Gravity and Strings

466 From eleven to four dimensions
16.3 Further reduction of N = 2A, d = 10 SUEGRA to nine dimensions
Now we consider the dimensional reduction of the action of the massless N = 2A, d = 10
supergravity Eq. (16.38) to nine dimensions. This reduction should give us the effective
field theory of the type-IIA superstring compactified on a circle, which we will later compare
with the effective theory of the type-IIB theory on another circle in order to find their
relations under T duality. 14
We could have reduced the 11-dimensional theory directly on a 2-torus, obtaining an
equivalent result with manifest invariance under global GL(2, R) transformations (in the
Einstein frame), according to the general arguments of Section 11.4. This would facilitate
the comparison with the reduction of the N = 2B, d = 10 theory in its manifestly
SL(2, R)-invariant form [691] since these two symmetries coincide in nine dimensions
[125], although they have very different (geometrical and non-geometrical) origins. However,
the compactification in two steps is necessary in order to obtain the T-duality relations
between the ten-dimensional fields, since these and their physical interpretation are much
simpler in the string frame, with string variables.
We start by reducing the bosonic NSNS sector of the action Eq. (16.38). Apart from
the fact that we are going to call x the compact coordinate, A (1) the KK vector, and A (2)
the winding vector, the result of this reduction was given in Eq. (15.25) and we can use it
directly. The only subtlety has to do with the normalization factor: after integration of the
compact coordinate x ∈ [0, 2πℓs], we obtain
where we have used
2πℓs ˆg 2 A
16πG (10)
NA
= 2πℓsg 2 A k0
16πG (10)
NA
=
g2 A
16πG (9) , (16.76)
NA
gA =ˆgAk − 1 2
0 , G (9)
NA = G(10)
NA /(2π Rx). (16.77)
Next, we perform the dimensional reduction of the bosonic RR sector.
16.3.1 Dimensional reduction of the bosonic RR sector
The task of reducing the RR field strengths is simplified very much by using the normalization
Eqs. (16.52). We find that the ten-dimensional odd-rank RR potentials split into the
following nine-dimensional RR potentials of odd and even rank,
Ĉ (2n−1) µ1···µ2n−1 = C (2n−1) µ1···µ2n−1 + (2n − 1)A(1) [µ1 C (2n−2) µ2···µ2n−1],
Ĉ (2n−1) µ1···µ2n−2x = C (2n−2) µ1···µ2n−2 , (16.78)
and the even-rank RR field strengths reduce to nine dimensions according to
ˆG (2n) a1···a2n = G(2n) a1···a2n , ˆG (2n) a1···a2n−1x = k −1 G (2n−1) a1···a2n−1 , (16.79)
14 T duality can also be established between the massive N = 2A, d = 10 supergravity and N = 2B, d = 10
supergravity [118, 691], but, due to lack of space, we will restrict ourselves to the simplest case.