Extremal black holes, nilpotent orbits and the true fake superpotential

TL;DR

This paper develops a group-theoretical framework to derive the fake superpotential for extremal black holes in supergravity, linking nilpotent orbits and Lagrangian submanifolds to solve for black hole properties.

Contribution

It introduces a systematic method to compute the fake superpotential W for extremal black holes using nilpotent orbits and Lagrangian parametrizations within supergravity models.

Findings

W is obtained by solving a sextic polynomial in W^2 for N=8 supergravity.

Explicit W formulas are derived for the one-modulus S^3 model and the STU model.

The nilpotency of the Noether charge is verified on explicit solutions.

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 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.