static black holes of dilaton-axion gravity become singular inthe extreme limit, which a prevents a direct determination of their near-horizon geometry. This is addressed by J first taking the near-horizon limit of extreme rotating NUT-less black holes, and then going to the static limit. The resulting four-dimensional geometry may be lifted to a 0 Bertotti-Robinson-like solution of six-dimensional vacuum gravity, which alsogives the 1 near-horizongeometryofextremeKaluza-Kleinblackholesinfivedimensions. 1 v The discovery of the AdS/CFT dualities1 stimulated the searchfor geometries 0 4 containingAdSsectors,whichtypicallyariseasthenear-horizonlimitofBPSblack 0 holes or p-branes in various dimensions. Here we discuss the near-horizon limit of 1 4-dimensional black holes arising in the truncated effective theory of the heterotic 0 string: dilaton-axion gravity with one Abelian vector field (EMDA). 1 The Einstein-frame metrics of the NUT-less rotating extreme black hole solu- 0 / tions of EMDA2 are given by c q Σr2 Γ 2Ma(r+a) 2 dr2 - ds2 = dt2 sin2θ dϕ dt Σ +dθ2 , (1) gr Γ − Σ (cid:18) − Γ (cid:19) − (cid:18) r2 (cid:19) : Σ=h a2sin2θ, Γ=h2 r2a2sin2θ, h=r2+2M(r+a), v − − i X M andabeingthemassandtherotationparameter. Thehorizonr =0reducingto r a point in the static case a=0, we carry out the near-horizonlimit in the rotating a casea=0. First,wetransformtoaframeco-rotatingwiththe horizon,andrescale time by6 t (r2/λ)t (r2 2aM). Then, we put r λx, cosθ y, and take the 0 0 → ≡ ≡ ≡ limit λ 0, arriving at → dx2 dy2 1 y2 ds2 =r2 (α+βy2) x2dt2 − (dϕ+xdt)2 , (2) 0(cid:20) (cid:18) − x2 − 1 y2(cid:19)− α+βy2 (cid:21) − with β = a/2M,α = 1 β. This is similar in form to the extreme Kerr-Newman near-horizon metric3, b−oth having the symmetry group SL(2,R) U(1), and co- × inciding in the extreme Kerr case M = a. Now take the static limit a 0 in (2), keeping r2 =2aM fixed. This yields (for r2 =1) the metric → 0 0 dx2 dy2 ds2 =x2dt2 (1 y2)(dϕ+xdt)2. (3) − x2 − 1 y2 − − − aSupportedbyRFBR 1 Thismetric,togetherwiththeassociatednear-horizondilatonφ,axionκandgauge potentialinthestaticlimit(independentoftheoriginalparametersoftheblackhole solution) φ=0, κ= y, A= y(dϕ+xdt)/√2, (4) − − constituteanewBertotti-Robinson-likesolutionofEMDA,theBREMDAsolution. This solution has properties remarkably similar to those of AdS2 S2: all × timelikegeodesicsareconfined,andtheKlein-Gordonequationisseparableinterms of the usual spherical harmonics. The isometry group SL(2,R) U(1), with U(1) beingtheremnantoftheSO(3)symmetryofS2,isgeneratedby×theKillingvectors L−1,L0,L1 andLφ =∂φ. The firstthreeoftheseconstitutethe sl(2,R)subalgebra of an infinite-dimensional algebra of asymptotic symmetries of (3), with generators n(n 1) n(n 1) L =t−n t+ − ∂ + x(n 1)∂ − ∂ (5) n (cid:20)(cid:18) 2x2t (cid:19) t − x− xt φ(cid:21) (where n Z) satisfying the Witt algebra [Ln,Lm] = (n m)Ln+m up to terms O(x−4). I∈t can be expected that a representation in term−s of asymptotic metric variationswillleadto the Virasoroextensionofthis algebrawith aclassicalcentral charge, opening the way for microscopic counting of the horizon microstates of dilaton-axion black holes. The BREMDA solution does have a close connection with AdS2 S2, which × is discovered by lifting it to 6 dimensions. EMDA in 4 dimensions may be shown 4 to derive from 6-dimensional vacuum gravity with 2 commuting spacelike Killing vectors ∂ , ∂ via a two-stepKaluza-Kleinreduction together with the assumption η χ ofa special relationbetween the Kaluza-Kleingaugefields, accordingto the ansatz ds2 =ds2 e−φθ2 eφ(ζ+κθ)2, (6) 6 4 − − θ =dχ+A dxµ, ζ =dη+B dxµ, G =e−φF˜ κF µ µ µν µν µν − (F =dA, G=dB). Usingthis ansatz,the BREMDAsolution(3)-(4)maybe lifted and rearranged,yielding the vacuum solution BR6 of 6-dimensional gravity dx2 dy2 ds26 =x2dt2− x2 − 1 y2 −(1−y2)dϕ2−(dχ+−√2xdt+√2ydϕ)2−dχ2−. (7) − This is the trivial 6-dimensional embedding of a 5-dimensional metric BR5, which can be shown to be the common near-horizon limit of all static, NUT-less black holes of 5-dimensional Kaluza-Klein theory. It enjoys the higher symmetry group SL(2,R) SO(3) U(1) U(1),butbreaksallfermionicsymmetries: nocovariantly × × × constant spinors exist. References 1. J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231 2. D.V. Gal’tsov and O.V. Kechkin, Phys. Rev. D 50 (1994) 7394 3. J. Bardeen and G.T. Horowitz, Phys. Rev. D 60 (1999) 104030 4. C.M Chen, D.V. Gal’tsov and S.A. Sharakin, in preparation 2