Gravity and Strings

298 The Kaluza–Klein black hole
which can be integrated to give global rescalings plus shifts of the coordinate z:
z ′ = az + b, a, b ∈ R. (11.23)
The former can be projected onto the directions orthogonal or parallel to the Killing vector.
In orthogonal directions they are just ( ˆd − 1)-dimensional GCTs,
In parallel directions they act only on z,
δɛ x µ = ɛ µ , ɛ µ = ˆ µ ˆν ˆɛ ˆν =ˆɛ µ . (11.24)
δz =−, = k −2 ˆk ˆν ˆɛ ˆν =ˆɛ z , (11.25)
which must correspond to some local internal symmetry of the lower-dimensional theory.
As we argued before, the ˆd-dimensional metric is going to give rise to the massless ( ˆd − 1)dimensional
fields (11.14). These fields should have good transformation properties under
this internal symmetry. In particular, the metric must be invariant under it and the vector
must transform under it in the standard way (because it is massless):
δ Aµ = ∂µ. (11.26)
Observe that the periodicity of has to be the same as the periodicity of z,inorder for it
to be a well-defined coordinate transformation. We know that the period of the U(1) gauge
parameters is related to the unit of electric charge, and we will see that this is also the case
in KK theories.
Using the above transformation law for the various components of the ˆd-dimensional
metric, we arrive at the conclusion that the lower-dimensional fields are the following natural
combinations of them:
gµν =ˆgµν −ˆgzµ ˆgzν/ ˆgzz, Aµ =ˆgµz/ ˆgzz, k =|ˆgzz| 1 2 =|ˆk ˆµ ˆk ˆµ| 1 2 . (11.27)
Equivalently, we can say that the higher-dimensional metric decomposes as follows:
ˆgµν = gµν − k 2 Aµ Aν, ˆgµz =−k 2 Aµ, ˆgzz =−k 2 . (11.28)
Furthermore, under the global transformations of the internal space Eq. (11.23), the metric
is invariant and only Aµ and k transform. The shifts of z have no effect on them and we
are left with a multiplicative R duality group that can be split according to R = R + × Z2.
Only R + acts on k,
and only Aµ transforms under the Z2 factor,
A ′ µ = aAµ, k ′ = a −1 k, a ∈ R + , (11.29)
A ′ µ =−Aµ. (11.30)
It is a general rule that, in dimensional reductions, global internal transformations give
rise to non-compact global symmetries of the lower-dimensional-theory action which