Gravity and Strings

368 Dilaton and dilaton/axion black holes
The most general BH-type solution of an N = 2 theory has to be duality-invariant and
thus has to be built out of the only invariants that the special-geometry formalism contains:
the Kähler potential and the chiral connection. In [382] it was realized that the metric for
extreme BHs in N = 2 theories can always be written in the form
ds 2 = e K dt 2 − e −K d x 2 , (12.88)
where the projective coordinates X are identified with real harmonic functions H that
are also related to the n + 1U(1) vector potentials of the theory. In [132] it was realized
that one could also use complex harmonic functions, and then the 1-form Ai that appears
in non-static SWIP BH-solutions
ds 2 = e K (dt 2 + Aidx i ) 2 − e −K d x 2 , (12.89)
is related to the chiral 1-form of the N = 2SUGRA theory by
ɛijk∂ j Ak = Ai. (12.90)
More precisely, N = 4, d = 4 SUGRA with only two vector fields corresponds to an
N = 2, d = 4 SUGRA with prepotential F = 2X 0 X 1 .The axidilaton is just τ = X 1 / X 0 .It
is a simple exercise to check that the above recipe, with
X 0 = iH2, X 1 = H1, (12.91)
gives the SWIP solutions.
It is natural to conjecture that the same (or a similar) recipe should work in more general
cases since the basic principle of correspondence between components of the metric and
special-geometry invariants should be valid. 13 However, in practice, the SWIP solutions remain
the only solutions whose complete explicit form is known. Also, from our experience
with the general (non-supersymmetric) SWIP solutions, it is to be expected that general
(non-supersymmetric) BH-type solutions of N = 2SUEGRA can also be constructed by
introducing non-extremality functions and a background metric.
13 The construction of extreme BH solutions of N = 2 SUEGRAs is reviewed in [43, 57, 708, 709].