9.2.4Attractor Mechanism, the Entropy and Other Special Geometry Invariants

One of the most important features of supergravity black-holes is the attractor mechanism discovered in the nineties by Ferrara and Kallosh for the case of BPS2 solutions [1, 2] and in recent time extended to non-BPS cases [7–14]. According to this mechanism the evolving scalar fields φi (τ ) flow to fixed values at the horizon of the black-hole (τ = −∞), which do not depend from their initial values at infinity radius (τ = 0) but only on the electromagnetic charges p, q.

In order to review the attractor mechanism, we must briefly recall the essential items of black hole field equations in the geodesic potential approach [6]. In this framework we do not consider all the fields listed in the table after (9.2.2). We introduce only the warp factor U (τ ) and the original scalar fields of D = 4 supergravity. The information about vector gauge fields is encoded solely in the set of electric and magnetic charges Q defined by (9.2.13). Furthermore for the sake of simplicity we focus on the case of an N = 2 theory where the 2n scalar fields span a special Kähler manifold and can be organized into n complex combinations zi . Under these conditions the correct field equations for an N = 2 black-hole are derived from the geodesic one dimensional field-theory described by the following Lagrangian:

where the geodesic potential V (z, z, Q) is defined by the following formula in terms of the matrix M4 introduced in (9.2.3):

The effective Lagrangian (9.2.35) is derived from the σ -model Lagrangian (9.2.1) upon substitution of the first integrals of motion corresponding to the electromagnetic charges (9.2.13) under the condition that the Taub-NUT charge, defined in (9.2.9), vanishes3 (n = 0). Indeed, when the Taub-NUT charge n vanishes, which

2In the supergravity framework BPS solutions are those that preserve a certain amount of supersymmetry, namely that admit a certain number of so named Killing spinors, i.e. of supersymmetry parameters such that supersymmetry transformations along them leave the chosen solution invariant.

3In [18] it was shown that every orbit of solutions contains a representative where the Taub-NUT charge is zero. Alternatively from a dynamical system point of view the Taub-NUT charge can be annihilated by setting a constraint which is consistent with the Hamiltonian and which reduces the dimension of the system by one unit. The problem of black hole physics is therefore equivalent to the sigma model based on an appropriate codimension one hypersurface in the Q manifold.