Gravity and Strings

314 The Kaluza–Klein black hole
KK vector that arises from the metric. On the other hand, we have to take into account that
the ˆd-dimensional vector representation of SO(1, ˆd − 1) gives rise to a vector and a scalar
of SO(1, d − 1) at each mass level:
ˆV (n)
ˆµ
16πG ( ˆd)
N
→ V (n)
µ , l(n) . (11.99)
Our previous experience tells us that, in the n = 0levels, the scalars l (n) will act as
Stückelberg fields for the vectors V (n)
µ ,giving rise to the mass terms for them that we
expect according to the general KK arguments. For n = 0weobtain a massless vector and
amassless scalar, Vµ and l. These are the only ones we keep in the dimensional reduction
of the theory. The massless scalar is associated with the spontaneous breaking of the ˆddimensional
gauge transformations δ ˆχ ˆV ˆµ = ∂ ˆµ ˆχ that depend on the coordinate z. Infact,
the only z-dependent gauge transformations that preserve the KK Ansatz are those linear
in z that shift the component ˆVz and they give rise to a global, non-compact symmetry
(duality) of the reduced theory.
Of course, we need to identify the lower-dimensional fields that transform correctly under
all the gauge symmetries in order to see all these arguments working. The action is
ˆS[ ˆg ˆµˆν, ˆV
1
ˆµ] = d ˆd
ˆx |ˆg| ˆR − 1 ˆG
2
, (11.100)
4
The reduction of the Einstein–Hilbert term goes exactly as before. We need only take care
of the Maxwell term. In accord with the Scherk–Schwarz formalism, we use flat indices
to identify fields that are invariant under the KK U(1) gauge transformations. Thus, the
massless d-dimensional vector field Vµ is, using the Vielbein Ansatz Eq. (11.33),
ea µ Vµ ≡êa ˆµ ˆV ˆµ =
ˆVµ − ˆVz Aµ ea µ ⇒ Vµ = ˆVµ − ˆVz Aµ. (11.101)
The ˆVz component becomes automatically the d-dimensional massless scalar l, and, thus,
we have the decomposition
ˆVz = l,
ˆVµ = Vµ + lAµ.
l = ˆVz,
Vµ = ˆVµ − ˆVz ˆgµz/ ˆgzz.
(11.102)
It is easy to check that the d-dimensional scalar and vector fields obtained in this way are
invariant under the KK U(1) δ transformations. Under the z-independent ˆd-dimensional
transformations, only Vµ transforms,
and, under the linear gauge transformations ˆχ = mz,
δχ Vµ = ∂µχ, χ =ˆχ(x), (11.103)
δml = m, δmVµ =−mAµ. (11.104)
Finally, under the rescalings of the z coordinate that rescale k and Aµ, only l transforms:
l ′ = a −1 l. (11.105)