The The Attractor MechanismAttractor Mechanisminin

Extremal Black HolesExtremal Black Holes

Alessio MARRANIAlessio MARRANIMuseo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”, Rome, ItalyMuseo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”, Rome, Italy

&&INFN – LNF, Frascati, ItalyINFN – LNF, Frascati, Italy

Prima Conferenza di Progetto del Prima Conferenza di Progetto del Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”,Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”,

Salone delle Conferenze, Ministero dell’Interno, Viminale, Rome,Salone delle Conferenze, Ministero dell’Interno, Viminale, Rome,30 November 200730 November 2007

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Collaborators:

Dr. S. Dr. S. BellucciBellucci (INFN -- LNF, Italy) Dr. A. Dr. A. CeresoleCeresole (Turin Univ. & INFN – Turin, Italy) Prof. S. Prof. S. FerraraFerrara (CERN, Switzerland & UCLA, USA & INFN -- LNF, Italy) Prof. M. Prof. M. GunaydinGunaydin (Penn State Univ., PA, USA) Dr. E. Dr. E. OraziOrazi (Turin Politecnico & INFN – Turin, Italy) Dr. A. Dr. A. ShcherbakovShcherbakov (JINR, Dubna, Russia & INFN – LNF, Italy) Dr. A. Dr. A. YeranyanYeranyan (Yerevan State Univ., Armenia & INFN – LNF, Italy)

Papers on Attractors:On some properties of the attractor equations, On some properties of the attractor equations, PLB 635, 172 (2006), PLB 635, 172 (2006), hep-th/0602161hep-th/0602161Charge orbits of symmetric special geometries and attractors,Charge orbits of symmetric special geometries and attractors, IJMP A21,IJMP A21, 5043 (20065043 (2006), ), hep-th/0606209hep-th/0606209 Mirror Fermat Calabi-Yau Threefolds and Landau-Ginzburg Black Hole Attractors,Mirror Fermat Calabi-Yau Threefolds and Landau-Ginzburg Black Hole Attractors, Riv. Nuovo Cim. 029, 1 (2006),Riv. Nuovo Cim. 029, 1 (2006), hep-th/0608091 hep-th/0608091Attractor Horizon Geometries of Extremal Black HolesAttractor Horizon Geometries of Extremal Black Holes,, XVII SIGRAV Conf. 2006,XVII SIGRAV Conf. 2006, hep-th/0702019 hep-th/0702019N=8 non-BPS Attractors, Fixed Scalars and Magic SupergravitiesN=8 non-BPS Attractors, Fixed Scalars and Magic Supergravities,, NPB 2007, in press, NPB 2007, in press, arXiv:0705.3866arXiv:0705.3866On the Moduli Space of non-BPS Attractors for N=2 Symmetric ManifoldsOn the Moduli Space of non-BPS Attractors for N=2 Symmetric Manifolds,, PLB 652, 111 (2007PLB 652, 111 (2007), ), arXiv:0706.1667arXiv:0706.16674d/5d Correspondence for the Black Hole Potential and its Critical Points4d/5d Correspondence for the Black Hole Potential and its Critical Points,, CQG 2007, in press, CQG 2007, in press, arXiv:0707.0964arXiv:0707.0964 Attractors with Vanishing Central ChargeAttractors with Vanishing Central Charge, , PLB 2007 (in press), PLB 2007 (in press), arXiv:0707.2730arXiv:0707.2730   Black Hole Attractors in Extended SupergravityBlack Hole Attractors in Extended Supergravity, , PASCOS07, PASCOS07, arXiv:0708.1268arXiv:0708.1268

Book : Supersymmetric mechanics. Vol. 2: The attractor mechanism and space time singularities,Supersymmetric mechanics. Vol. 2: The attractor mechanism and space time singularities,

LNP 701, Springer – Verlag, 2006LNP 701, Springer – Verlag, 2006

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N=2, d=4 Supergravity coupled to nN=2, d=4 Supergravity coupled to nVV Abelian vector multiplets: Abelian vector multiplets:

Maxwell-Einstein Supergravity Theory (Maxwell-Einstein Supergravity Theory (MESGTMESGT))

