Gravity and Strings

11.2 KK dimensional reduction on a circle S 1 297
Furthermore, technically, the dimensional-reduction procedure requires that we use the
coordinate z.
Our starting point, therefore, is a ˆd-dimensional 7 metric ˆg ˆµˆν independent of z.
It is sometimes convenient to give a coordinate-independent characterization of the metrics
we are going to deal with. These are metrics admitting a spacelike Killing vector ˆk ˆµ .If
the metric admits the Killing vector ˆk ˆµ then its Lie derivative with respect to it vanishes:
Lˆk ˆg ˆµˆν = 2 ˆ∇( ˆµ ˆk ˆν) = 0 (11.17)
(this is just the Killing equation, see Section 1.5) and this is the condition we would impose
on other fields, if we had them.
To this local condition we have to add a global condition: that the integral curves of the
Killing vector are closed. z will be the coordinate parametrizing those integral curves (the
“adapted coordinate”) and it can be rescaled to make it have period 2πℓ. This global condition
will not be explicitly used in most of what follows, but only it guarantees consistency.
In adapted coordinates ˆk ˆµ = δz ˆµ .
It is reasonable to think of the hypersurfaces orthogonal to the Killing vector as the ddimensional
spacetime of the lower-dimensional theory. Then, the first object of interest is
the metric induced on them. This is
ˆ ˆµˆν ≡ˆg ˆµˆν + k −2 ˆk ˆµ ˆk ˆν, k 2 ≡−ˆk ˆµ ˆk ˆµ. (11.18)
ˆ ˆµ ˆν =ˆg ˆµ ˆρ ˆ ˆρ ˆν can be used to project onto directions orthogonal to the Killing vector and
−k −2 ˆk ˆµ ˆk ˆν to project onto directions parallel to it. In adapted coordinates, due to the orthogonality
of ˆ and ˆk, wehave
k =|ˆk ˆµ ˆkµ| 1 2 =|ˆgzz|, ˆ ˆµz = 0. (11.19)
The remaining components define the ( ˆd − 1)-dimensional metric
gµν ≡ ˆµν. (11.20)
To understand why this is the right definition of the ( ˆd − 1)-dimensional metric instead of
just ˆgµν (apart from the reason to do with orthogonality to the Killing vector), we need
to examine the effect of ˆd-dimensional GCTs on it. Under the infinitesimal GCTs δˆɛ ˆx ˆµ =
ˆɛ ˆµ ( ˆx), the ˆd-dimensional metric transforms as follows:
δˆɛ ˆg ˆµˆν =−ˆɛ ˆλ ∂ˆλ ˆg ˆµˆν − 2 ˆgˆλ( ˆµ ∂ˆν)ˆɛ ˆλ . (11.21)
For the moment, we are interested only in ˆd-dimensional GCTs that respect the KK
Ansatz, i.e. that do not introduce any dependence on the internal coordinate z. These fall
into two classes: those with infinitesimal generator ˆɛ ˆµ independent of z and those generated
by a z-dependent ˆɛ ˆµ . The latter act only on z and they are found to be only
δz = az, a ∈ R, (11.22)
7 All ˆd-dimensional objects carry a hat, whereas d = ( ˆd − 1)-dimensional ones do not. The ˆd-dimensional
indices split as follows: ˆµ = (µ, z) (curved) and â = (a, z) (tangent-space indices).