It is shown that extremal magnetic black hole solutions of N = 2 supergravity
coupled to vector multiplets $X^\Lambda$ with a generic holomorphic
prepotential $F(X^\Lambda)$ can be described as supersymmetric solitons which
interpolate between maximally symmetric limiting solutions at spatial infinity
and the horizon. A simple exact solution is found for the special case that the
ratios of the $X^\Lambda$ are real, and it is seen that the logarithm of the
conformal factor of the spatial metric equals the Kahler potential on the
vector multiplet moduli space. Several examples are discussed in detail.Comment: 10 pages, minor correction

A. Ceresole

Andrew Strominger

B. de Wit

E. Cremmer

G.W. Gibbons

L. Castellani

P. Breitenlohner

Renata Kallosh

Sergio Ferrara

English

arXiv

Crossref

<i>N</i>=2 extremal black holes

It is shown that extremal magnetic black hole solutions of N = 2 supergravity coupled to vector multiplets X^\Lambda with a generic holomorphic prepotential F(X^\Lambda) can be described as supersymmetric solitons which interpolate between maximally symmetric limiting solutions at spatial infinity and the horizon. A simple exact solution is found for the special case that the ratios of the X^\Lambda are real (imaginary), and it is seen that the logarithm of the conformal factor of the spatial metric equals the K\"{a}hler potential on the vector multiplet moduli space. Several examples are discussed in detail

