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FIRST ORDER FORMALISM FOR NON-SUPERSYMMETRIC MULTI BLACK HOLE CONFIGURATIONS

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. Purpose.

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FIRST ORDER FORMALISM FOR NON-SUPERSYMMETRIC MULTI BLACK HOLE CONFIGURATIONS

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

  2. 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) multiple black hole configuration supersymmetric and non-supersymmetric rotating black holes Supersymmetry in Integrable Systems 2012

  3. 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

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

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

  6. 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

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

  8. 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

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

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

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

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

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

  14. 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 liare equal to? Supersymmetry in Integrable Systems 2012

  15. Constructing flow-defining functions 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 G.Bossard’11 Supersymmetry in Integrable Systems 2012

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

  17. Composite Full set of equations Supersymmetry in Integrable Systems 2012

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

  19. Properties We showed that solutions Rasheed-Larsen black holes 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

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

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