Extremal black holes, nilpotent orbits and the true fake superpotential

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.

An inverse scattering formalism for STU supergravity

“…Matrices g satisfying g T ηg = η belong to SO (4,4). The subgroup K = SO(2, 2) × SO(2, 2) then satisfies the further constraint that it leaves invariant [22] …”

“…We therefore make the ansätze generalizing the ones used in [13,14] A + (t) = 1 1 − 22) with the parametrization of matrices C k as follows…”

Multi-centered first order formalism

“…based on the first order reformulation of the scalar equations of motion, was introduced in [18], and then developed in various works [19][20][21][22][23][24]. The possibility to switch from second order to first order differential equations of motion -without doubling their number -has an applicative relevance.…”

Duality covariant non-BPS first order systems

“…The simplest generalisation is the class of extremal black holes, which are still characterised by a vanishing Hawking temperature, but do not preserve any supersymmetry. The corresponding static solutions are known to be described by first order equations as well [10][11][12][13][14][15][16][17][18][19][20][21], although the latter are then not a direct consequence of supersymmetry.…”

“…In contrast, we will focus on the under-rotating (or ergo-free) black holes, which then admit a flat three-dimensional base, and include the static extremal black holes [22][23][24]. Single centre under-rotating non-BPS black holes have been studied throughout the last decade or so, from various aspects and using various techniques, see for example [10,12,15,[17][18][19][25][26][27][28] and references therein for some JHEP09 (2012)100 developments. Using the seed solution of [11,13,29] combined with a general duality transformation as explained in [30], one can construct any desired solution, but a manifestly duality covariant formulation was lacking.…”