Ferrara, Sergio

Kallosh, Renata

Strominger, Andrew

CERN Document Server

arXiv:hep-th/9508072v3  16 Feb 1996CERN-TH/95-211SU-ITP-95-16hep-th/9508072August 15, 1995N = 2 EXTREMAL BLACK HOLESSergio Ferrara1Theory Division, CERN, 1211 Geneva 23, SwitzerlandRenata Kallosh2Physics Department, Stanford University, Stanford CA 94305-4060, USAAndrew Strominger3Department of Physics, University of California, Santa Barbara, CA 93206-9530, USAABSTRACTIt is shown that extremal magnetic black hole solutions of N = 2 supergravitycoupled to vector multiplets XΛ with a generic holomorphic prepotential F (XΛ) canbe described as supersymmetric solitons which interpolate between maximally sym-metric limiting solutions at spatial infinity and the horizon. A simple exact solutionis found for the special case that the ratios of the XΛ are real, and it is seen that thelogarithm of the conformal factor of the spatial metric equals the Ka¨hler potential onthe vector multiplet moduli space. Several examples are discussed in detail.1E-mail: ferraras@cernvm.cern.ch2E-mail: kallosh@physics.stanford.edu3E-mail: andy@denali.physics.ucsb.edu11 IntroductionBlack holes seem to be a never-ending source of surprises. While much has been learned abouttheir behavior, much remains to be understood – even at the classical level. In this paper westudy the classical supersymmetric solutions in a general theory with N = 2 supersymmetry.Previous work on this subject can be found in [1, 2]. The solutions appear to have a richerstructure than the more thoroughly studied N = 4 case. In section 2 we recall some aspects of N= 2 supergravity. In section 3 the magnetic solutions are described in terms of trajectories in thespecial geometry of the N = 2 moduli space which terminate at a supersymmetric fixed point atthe horizon. In section 4 we find that the equations can be integrated for a restricted but largeclass of cases. An intriguing relation between the Ka¨hler potential on the moduli space and themetric conformal factor emerges. Some simple examples are worked out in detail in section 5.We do not attain a complete characterization of the classical geometry of N = 2 black holes inthis paper, but we hope that our results prove useful for future efforts in this direction.2 Special Geometry and N = 2 SupersymmetryWe study N = 2 supergravity coupled to n N = 2 vector multiplets in the framework of specialgeometry [3]–[6]. In this section some formulae that will be needed in the following are recalled.Further details can be found in [4] whose notation we adopt. The supergravity theory is definedin terms of a projective covariantly holomorphic section (XΛ(φi),− i2FΛ(φi)), Λ = 0, 1, ..., n, i =1, ..., n, of an Sp(2n+2) vector bundle over the moduli space parametrized by φi. (We note thatalternate conventions are often employed in which the definition of FΛ differs by a factor of 2i.)In some cases the theory can be described in terms of a covariantly holomorphic function F (X)of degree two:FΛ(φi) = FΛ(X(φi)) =∂∂XΛF (X) . (1)Given a prepotential F (X), or a covariantly holomorphic section (XΛ, − i2FΛ), one can constructthe entire scalar and vector parts of the action.It is convenient to introduce the inhomogeneous coordinatesZΛ =XΛ(φi)X0(φi), Z0 = 1 . (2)We assume ZΛ(φi) to be invertible, so that, in special coordinates,∂ZΛ∂φi= δΛi . In this case thecomplex scalars Z i = φi (i = 1, ..., n) represent the lowest component of the n vector multipletsof N = 2 supersymmetry. The Ka¨hler potential determining the metric of these fields isK(Z, Z¯) = 2 ln |X0| = − ln(NΛΣ(Z, Z¯)ZΛZ¯Σ)= − ln12[f(Z) + f¯(Z¯) +12(Z i − Z¯ i)(f¯i − fi)] , (3)2where NΛΣ =14(FΛΣ + F¯ΛΣ) and f(Z) = (X0)−2F (X). In the conformal gauge [4]NΛΣXΛX¯Σ = 1 . (4)The graviphoton field strength, as well as the field strengths of the n Abelian vector multiplets,are constructed out of n+1 field strengths FˆΛµν = ∂µWΛν − ∂νWΛµ . The graviphoton field