Gravity and Strings

310 The Kaluza–Klein black hole
and charge13 ˜q = pzk −1
0 . (11.85)
The following identity, known as a Bogomol’nyi identity, issatisfied: 14
M =|˜q|. (11.86)
This is similar to the identity satisfied by the electric charge and mass of an ERN BH,
between the mass and the NUT charge of an extreme Taub–NUT solution, or between the
action and the second Chern class of instantons. We will see that this is not a coincidence. In
Chapter 13 we will see that all of them are Bogomol’nyi identities (saturated Bogomol’nyi
bounds) signaling the presence of residual supersymmetries in the background.
If Z is a compact coordinate with period 2πℓ then the single-valuedness of the wave
function implies that the momentum pz would be quantized,
pz = n/ℓ (11.87)
(in natural units), and so would the mass and charge of the corresponding KK mode be, as
we know. Actually, since k0 = Rz/ℓ,wefind
M =|n|/Rz, ˜q = n/Rz. (11.88)
To finish this section we can try to see how far one can go without assuming that there
is an isometry in the direction of the compact coordinate z. Using the split Eq. (11.28),
we can equally well arrive at the action (11.73) but now with the fields having periodic
dependences on z.Now we should proceed to Fourier-expand all of them. This is not trivial,
though, since we do not know how to expand Z because it is not a periodic function of Z
(although ˙Z is).
11.2.4 Electric–magnetic duality and the KK action
As in the case of the four-dimensional Einstein–Maxwell theory, the four-dimensional KK
theory has an electric–magnetic symmetry, but, instead of being a continuous symmetry
(at the classical level), it is a discrete Z2 symmetry. The duality transformation has to be
defined very carefully in order to give consistent results. When this is done, the duality can
be used to construct new solutions of the same theory. In general the duality transformation
is not a symmetry, but relates two different theories or different degrees of freedom of the
same theory.
We start by performing a Poincaré-duality transformation on the (modified-Einsteinframe)
KK action. We remind the reader that the replacement of ˜F by its dual in the action
leads in general to an action with the wrong sign for the kinetic term, which does not give
rise to the dual equations of motion. This is why one has to follow the Poincaré-duality procedure
explained in Section 8.7.1. Only the term involving the vector field in Eq. (11.55)
13 q = pz for the untilded Aµ field.
14 M =|q|k −1
0 for the untilded Aµ field.