String Theory and M-Theory

8.1 Low-energy effective actions 317
is not ¡ SL(2, ) invariant. The transformation of the scalar curvature term
under this change of variables is given by
1
2κ2
d 10 x √ −g e −2Φ R → 1
2κ2
d 10 x √ −g(R − 9
2 ∂µ Φ∂µΦ), (8.69)
where the string-frame metric is used in the first expression and the Einsteinframe
metric is used in the second one.
Using the quantities defined above, the type IIB supergravity action can
be recast in the form
S = 1
2κ2
d 10 x √ −g
− 1
8κ2
R − 1
12 HT µνρMH µνρ + 1
4 tr(∂µ M∂µM −1
)
d 10 x √ −g| F5| 2
+
εijC4 ∧ H (i)
3
∧ H(j)
3
, (8.70)
where the metric g E is used throughout. This action is manifestly invariant
under global SL(2, ¡ ) transformations.
The self-duality equation, F5 = ⋆ F5, which is imposed as a constraint in
this formalism, is also SL(2, ¡ ) invariant. To see this, first note that the
Hodge dual that defines ⋆ F5 is invariant under a Weyl rescaling, so that it
doesn’t matter whether it is defined using the string-frame metric or the
Einstein-frame metric. The definition of F5 in Eq. (8.57) can be recast in
the manifestly SL(2, ¡ ) invariant form
F5 = F5 + 1 (i)
εijB 2
2
The invariance of the self-duality equation then follows.
Type I supergravity
Field content
∧ H(j) 3 . (8.71)
As explained in Chapter 6, type I superstring theory arises as an orientifold
projection of the type IIB superstring theory. This involves a truncation
of the type IIB closed-string spectrum to the left–right symmetric states
as well as the addition of a twisted sector consisting of open strings. The
massless closed-string sector is N = 1 supergravity in ten dimensions and
the massless open-string sector is N = 1 super Yang–Mills theory with gauge
group SO(32) in ten dimensions. Therefore, the low-energy effective action
should describe the interactions of these two supermultiplets to leading order
in the α ′ expansion.