• DOI:10.1007/JHEP01(2010)038
• Corpus ID: 15480901

Extremal black holes, nilpotent orbits and the true fake superpotential

@article{Bossard2009ExtremalBH,
  title={Extremal black holes, nilpotent orbits and the true fake superpotential},
  author={Guillaume Bossard and Yann Michel and Boris Pioline},
  journal={Journal of High Energy Physics},
  year={2009},
  volume={2010},
  pages={1-66}
}

• G. Bossard, Y. Michel, B. Pioline
• Published 12 August 2009
• Mathematics
• Journal of High Energy Physics

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 W2 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 S3 model, W2 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.