Gravity and Strings

16.1 Dimensional reduction from d = 11 to d = 10 459
to fix the form of the corresponding field strength. For RR potentials, a convenient general
form for the field strengths was proposed in [112, 470] and used explicitly in [136, 691]. In
differential-forms and component language, it is
ˆG (2n) = dĈ (2n−1) − ˆHĈ (2n−3) ,
ˆG (2n)
= 2n ∂Ĉ (2n−1) − 1
2 (2n − 1)(2n − 2)∂ ˆBĈ (2n−3)
.
(16.49)
The differential-forms language considerably simplifies the expressions. The gauge
transformations of the RR form potentials are
δĈ (2n−1) = d ˆ (2n−2) − d ˆB ˆ (2n−4) . (16.50)
This normalization is extremely useful because it can be generalized to the type-IIB
and massive type-IIA fields. For massless type-II supergravities, we can also introduce the
notation
Ĉ = Ĉ (0) + Ĉ (1) + Ĉ (2) + ···,
ˆG = ˆG (1) + ˆG (2) + ···,
ˆ (·) = ˆ (0) + ˆ (1) (16.51)
+ ···,
with which we can write 7 (as we are going to show)
ˆH = d ˆB,
ˆG = dĈ − ˆH ∧ Ĉ,
δ ˆB = d ˆ,
δĈ = d ˆ (·) − d ˆB ∧ ˆ (·) .
(16.52)
Now we have to prove that it is indeed possible to have magnetic RR potentials with
field strengths of that kind. First of all, the field strengths ˆG (2) and ˆG (4) we are already
using conform to this normalization. Their Bianchi identities are
d ˆG (2) = 0, d ˆG (4) − ˆH ∧ ˆG (2) = 0, (16.53)
and, from the action, the equations of motion are found to be
d ⋆ ˆG (2) + H ∧ ⋆ ˆG (4) = 0, d ⋆ ˆG (4) − ˆH ∧ ˆG (4) = 0. (16.54)
The Bianchi identities for the dual field strengths ˆG (8) and ˆG (6) are, according to the
general normalization,
d ˆG (8) − ˆH ∧ ˆG (6) = 0, d ˆG (6) − ˆH ∧ ˆG (4) = 0. (16.55)
By comparison with the equations of motion for the electric potentials, we find the relations
ˆG (8) =− ⋆ ˆG (2) , ˆG (6) =+ ⋆ ˆG (4) . (16.56)
7 The RR gauge transformations are also written, after a redefinition of the gauge parameters, in the form
δĈ = d ˆ (·) e ˆB .