Gravity and Strings

17.3 Dimensional reduction of N = 2B, d = 10 SUEGRA and type-II T duality 493
This is clearly enough to conclude that the reduction of the N = 2A and N = 2B theories
to d = 9gives the same nine-dimensional theory. However, just as a check, we can complete
the reduction of the action of the N = 2B theory and see that it coincides with Eq. (16.84).
We are only going to outline how this is done: the NSD action can be reduced to d = 9
straightforwardly using the above formulae, but we obtain a theory with 5- and 4-form RR
field strengths that originate from the ten-dimensional self-dual 5-form field strength. The
nine-dimensional 5- and 4-forms are related by Eq. (17.31), which is a constraint of the
action that we have to eliminate in order to arrive at the action Eq. (16.84). To eliminate
consistently the constraint and the 5-form, we first Poincaré-dualize it into a second 4-form,
following the standard procedure. Finally, we identify the two 4-forms and the result is
Eq. (16.84) with a single 4-form and with the correct sign in the Chern–Simons term.
This result allows us to map fields of one ten-dimensional theory onto fields of the other
ten-dimensional theory (which is always independent of one coordinate). This mapping
is the generalization of Buscher’s T-duality rules to type-II theories [125, 691] that we
describe in the next section, but it is worth making some preliminary remarks.
1. The rules reflect the T-duality rules for D-branes that we discussed on page 428.
2. We could have reduced the manifestly S-duality-invariant action Eq. (17.22) and we
would have obtained a manifestly SL(2, R)-invariant action in d = 9. As usual in KK
compactification, the action would also be invariant under a group R + of rescalings
of the internal dimension and other Z2 factors which combine into GL(2, R), which
is the invariance group that one obtains in the reduction from d = 11 to d = 9. The
IIB S duality now has a geometrical interpretation in the IIA theory.
3. It is possible to use the full SL(2, R) invariance of the action to perform a GDR of
the theory [426, 691]. The result is a family of massive supergravity theories that depends
on three mass parameters transforming in the adjoint of SL(2, R) that fit into
a symmetric mass matrix. One of the theories, which depends on a single parameter,
is precisely the theory one would obtain by reducing Romans’ theory to d = 9
[118] 3 ,but there are other theories that cannot be obtained from known 11- and
ten-dimensional supergravities. Most of the d = 9 theories obtained in this way are
gauged supergravities [258, 726] and the gauge group is determined by the conjugacy
class of the chosen mass matrix [120]. While there is a simple string interpretation
for Romans’ theory based on the D8-brane, the remaining massive/gauged theories
have a less-conventional interpretation that is based on non-conventional extended
branes.
4. Although only low-rank RR potentials appear in the action, we have extended the
relation to the high-rank magnetic RR potentials. This implies the existence of new
high-rank RR potentials unrelated to the electric ones: the IIB Ĉ (8) can be related
to the IIA Ĉ (7) ,butalso to aĈ (9) that exists in Romans’ theory only since it is the
3 This is the theory that one obtains with the GDR Ansatz studied in Section 11.5.3. Observe that the mass
parameter is naturally quantized since it is a winding number. This implies that the mass parameter of
Romans’ theory must also be quantized, if we insist on identifying these theories as string theory indicates.
This GDR Ansatz is related to the RR 9-form potential we are going to discuss next.