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Extremal black holes, nilpotent orbits and the true fake superpotential


Bossard,  Guillaume
Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Bossard, G., Michel, Y., & Pioline, B. (2010). Extremal black holes, nilpotent orbits and the true fake superpotential. General Relativity and Gravitation, 42(3), 539-565. doi:10.1007/s10714-009-0871-1.

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 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.