Abstract
We provide an intrinsic classification of the large and small orbits for N = 2 , 4D extremal black holes on symmetric spaces which does not depend on the duality frame used for the charges or on the special coordinates. A coordinate independent formula for the fake superpotential W, which (at infinity) represents the black hole ADM mass, is given explicitly in terms of invariants of the N = 2 special geometry.
Highlights
Black hole solutions ofgravity theories with abelian vector and scalar fields, arising at low energy from superstring or M-theory, are presently at the centre of a broadening field of research
The highest root of the cubic is λ1 and for small black holes it will coincide with the fake superpotential WnonBPS in various non-BPS orbits to be discussed below
There is only one small orbit, arising for i1 = i2 everywhere; it has W = i1 and it is BPS. This approach gives a clean intrinsic classification of both large and small orbits of N = 2 black holes for special geometries based on symmetric spaces G/H entirely in terms of the H invariants
Summary
Black hole solutions of (super-)gravity theories with abelian vector and scalar fields, arising at low energy from superstring or M-theory, are presently at the centre of a broadening field of research. For fixed values of the I4 invariant in 4d and of analogous cubic invariants I3 existing in 5d, the charge vector Q for supergravity theories based on symmetric spaces describes orbits whose nature determines the amount of supersymmetry preserved by the attractor points [9]. Small orbits arise for vanishing horizon area, when I4 = 0, they have zero entropy and they correspond to solutions with no attractor behaviour: the scalars fields never reach a fixed point at finite distance in moduli space. The highest root of the cubic is λ1 and for small black holes it will coincide with the fake superpotential WnonBPS in various non-BPS orbits to be discussed below.
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