Gravity and Strings

452 From eleven to four dimensions
Now we are going to perform the dimensional reduction. For many purposes it is enough
to reduce the bosonic fields by setting to zero all fermions in the action and also reducing
the supersymmetry transformation laws, and this is what we are going to do.
16.1.2 Reduction of the bosonic sector
The action for the bosonic fields is
ˆS =
1
16πG (11)
N
d11 ⎡
ˆx | ˆg| ⎣ ˆR − 1
ˆG
2 · 4!
2 − 1
(144) 2
⎤
1
ˆɛ ˆG ˆGĈ ⎦, (16.9)
| ˆg|
and the equations of motion are
ˆR
1
− ˆG ˆα1 ˆα2 ˆα3 ˆG
1
− ˆµˆν ˆµ ˆν ˆα1 ˆα2 ˆα3 12
12 ˆg ˆG ˆµˆν
2
= 0,
∇ ˆµ
ˆG ˆµˆν ˆρ ˆσ 1
−
32 · 27 1
ˆɛ
| ˆg|
ˆν ˆρ ˆσ ˆµ 1··· ˆµ 4 ˆν1···ˆν4 ˆG ˆG = 0.
ˆµ1··· ˆµ 4 ˆν1···ˆν4
(16.10)
The equation of motion of the 3-form potential can be rewritten in this way:
⋆
∂ ˆG + 35
Ĉ ˆG = 0. (16.11)
2
This equation has the form of a Bianchi identity and we could identify the expression in
parentheses with 7∂ ˆ ˜C where ˆ ˜C is a 6-form potential that is the dual of the 3-form potential.
This implies that the field strength of the dual 6-form is [22, 119, 136]
⋆ ˆG = 7(∂ ˆ ˜C − 10Ĉ∂Ĉ) ≡ ˆ ˜G. (16.12)
This field strength is obviously invariant under the gauge transformations
δ ˆ˜χ
ˆ ˜C = 6∂ ˆ ˜χ, (16.13)
where ˆ ˜χ is a 5-form. However, in its definition the 3-form appears explicitly and, to make
it invariant under the 3-form gauge transformations (16.4), ˆ ˜C has to transform as follows:
δ ˆχ
ˆ ˜C =−30∂ ˆχĈ. (16.14)
This procedure for defining the dual of some field in which the original field has not been
completely eliminated, by making use of the equations of motion, is usually referred to as