String Theory and M-Theory

8.1 Low-energy effective actions 311
Action
The bosonic action in the string frame for the D = 10 type IIA supergravity
theory is obtained from the bosonic D = 11 action once the integration over
the compact coordinate is carried out. The result contains three distinct
types of terms
The first term is
SNS = 1
2κ2
S = SNS + SR + SCS. (8.39)
d 10 x √ −g e −2Φ
R + 4∂µΦ∂ µ Φ − 1
2
|H3| . (8.40)
2
Note that the coefficient is 1/2κ 2 , which does not contain any powers of the
string coupling constant gs. This string-frame action is characterized by the
exponential dilaton dependence in front of the curvature scalar. By a Weyl
rescaling of the metric, this action can be transformed to the Einstein frame
in which the Einstein term has the conventional form. This is a homework
problem.
The remaining two terms in the action S involve the R–R fields and are
given by
SR = − 1
4κ2
SCS = − 1
4κ 2
d 10 x √
−g |F2| 2 + | F4| 2
, (8.41)
B2 ∧ F4 ∧ F4. (8.42)
As a side remark, let us point out the following: a general rule, discussed in
Chapter 3, is that a world sheet of Euler characteristic χ gives a contribution
with a dilaton dependence exp(χΦ), which leads to the correct dependence
on the string coupling constant. All terms in the classical action Eq. (8.39)
correspond to a spherical world sheet with χ = −2, because they describe
the leading order of the expansion in gs. Notice, however, that the terms
SR and SCS, which involve R–R fields, do not contain the expected factor
of e −2Φ . This is only a consequence of the way the R–R fields have been
defined. One could rescale C1 and F2 by C1 = e −Φ C1 and F2 = e −Φ F2,
where F2 = d C1 − dΦ ∧ C1 and make analogous redefinitions for C3 and
F4. Then the factor of e −2Φ would appear in all terms. However, this field
redefinition is not usually made, so the action that is displayed is in the form
that is most commonly found in the literature.
Supersymmetry transformations
Let us now examine the supersymmetry transformations of the fermi fields
to leading order in these fields. We first rewrite the gravitino variation in