Gravity and Strings

300 The Kaluza–Klein black hole
As a general rule, one cannot naively truncate actions by setting some fields to specific
values. Doing this in the equations of motion (the correct procedure) would leave us with
constraints that must be satisfied and cannot be obtained from the truncated actions. In
other words, one cannot reproduce all the truncated equations of motion from a truncated
action.
When will a truncation in the action be consistent? Also as a general rule, if there is a
discrete symmetry in the action, eliminating only the fields which are not invariant under
it will always be consistent. From this point if view, since there is no discrete symmetry
acting on k, the inconsistency of its elimination is not surprising. On the other hand, there
is a Z2 symmetry that acts only on Aµ and it is easy to see that it is consistent to eliminate
only this field. For instance, this truncation is used to obtain N = 1, d = 10 supergravity
from N = 1, ˆd = 11 supergravity (or the heterotic string from M theory) and can be related
to dimensional reduction over the orbifold S1 /Z2 (a segment of a line, with two boundaries)
instead of on the circle S1 .
Performing the dimensional reduction on the equations of motion is in general a quite
lengthy calculation (which we will nevertheless perform in Section 11.5). Furthermore, the
above decomposition of higher-dimensional fields into lower-dimensional ones cannot be
used in the presence of fermions.
In [836] Scherk and Schwarz described a systematic procedure for performing the dimensional
reduction in the action and using the Vielbein formalism so it can also be applied
to fermions. Another advantage of using Vielbeins is that we can work with objects
that have only Lorentz indices and are, therefore, scalars under GCTs. Since some of the
GCTs become internal gauge transformations, those objects are automatically GCT-scalars
and gauge-invariant.
The first thing to do is to reexpress the relations Eqs. (11.27) and (11.28) in terms of Vielbeins.
Using local Lorentz rotations, one can always choose an upper-triangular Vielbein
basis of the form
ê ˆµ â
=
eµ a
kAµ
,
0 k
êâ ˆµ
=
ea µ −Aa
0 k−1
, (11.33)
where Aa = ea µ Aµ and we will assume that all d-dimensional fields with Lorentz indices
have been contracted with the d-dimensional Vielbeins.
This choice of Vielbein basis breaks the ˆd-dimensional local Lorentz invariance to the
d = ( ˆd − 1)-dimensional one, which is the subgroup that preserves our choice. If there were
other symmetries (such as supersymmetry) acting on the Vielbeins, we would have to add
to them compensating Lorentz transformations in order to preserve the choice of Vielbeins.
Next, we find the non-vanishing components of ˆ â ˆbĉ ,
where
ˆabc = abc,
ˆabz =− 1
2 kFab,
ˆazz =− 1
2 ∂a ln k, (11.34)
Fab = ea µ eb ν Fµν, Fµν = 2∂[µ Aν], (11.35)
is the vector-field strength. With these we find the non-vanishing components of the spin