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On Symmetries of Extremal Black Holes with One and Two Centers

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Black Objects in Supergravity

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 144))

Abstract

After a brief introduction to the Attractor Mechanism, we review the appearance of groups of type \(E_{7}\) as generalized electric-magnetic duality symmetries in locally supersymmetric theories of gravity, with particular emphasis on the symplectic structure of fluxes in the background of extremal black hole solutions, with one or two centers. In the latter case, the role of an “horizontal” symmetry \(SL_{h}\left( 2,\mathbb{R }\right) \) is elucidated by presenting a set of two-centered relations governing the structure of two-centered invariant polynomials.

Based on Lectures given by SF and AM at the School “Black Objects in Supergravity” (BOSS 2011), INFN—LNF, Rome, Italy, May 9-13 2011. To appear in the Proceedings.

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Notes

  1. 1.

    We recall that a point \(x_{fix}\) where the phase velocity \(v\left( x_{fix}\right) \) vanishes is called a fixed point, and it gives a representation of the considered dynamical system in its equilibrium state,

    $$\begin{aligned} v\left( x_{fix}\right) =0. \end{aligned}$$

    The fixed point is said to be an attractor of some motion \(x\left( t\right) \) if

    $$\begin{aligned} lim_{t\rightarrow \infty }x(t)=x_{fix}. \end{aligned}$$
  2. 2.

    Here \(U\)-duality is referred to as the “continuous” symmetries of [30, 31]. Their discrete versions are the \(U\)-duality non-perturbative string theory symmetries introduced by Hull and Townsend [32].

  3. 3.

    For related results in terms of a map formulated in the “\(4D/5D\) special coordinates” symplectic frame (and thus manifestly covariant under the \(d=5\) \(U\)-duality group \(G_{5}\)), see e.g. [54, 55].

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Acknowledgments

The reported results were obtained in different collaborations with Laura Andrianopoli, Leron Borsten, Anna Ceresole, Riccardo D’Auria, Mike Duff, G. W. Gibbons, Murat Günaydin, Renata Kallosh, Emanuele Orazi, William Rubens, Raymond Stora, A. Strominger, Mario Trigiante, and Armen Yeranyan, which we gratefully acknowledge. This work is supported by the ERC Advanced Grant no. 226455, “Supersymmetry, Quantum Gravity and Gauge Fields” (SUPERFIELDS).

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Authors and Affiliations

  1. Physics Department, Theory Unit, CERN, CH-1211 , Geneva 23, Switzerland

    Sergio Ferrara & Alessio Marrani

  2. INFN-Laboratori Nazionali di Frascati, Via Enrico Fermi 40, I-00044 , Frascati, Italy

    Sergio Ferrara

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  1. Sergio Ferrara
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Correspondence to Sergio Ferrara .

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  1. Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, via E. Fermi 40, Frascati, 00044, Roma, Italy

    Stefano Bellucci

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Ferrara, S., Marrani, A. (2013). On Symmetries of Extremal Black Holes with One and Two Centers. In: Bellucci, S. (eds) Black Objects in Supergravity. Springer Proceedings in Physics, vol 144. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00215-6_9

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