strengthisT+µν =4NΛΣXΛNIJXIXJFˆ+Σµν , (5)where the superscript + (-) denotes the (anti)-self-dual part. This defines the central charge ofthe theory, since it enters into gravitino transformation rules. The vector field strengths whichenter the gaugino supersymmetry transformations areF+Λµν = Fˆ+Λµν −14XΛ T+µν . (6)The anti-self-dual vector field strengths Fˆ−Λ are part of the symplectic vector(Fˆ−Λ−2iN¯ΛΣ Fˆ−Σ ≡ iG−Λ), (7)where again the expression for the matrix NΛΣ is derived from the prepotential F (X) [4]–[6]. Thevector part of the action is then proportional toRe Fˆ−ΛG−Λ , (8)and the graviphoton field strength can be written in the manifestly symplectic formT−µν = 2XΛG−Λµν + FΛFˆ−Λµν . (9)The Lagrangian for the scalar components of the vector multiplets is defined by the Ka¨hlerpotential asgij¯ ∂µφi ∂ν φ¯j¯ gµν , (10)where gµν is the space-time metric, andgij¯ = ∂i∂j¯K(φ, φ¯) . (11)The gravitino supersymmetry transformation law, to leading order in fermi fields, isδψαµ = 2∇µǫα −116γνλT−νλγµǫαβǫβ + iAµǫα , (12)where α, β = 1, 2 are SU(2) indices and Aµ =i2NΛΣ[X¯Λ∂µXΣ − (∂µX¯Λ)XΣ]. The gauginotransformation law isδΩΛα = 2γµ∇µXΛǫα +12γνλF+Λνλ ǫαβǫβ + 2iγµAµǫα . (13)3BPS states of the N = 2 theory have a mass equal to the central charge z. It follows from thesupersymmetry transformation rules that this is simply the graviphoton charge [6]M = |z| = |q(e)Λ XΛ −i2qΛ(m)FΛ| = eK/2|q(e)0 + q(e)i Zi +i2(q0(m)Zi − qi(m))fi − iq0(m)f | , (14)where q(e) and q(m) are electric and magnetic charges associated to iG and Fˆ and comprise asymplectic vector. Duality transformations of the N = 2 theory correspond to different choicesof the symplectic representative (XΛ,− i2FΛ) of the symplectic geometry.3 Magnetic N = 2 BPS Black HolesIn this section we discuss the general form of the supersymmetric magnetic black hole solutionsand their interpretation as interpolating solitons. It has been shown by Tod [2] in general that,for N = 2 theories, a static metric admitting supersymmetries can be put in the formds2 = −e2Udt2 + e−2Ud~x2 . (15)For spherically symmetric black hole solutions U will be a function only of the radial coordinate r.By solving the Bianchi identities dFˆΛ = 0 we then find for the radial component of the magnetic4field strengths FˆΛr ≡ 2ǫrθφFˆΛθφ :FˆΛr =qΛr2eU(r) . (16)Inserting (15) and (16) into the gravitino transformation law, and demanding that the variationvanish for some choice of ǫ, we derive the following first order differential equation:4U ′ = −√√√√(Z¯Nq)(ZNq)(Z¯NZ)(ZNZ)(Z¯NZ¯)eU , (17)where U ′ ≡ ∂U∂ρ, ρ ≡ 1/r, and we employ the notation (ZNq) ≡ ZΛNΛΣ qΣ.Equation (17) may be viewed as determining U as a function of the moduli fields ZΛ. Avanishing gaugino transformation further requires that the moduli fields obey(ZΛ)′ = −eU4√√√√(ZNZ)(Z¯Nq)(Z¯NZ)(Z¯NZ¯)(ZNq)(ZΛq0 − qΛ). (18)Differentiating again with respect to ρ and substituting (17) leads to the second order differentialequation(ZΛ)′′ −((ZNq)(ZNZ)+ q0)((ZΛ)′)2ZΛq0 − qΛ+12(ln(ZNZ)(Z¯Nq)(Z¯NZ)(Z¯NZ¯)(ZNq))′(ZΛ)′ = 0 . (19)4 It follows from equation (7) that for a generic prepotential F the electric charge will be nonzero despite thefact that Fˆ is a magnetic field strength. In general this electric charge will not respect the charge quantizationcondition. This can be avoided by restricting the prepotential as in the examples in section 5.4This equation is independent of U and can be viewed as a generalized geodesic equation whichdescribes how Z evolves as one moves into the core of the black hole. Initial conditions for Zare specified at infinity (ρ = 0) corresponding to the asymptotic values of the field. The firstderivative of Z is then fixed in terms of the charge of the black hole by the supersymmetryconstraint (18). Z will then evolve until it runs into a fixed point. It is evident from (18) and(19) that these fixed points are atZΛfixed =qΛq0, (20)where (ZΛ)′ = 0. Each fixed point is typically surrounded