Gravity and Strings

362 Dilaton and dilaton/axion black holes
that we can arrange into a twelve-dimensional vector q . q transforms linearly under S and
T-duality transformations S and R:
q ′ = S ⊗ Rq. (12.60)
The charges8 must enter into the metric in duality-invariant combinations because the
metric is duality-invariant. There are only two such invariants that are quadratic and quartic
in the charges:
I2 ≡ q T M −1
0 ⊗ I6×6q, I4 ≡ det
n=6
q
n=1
(n) (n) T
q
. (12.62)
Here M0 is the asymptotic value of the scalar matrix M. Thus, I2 is moduli-dependent
and I4 is moduli-independent. On the other hand, I4 vanishes when only one vector field
is non-trivial and, therefore, starting from the most general charge configuration with only
one vector field and I4 = 0, we cannot generate the most general charge configuration with
I4 = 0byS-and T-duality transformations. The generating solution has to have both I2 and
I4 generically non-vanishing.
To attain a better understanding, we can try to construct the most general solution starting
from the d = 4, a = 1 dilaton BH solutions we studied in the previous section. We simply
have to observe that the equations of motion of the axion/dilaton model coincide 9 with those
of the four-dimensional a = 1 model if the axion a = 0 and F ⋆ F = 0. Then, the purely electric
BH Eq. (12.23) provides a solution of the axion/dilaton model with one independent
charge and one non-trivial modulus (ϕ0). By performing one SO(2) S-duality transformations
Eq. (12.47), we can generate a solution that has electric and magnetic charge. As
in the Einstein–Maxwell case, the SO(2) parameter becomes a new independent charge.
A non-trivial axion is generated. Further SL(2, R) transformations only shift ϕ0 and add
an asymptotic value to the axion a0. Inthis way we have obtained the most general axion/dilaton
BH solution with one vector field [850], but it has the same metric as the purely
dilatonic BH.
This solution is also a solution of N = 4, d = 4SUEGRA with five vanishing vector
fields. We could excite them by performing SO(6)/SO(5) T-duality rotations that do not
leave the charge vector invariant. However, in this way we can obtain only solutions in
which all the magnetic charges are proportional to all the electric charges with the same
proportionality factor. We would have added only five new independent parameters to the
solution and the metric would still be the same (because I4 = 0).
A more general solution with two non-vanishing charges in different vectors q (1) and
p (2) was found in [432] and, later on, studied in [612]. It has a different metric (and
8 Observe that, the axion being a local θ-parameter, it induces a Witten effect on the charges, as explained in
Section 8.7.4. Furthermore, the DSZ quantization condition takes the manifestly SL(2, R)-invariant form
(n) T
q 1 η q (n)
2 = m/2, m ∈ Z. (12.61)
(q (n) is canonically normalized, but p (n) is 1/(4π) times the canonical magnetic charge. The product of the
canonical charges is quantized in integer multiples of 2π.)
9 The vector fields have a different normalization.