Field content :Field content : Supergravity MultipletSupergravity Multiplet

VielbeinVielbeinSU(2)SU(2) doublet of gravitinosdoublet of gravitinos

GraviphotonGraviphoton

U(1)U(1) gauge boson gauge boson

Doublet of gauginosDoublet of gauginos

ComplexComplexscalar fieldsscalar fields(MODULI)(MODULI)

nnVV Abelian Vector Multiplets Abelian Vector Multiplets

OverallOverall Gauge SymmetryGauge Symmetry

No Hypermultiplets will be considered : they decouple from the systemNo Hypermultiplets will be considered : they decouple from the system

Only scalars from vector multiplets are relevant for the ATTRACTOR MECHANISMOnly scalars from vector multiplets are relevant for the ATTRACTOR MECHANISM

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Bekenstein – Hawking Entropy – Area FormulaBekenstein – Hawking Entropy – Area Formula (Macroscopic Approach to Black Hole ThermodynamicsMacroscopic Approach to Black Hole Thermodynamics)

What is the Attractor Mechanism?What is the Attractor Mechanism?

We consider anWe consider an Extremal (Extremal (T=0T=0), dyonic, asymptotically flat,), dyonic, asymptotically flat, spherically symmetric, static Black Hole (BH)spherically symmetric, static Black Hole (BH)

A prioriA priori the BH entropy will depend on the following variables : the BH entropy will depend on the following variables :

BH Electric chargesBH Electric chargesBH Magnetic chargesBH Magnetic charges

Values of the moduli fields at the Event HorizonValues of the moduli fields at the Event Horizonof the black holeof the black hole::

they willthey will in generalin general depend on depend on thethe initial datainitial data of their deterministic, classicalof their deterministic, classicalevolution dynamics, i.e. on theevolution dynamics, i.e. on the asymptoticalasymptoticalvaluesvalues

Notice :Notice :

are are UNCONSTRAINEDUNCONSTRAINED;;they can take they can take anyany possible possiblecomplex valuecomplex value

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Can the moduli be Can the moduli be stabilizedstabilized at the Event Horizon of the BH?at the Event Horizon of the BH?

Can theyCan they be made be made INDEPENDENTINDEPENDENT on the on the UNCONSTRAINED asymptoticalUNCONSTRAINED asymptotical values values ??

Regardless Regardless of the initial conditions, the Horizon values depend of the initial conditions, the Horizon values depend ONLY ONLY on the on the charges, but nevertheless the evolution remains charges, but nevertheless the evolution remains DETERMINISTICDETERMINISTIC!!

ATTRACTOR MECHANISMATTRACTOR MECHANISM::In approaching the Event Horizon, the moduli completelyIn approaching the Event Horizon, the moduli completely lose memorylose memory of the initialof the initialdata, and take values dependentdata, and take values dependent ONLYONLY on theon the electric/magnetic charges of the BHelectric/magnetic charges of the BH::

S. Ferrara, R. Kallosh, Phys.Rev. D54 (1996),1514,S. Ferrara, R. Kallosh, Phys.Rev. D54 (1996),1514, Phys.Rev. D54 (1996),1525,Phys.Rev. D54 (1996),1525,

S. Ferrara, G. Gibbons, R. Kallosh, Nucl.Phys. B500 (1997),75S. Ferrara, G. Gibbons, R. Kallosh, Nucl.Phys. B500 (1997),75

Conserved charges, from

gauge-inv.

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Thus, which is the criterion to determine the purely charge-dependent configs.Thus, which is the criterion to determine the purely charge-dependent configs. of the moduli? of the moduli? How can the ATTRACTOR MECHANISM be implemented?How can the ATTRACTOR MECHANISM be implemented?

Critical implementationCritical implementation (Ferrara, Gibbons, Kallosh, Nucl. Phys. B500 (1997), 75)(Ferrara, Gibbons, Kallosh, Nucl. Phys. B500 (1997), 75)::

actually are non-degenerate critical points actually are non-degenerate critical points of an of an “effective black hole potential”“effective black hole potential”

Regular contravariant metric of theRegular contravariant metric of theSpecial Kaehler moduli space :Special Kaehler moduli space : Covariant derivative of Z:Covariant derivative of Z:

Z is the N=2 Kaehler-covariantly holomorphic “central charge function”Z is the N=2 Kaehler-covariantly holomorphic “central charge function”

Horizon moduli configs.Horizon moduli configs.characterized as critical pts.characterized as critical pts.of Vof VBH BH

CLASSICAL BLACK HOLE ENTROPYCLASSICAL BLACK HOLE ENTROPY

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General classification of BH Attractors in N=2, d=4 MESGT:General classification of BH Attractors in N=2, d=4 MESGT:

