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FIRST ORDER FORMALISM FOR NON-SUPERSYMMETRIC MULTI BLACK HOLE CONFIGURATIONS A.Shcherbakov LNF INFN Frascati (Italy) in collaboration with A.Yeranyan Supersymmetry in Integrable Systems - SIS'12

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Purpose In the framework of N=2 D=4 supergravity, construct the first order equation formalism governing the dynamics of the graviton, scalar and electromagnetic fields in the background of extremal black hole(s) 1.multiple black hole configuration 2.supersymmetric and non-supersymmetric 3.rotating black holes Supersymmetry in Integrable Systems 2012 2

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Why equations and not solutions? The main goal – to find a solution. The equations of motion are coupled non-linear differential equations of the second order. The known solutions are just particular ones. Why not to rewrite the equations of motion in an easier-to-solve manner? Supersymmetry in Integrable Systems 2012 3

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Results Equations Two possible cases A. B. Supersymmetry in Integrable Systems 2012 4

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Setup Einstein gravity coupled to electromagnetic fields in a stationary background With N=2 D=4 SUSY, the σ-model metric G aā and couplings μ ΛΣ and ν ΛΣ are expressed in terms of a holomorphic prepotential F=F(z). Supersymmetry in Integrable Systems 2012 5

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Reduction to three dimensions Reduction is performed in Kaluza-Klein manner metric vector-potential Three dimensional vector potentials a and w can be dualized in scalars Supersymmetry in Integrable Systems 2012 6

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If the three dimensional space is flat, the equations of motion read with an additional constraint These equations contain the following objects Equations of motion Supersymmetry in Integrable Systems 2012 7 divergenless

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Black hole potential In the case of a single non-rotating black hole tensorial black hole potential reduces to a singlet For N=2 D=4 SUGRA where Supersymmetry in Integrable Systems 2012 8

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Black hole potential Single non rotating BHGeneral case Supersymmetry in Integrable Systems 2012 9 rotation & Maxwell Recall hints to introduce

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Equations of motion (summary) The equations of motion has the following form with the constraint where Supersymmetry in Integrable Systems 2012 10

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Present state of art supersymmetric, single centersupersymmetric, multi center non-supersymmetric, single centernon-supersymmetric, multi center Supersymmetry in Integrable Systems 2012 11

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Supersymmetric single o multi-center Single center Natural splitting Entropy Supersymmetry in Integrable Systems 2012 12 Multi center S.Ferrara, G.Gibbons, R.Kallosh ‘97F.Denef ‘00

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Non supersymmetric single center Analogous description for non-BPS black holes Entropy Supersymmetry in Integrable Systems 2012 13 A.Ceresole G. Dall’Agata ‘07 S.Bellucci, S.Ferrara, A.Marrani, A.Yeranyan ‘08 Example of a fake superpotential

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Constructing the first order equations General form of the first order equations plus other equations (if any). The algebraic constraint imposes a relation What functions W, Pi and li are equal to? Supersymmetry in Integrable Systems 2012 14

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As a starting point, let us consider the spatial infinity and the supersymmetric flow. Wi and Pi are defined by ADM mass M, NUT charge N and scalar charges π At spatial infinity Phase restoration Constructing flow-defining functions Supersymmetry in Integrable Systems 2012 15 G.Bossard’11

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Now let us generalize the consideration for the whole space: Constructing flow-defining functions To pass to a non-supersymmetric solution, “charge flipping” is needed. Supersymmetry in Integrable Systems 2012 16 G.Bossard’11 D0 D2 D4 D6 Toy example: 1. Composite 2. Almost BPS A.Yeranyan ‘12

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Composite Full set of equations Supersymmetry in Integrable Systems 2012 17

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Almost BPS Supersymmetry in Integrable Systems 2012 18 Full set of equations

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Properties We showed that solutions 1.Rasheed-Larsen black holes 2.magnetic/electric multi-black hole satisfy the corresponding equations of motion. Let us stress that all these solutions are particular ones and not general. Appearance of the phases demonstrates how the concept of “flat directions” gets generalized for multi-black hole configurations. Supersymmetry in Integrable Systems 2012 19

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THANK YOU! - I think you should be more explicit here in step two… Supersymmetry in Integrable Systems 2012 20

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