Gravity and Strings

11.2 KK dimensional reduction on a circle S 1 313
self-dual point in moduli space, a well-known phenomenon in the context of T duality, in
which there is an enhancement of gauge symmetry at the self-dual points.
We should also stress that the electric–magnetic-duality transformation acts on the KK
frame and ˆd metric in a highly non-trivial way. Also, since it is only a discrete Z2 transformation
even at the classical level, we cannot use it to construct dyonic solutions, although
some dyonic solutions can be found.
Afinal remark: the dual KK action Eq. (11.91) in d = 5isidentical to the fivedimensional
string effective action up to k0 factors, Eq. (15.13), with the identification
˜k = e φ . Evidently, in the Einstein frame the two actions would be absolutely identical with
the identification k = e φ . Then, if we are careful enough with factors of k0, wecan identify
any solution of the five-dimensional string effective action involving only the dilaton,
the Kalb–Ramond 2-form (these fields are introduced in Part III), and the metric in sixdimensional
pure gravity.
11.2.5 Reduction of the Einstein–Maxwell action and N = 1, d = 5 SUGRA
Although the beauty of Kaluza–Klein theories is that they geometrize other interactions,
unifying all of them in gravity, it is possible, and sometimes necessary, to introduce other
fields in ˆd dimensions. For instance, in the compactification of supergravity theories we
have to include at the very least all the fields that enter into the supermultiplet in which
the graviton lies. In higher dimensions, apart from gravitinos, the minimal supergravity
multiplet necessarily contains other fermions plus scalars and k-form fields. In Part III we
are going to reduce several of these supergravity theories but now we want to see in a
simple example (N = 1, d = 5 SUGRA) how the Scherk–Schwarz formalism works in the
presence of matter fields.
In ˆd = 5 the minimal SUGRA [261] has a metric, a vector field, and a pair of symplectic
Majorana gravitinos that are associated with eight real supercharges. The action of the
bosonic sector is essentially the Einstein–Maxwell action with an extra topological (in the
sense of metric-independent) cubic Chern–Simons term:
ˆS =
d5 ˆx
|ˆg| ˆR − 1
4 ˆG 2 + 1
12 √ 3
ˆɛ
|ˆg|
ˆG ˆG ˆV
, (11.98)
where ˆG = 2∂ ˆV is the 2-form field strength of the vector ˆV . The field strength and the
action (up to a total derivative) are invariant under the gauge transformations δ ˆχ ˆV = ∂ ˆχ.
We want to reduce this theory on a circle, but with the same effort we can first perform
the reduction of the ˆd-dimensional Einstein–Maxwell theory (without any topological term)
on a circle and then apply the results to our case.
Before we dimensionally reduce the action of the ˆd-dimensional Einstein–Maxwell theory,
it is convenient to know the spectrum of new states that appear when we consider a
massless spin-1 particle on a circle. According to general arguments, we expect an infinite
tower of states with masses proportional to the inverse of the compactification radius. Furthermore,
we know that these massive states will be electrically charged under the massless