1.1. ½-BPS Attractors½-BPS Attractors:: they preserve the they preserve the maximum number of SUSYsmaximum number of SUSYs (4 out of 8), and (4 out of 8), and they do they do saturatesaturate the the Bogomol’ny – Prasad – SommerfeldBogomol’ny – Prasad – Sommerfeld ( (BPSBPS) bound:) bound:

Characterizing conditions:Characterizing conditions:

Known since the mid 90’s, starting from the cited seminal paper by Ferrara, Gibbons and Kallosh. Known since the mid 90’s, starting from the cited seminal paper by Ferrara, Gibbons and Kallosh.

2. non-BPS Attractors with 2. non-BPS Attractors with non-vanishingnon-vanishing central charge central charge: they do not preserve anySUSY of the asymptotical Minkowskian bkgd.,and do NOT saturateNOT saturate the BPS bound:

Characterizing conditions:Characterizing conditions:

Recently discovered (Goldstein et al., hep-th/0507096, Tripathy and Trivedi, hep-th/0511117, and many others…),they correspond to BH backgrounds breaking all SUSYs, but in the framework of a SUSY theory : important phenomenological implications!important phenomenological implications!

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Characterizing conditions:Characterizing conditions:

3. non-BPS Attractors with 3. non-BPS Attractors with vanishingvanishing central charge central charge: they do not preserve anySUSY of the asymptotical Minkowskian background,and do NOT saturate the BPS bound:

It can be traced back to the It can be traced back to the regularityregularityof the SKG of the moduli spaceof the SKG of the moduli space

Until Until June 2006June 2006, the unique explicit example of such a kind of extremal BH Attractors was given by , the unique explicit example of such a kind of extremal BH Attractors was given by GiryavetsGiryavetsin hep-th/0511215.in hep-th/0511215.

Such a kind of Attractors turns out to be really interesting, since it gives rise to a BH background breaking allSUSYs in the framework of N=2, d=4 MESGT, butbut without central extensionwithout central extension of the N=2 SUSY algebrapertaining the asymptotical Minkowskian background.

Homogeneous Symmetric Special Kaeheler Geometries;Homogeneous Symmetric Special Kaeheler Geometries;

Special Kaeheler Moduli Spaces arising from compactifications of d=10 Superstrings on Fermat CY3;Special Kaeheler Moduli Spaces arising from compactifications of d=10 Superstrings on Fermat CY3;

Since then, the Since then, the non-BPS, Z=0 Attractorsnon-BPS, Z=0 Attractors have been studied by S.Bellucci, S.Ferrara, A.M., E. Orazi have been studied by S.Bellucci, S.Ferrara, A.M., E. Oraziand A. Shcherbakov in a number of frameworks:and A. Shcherbakov in a number of frameworks:

Peculiar Homogeneous Symmetric Models, the so-called stPeculiar Homogeneous Symmetric Models, the so-called st22 (n (nV V =2) and stu (n=2) and stu (nVV =3) Models. =3) Models.

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Outlook and further developments:Outlook and further developments:• Study of BH Attractors in particular classes of Special Kaeheler Geometries (SKGs) of the scalar manifold of Maxwell-Einstein supergravities, such as:

SUSY extension of the moduli space: What is the supersymmetrized analogue of the Attractor Mechanism?

• Analysis of the (SK?) geometries of the moduli spaces arising from compactifications on (CY) Supermanifolds

SKGs with “deformed” periods, arising from dimensional compactifications on complex 3-folds which are CY3s only LOCALLYonly LOCALLY (for recent studies on SKG related to LOCAL CY3s, see e.g. Bilal and Metzger, hep-th/0503173)

• Going beyond the Static case :Going beyond the Static case : RotatingRotating and/or and/or Asymptotically non-flatAsymptotically non-flat (AdS) (AdS) and/or and/or with non-vanishing Cosmological Constantwith non-vanishing Cosmological Constant Extremal BHsExtremal BHs

for recent advances on for recent advances on compactifications on supermanifoldscompactifications on supermanifolds,, see e.g.see e.g. Grassi and Marescotti, hep-th/0607243 Grassi and Marescotti, hep-th/0607243

Need for extension of Symplectic Geometry on Supermanifolds (recently studied, see Lavrov and Radchenko, arXiV : 0708.3778)

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Thank You!