by a finite basin of attraction. Thelimiting form of the metric at such a fixed point is found by integrating (17),e−U =c4ρ , (21)where the constant c is given byc =√qΛNΛΣ qΣ = q0 e−K(ZΛfixed)/2 . (22)This corresponds to the maximally symmetric charged Robinson-Bertotti universe. Thus, as in[7], the extremal black holes may be viewed as solitons which interpolate between maximallysymmetric vacua at infinity and the horizon.The locations of the fixed points (20) depend on the charges but not on the asymptotic valuesof the moduli fields. Thus if the the asymptotic values of those fields are adiabatically changed,the geometry of the black hole near the horizon remains fixed. Symplectic invariance implies asimilar structure for many of the dyonic and electrically charged extremal black holes.4 Space-Time Geometry from Ka¨hler GeometryIn this section we consider a remarkably simple special class of solutions which exist for a genericprepotential. We will work in a symplectic basis in which q0 = 0 in order to simplify the equations.In such a basis the fixed points ZΛfixed move to infinite coordinate values. Thus solutions for whichthe moduli field is constant at the fixed point (which corresponds to the Reissner-Nordstro¨msolution) cannot be described in this basis5. In the q0 = 0 basis it is straightforward to checkthat (17) and (18) are solved bye2U(ρ) = eK(Z,Z¯)−K∞ , (23)andZ i = Z i∞+qi4ρ e−K∞/2 , (24)5However, by using the manifestly symplectic constraint [6] instead of the superconformal one [4], one candescribe the Reissner-Nordstro¨m configuration in terms of the Ka¨hler potential, see example 5.4.25provided that the asymptotic value of Z i is restricted to obeyZ i∞= Z¯ i∞. (25)Alternatively we may haveZ i = Z i∞+ iqi4ρ e−K∞/2 , (26)provided that the asymptotic value of Z i is restricted to obeyZ i∞= −Z¯ i∞. (27)The restrictions imply that these solutions exist only at special points in the moduli space whereall Z i are either real or imaginary. More general solutions may be obtained from these bysymplectic transformations.Thus the space-time metric isds2 = eK(Zi,Z¯i=Zi)−K∞ dt2 − e−K(Zi,Z¯i=Zi)+K∞ d~x2 , (28)where each Z i solves a three-dimensional harmonic equation and is given in eq. (24) or (26).Hence the logarithm of the spatial conformal factor is identified with the moduli space Ka¨hlerpotential.5 Examples of N ≥ 2 BPS states5.1 Calabi-Yau magnetic black holesThe prepotential isF = idABCXAXBXCX0. (29)We consider pure imaginary ZA and real dABC (corresponding to the classical Calabi-Yau modulispace)e−K(Z,Z¯) = −2dABC ImZA ImZB ImZC , (30)where ImZA = 12i(ZA − Z¯A) = Im(ZA)∞ +qA(m)re−K∞/2.ds2 =(dABC ImZA ImZB ImZC[dABC ImZA ImZB ImZC ]∞)−1dt2 −(dABC ImZA ImZB ImZC[dABC ImZA ImZB ImZC ]∞)d~x2 . (31)65.2 Massive and massless SU(1,n)SU(n) supersymmetric white holesThe prepotential isF (X0, X1) = (X0)2 − (X i)2 , e−K(Z,Z¯) = 1− |Z i|2 . (32)Here the Z i are real:Z i = Z i∞+qire−K∞/2 , (33)ds2 =(1− |Z|21− |Z∞|2)−1dt2 −(1− |Z|21− |Z∞|2)d~x2 . (34)In particular in the simplest case of i = 1 we getgtt = g−1rr =(1−2Z∞ qreK∞/2 −q2r2)−1. (35)To satisfy the supersymmetry bound we requireM = −Z∞ q√1− |Z∞|2≥ 0 , (36)and the geometry isgtt = g−1ii =(1 +2Mr−q2r2)−1. (37)These configurations are non-trivial in the limit when the ADM mass tends to zero. It will beinteresting to understand if or when such states can arise in a physical theory. The limit M → 0can be achieved via Z∞ → 0. In this limit the scalar field becomes inversely proportional to theradius r,Z(r) =qr. (38)For N = 4 BPS states the analogous massless states have been studied recently [8]–[10].The configuration exhibits a repulsive (i.e. antigravitating) singularity and was referred to asa supersymmetric white hole [9]. The difference between the metric (37) for N= 2 and thecorresponding metric obtained in [8]–[9] for N = 4 is that gN=4tt = (gN=2tt )1/2. This does notchange the repulsive nature of the