Gravity and Strings

358 Dilaton and dilaton/axion black holes
Here we have used the S-dual potential õ which is the potential related to the S-dual
field strength,
˜Fµν = 2∂[µ Ãν], (12.40)
whose existence is ensured by the equation of motion of Aµ, which is just the Bianchi identity
for ˜Fµν. Knowledge of the electric components Ftr and ˜Ftr and of the metric and dilaton
is enough to find all the components Fµν, but this form of presenting the result is more elegant
and convenient since it exhibits the symmetries of the theory acting on the solution.
In particular, we see that S duality interchanges Aµ and õ and q and p/(16πGN (4) ), and
takes ϕ0 to −ϕ0, which also takes to −.
The purely electric dilaton BH solutions with a = 1 are recovered when H2 = 1andthe
purely magnetic ones when H1 = 1. When H1 = H2 = H the scalar becomes trivial and we
recover the RN solutions. Thus, these solutions are the most general from the point of view
of electric–magnetic duality.
As usual, when W = 1, H1 and H2 can be arbitrary harmonic functions in threedimensional
Euclidean space. They may but need not have coincident poles and, thus, the
solutions describe electric and magnetic monopoles and dyons in static equilibrium.
Solutions of the four-dimensional (a = 1)-model with primary scalar hair and electric
charge have been presented in [19] and probably can be generalized to all values of a and
to higher dimensions. We will not pursue this issue any further.
12.2 Dilaton/axion black holes
The a-model is a good starting point from which to study BH solutions of supergravity/superstring
theories, but it is clearly too simple. It is natural to introduce successive
generalizations to this model that make it closer to the real thing. In higher dimensions we
can introduce differential-form potentials of higher rank, but these are associated with extended
objects. In four dimensions we can introduce, as a first step, additional vector fields,
all of them coupled in the same way to the scalar field. Then, we can introduce new scalars
or different couplings of the scalar(s) to the vector fields. We would have an action of the
form
1
S =
d 4 x |g| R + 1
2 gij∂µϕi ∂ µ ϕ j − 1
4 MijF i µν F j µν , (12.41)
16πG (4)
N
where g ij (ϕ) and Mij(ϕ) are some square matrices depending on the scalars. g ij can be
interpreted as the inverse metric of some space of which the scalars ϕi are the coordinates.
The scalar kinetic term is a σ -model.
A good example of an action of this kind is provided by the four-dimensional KK action
that one obtains from ˆd = 4 + N dimensions by compactification on T N , Eq. (11.195). The
scalars parametrize an R + × SL(N, R)/SO(2) coset space.
There is another kind of couplings of scalars to vectors that we can introduce in four
dimensions: couplings of the form
− 1
4 Nij(ϕ)F i µν ⋆ F j µν . (12.42)