Gravity and Strings

17.2 Type-IIB S duality 489
SU(1,1)/U(1), U(1) being the maximal compact subgroup of SU(1,1), and transform under
a combination of a global SU(1,1) transformation and a local U(1) transformation that
depends on the global SU(1,1) transformation. They are combinations of the dilaton and
the RR scalar. The group SU(1,1) is isomorphic to SL(2, R), the conjectured classical Sduality
symmetry group for the type-IIB string theory [583]. A simple field redefinition
[125] is enough to rewrite the action in terms of two real scalars parametrizing the coset
space SL(2, R)/SO(2), which can now be identified with the dilaton and the RR scalar.
Now the S-duality symmetry becomes manifest only when we rescale the metric to work
in the Einstein frame:
ϕ
−
ˆjE µν = e 2 jµν. (17.13)
However, the RR potentials we are working with here are not the most appropriate to
exhibit manifest SL(2, R) symmetry. In fact, they have been chosen because they are the
most appropriate to study T duality and the worldvolume effective actions of D-branes. In
particular, while the NSNS and RR 2-forms we are using form an SL(2, R) doublet (as
we are going to see), their field strengths do not. Furthermore, our self-dual RR 4-form
potential Ĉ (4) is not SL(2, R)-invariant. Thus, for the purpose of exhibiting the SL(2, R)
symmetry it is convenient to perform the following field redefinitions: 1,2
(2)
ˆB Ĉ
=
ˆB
, ˆD = Ĉ (4) − 3 ˆBĈ (2) . (17.14)
These new fields undergo the following gauge transformations:
and have field strengths
where η is the 2 × 2matrix
δ ˆ B = 2 ˆ , δ ˆD = 4∂ ˆ + 2 ˆ T η ˆ H, (17.15)
ˆ H = 3∂ ˆ B,
ˆF = ˆG (5) =+ ⋆ ˆF = 5
η = iσ 2 =
∂ ˆD − ˆB T η ˆ
H ,
(17.16)
0 1
=−η
−1 0
−1 =−η T . (17.17)
Given the isomorphism SL(2, R) ∼ Sp(2, R),itplays the role of an invariant metric:
SηS T = η, ⇒ ηSη T = (S −1 ) T , S ∈ SL(2, R). (17.18)
Next, we define the complex scalar ˆτ that parametrizes the coset space SL(2, R)/SO(2),
ˆτ = Ĉ (0) + ie −ˆϕ , (17.19)
1 In our conventions all fields are either invariant or transform covariantly as opposed to contravariantly.
2 The complete relations (including fermions) between the formulation of the N = 2B theory in “stringy
variables” that we have introduced in the previous section and the manifestly S-duality-covariant formulation
that we are going to introduce in this section can be found in [426].