singularity.5.3 SO(2,1)SO(2) ×SO(2,n)SO(2)×SO(n) BPS statesHere again we consider pure imaginary Z.F = −iXsX0((X1)2 −n∑a=2(Xa)2), e−K(Z,Z¯) = 2ImZs((ImZ1)2 −n∑a=2(ImZa)2). (39)7The metric of the BPS configuration defined by the Ka¨hler potential is given bygtt = g−1ii = ImZs((ImZ1)2 − (ImZa)2)[ImZs((ImZ1)2 − (ImZa)2)]∞−1. (40)The configurations presented in this example may contain both types of supersymmetricstates, those with attractive singularities and those with the repulsive ones, depending on thechoice of the parameters describing the harmonic functionsZ i = Z i∞+ iqi(m)re−K∞/2 , i = {s, 1, a = 2, . . . , n}. (41)Note that this example actually provides one of the particular choices of Calabi-Yau magneticblack holes.5.4 N=4, 2 pure supergravity black holes from Ka¨hler geometry per-spectiveHere we will analyse some previously known supersymmetric black hole solutions in the frameworkof the manifestly symplectic formalism [6].The Ka¨hler potential is different from the conformal gauge (4) (here XΛ, FΛ are holomorphicsections) and is given bye−K(X,X¯) = X¯ΛNΛΣXΣ . (42)5.4.1 SL(2,Z) axion-dilaton dyonsIt is instructive to analyse the known SL(2, Z)-invariant axion-dilaton dyonic black hole solution[11] of N = 4 supergravity as a particular solution of N = 2 supergravity coupled to an N =2 vector multiplet. These solutions have complex moduli. The prepotential for this theory isF (X) = 2X0X1. The holomorphic section includes(XΛ− i2FΛ)=⇒(X0−iX1),(X1−iX0). (43)For the axion-dilaton black holeX0 andX1 can be identified with two complex harmonic functionsH1, H2 as follows:X1 = H1(~x) , X0 = iH2(~x) . (44)The Ka¨hler potential ise−K(X,X¯) = i (H¯1H2 − H¯2H1) = g−1tt (~x) , (45)in agreement with the metric of the axion-dilaton black hole found in [11].85.4.2 Reissner-Nordstro¨m solutionThe prepotential for pure N = 2 supergravity is F (X) = (X0)2. The Ka¨hler potential of thistheory in the manifestly symplectic formalism [6] is given bye−K(X,X¯) = X0X¯0 = V −1(~x)V¯ −1(~x) = e−2U(~x) , (46)where X0 = V −1(~x) is a real (imaginary) harmonic function for electric (magnetic) Reissner-Nordstro¨m extremal supersymmetric black hole [1, 2].In conclusion, we have found a simple relation between the special geometry describing thecouplings of scalars and vectors in extended locally supersymmetric theories and the space-timegeometry of the black-hole-type solutions in these theories. It is likely that more general solutionswith complex moduli will be found in this framework.AcknowledgementsWe are grateful to the Aspen Center for Physics where this work was initiated. S.F. was supportedby DOE grants DE-AC0381-ER50050 and DOE-AT03-88ER40384,Task E, and by EEC ScienceProgram SC1*CI92-0789. R.K. was supported by NSF grant PHY-8612280. A.S. was supportedby DOE grant DOE-91ER40618.References[1] G.W. Gibbons and C.M. Hull, Phys. Lett. 109B, 190 (1982);P. Breitenlohner, D. Maison, and G.W. Gibbons, Commun. Math. Phys. 120, 295 (1988).[2] K.P. Tod, Phys. Lett. 121B, 241 (1983).[3] E. Cremmer, C. Kounnas, A. Van Proeyen, J.P. Deredinger, S. Ferrara, B. de Wit, andL. Girardello, Nucl. Phys. B250, 385 (1985).[4] B. de Witt, P.G. Lawers, and A. Van Proeyen, Nucl. Phys. B255, 569 (1985).[5] A. Strominger, Commun. Math. Phys. 133, 163 (1990).[6] L. Castellani, R. D’Auria, and S. Ferrara, Class. Quantum Grav. 1, 317 (1990);A. Ceresole, R. D’Auria, S. Ferrara, and A. Van Proeyen, Nucl. Phys. B444, 92 (1995).[7] G.W. Gibbons and P.K. Townsend, Phys. Rev. Lett. 71, 3754 (1993).9[8] K. Behrndt, preprint HUB-EP-9516, hep-th/9506106; R. Kallosh, preprint SU-ITP-95-12,hep-th/9506113.[9] R. Kallosh and A. Linde, preprint SU-ITP-95-14, hep-th/9507022.[10] M. Cveticˇ and D. Youm, UPR-674-T preprint, hep-th/9507160.[11] R. Kallosh and T. Ort´ın, Phys. Rev. D48, 742 (1993).10

N = 2 extremal black holes

N=2 Extremal Black Holes

N=2 Extremal Black Holes

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