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Extremal black holes, nilpotent orbits and the true fake superpotential

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Abstract

Dimensional reduction along time offers a powerful way to study stationary solutions of 4D symmetric supergravity models via group-theoretical methods. We apply this approach systematically to extremal, BPS and non-BPS, spherically symmetric black holes, and obtain their “fake superpotential” W. The latter provides first order equations for the radial problem, governs the mass and entropy formula and gives the semi-classical approximation to the radial wave function. To achieve this goal, we note that the Noether charge for the radial evolution must lie in a certain Lagrangian submanifold of a nilpotent orbit of the 3D continuous duality group, and construct a suitable parametrization of this Lagrangian. For general non-BPS extremal black holes in \( \mathcal{N} \) = 8 supergravity, W is obtained by solving a non-standard diagonalization problem, which reduces to a sextic polynomial in W 2 whose coefficients are SU(8) invariant functions of the central charges. By consistent truncation we obtain W for other supergravity models with a symmetric moduli space. In particular, for the one-modulus S 3 model, W 2 is given explicitely as the root of a cubic polynomial. The STU model is investigated in detail and the nilpotency of the Noether charge is checked on explicit solutions.

Keywords

Supersymmetry and Duality Black Holes in String Theory Black Holes 
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© The Author(s) 2010

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  1. 1.AEI, Max-Planck-Institut für GravitationsphysikPotsdamGermany
  2. 2.Laboratoire de Physique Théorique et Hautes EnergiesParis cedex 05France

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