Gravity and Strings

490 The type-IIB superstring and type-II T duality
and the 2 × 2symmetric SL(2, R) matrix ˆMij,given in Eq. (11.207) in terms of ˆτ, that
satisfies the property
ˆM −1 = η ˆMη T . (17.20)
Under S ∈ SL(2, R) given by Eq. (11.205) the new variables that we have defined transform
as follows:
ˆM ′ = S ˆMS T , ˆτ ′ =
α ˆτ + β
γ ˆτ + δ ,
ˆ B ′ = S ˆ B, (17.21)
and the 4-form ˆD and the Einstein metric ˆjE are inert.
Now, it is a simple exercise to rewrite the NSD N = 2B action in the following manifestly
S-duality-invariant form:
ˆSNSD =
ˆg 2 B
16πG (10)
NB
d10 ˆx
|ˆjE| ˆR( ˆjE) + 1
4Tr
∂ ˆM ˆM −1
2 + 1 ˆH 2 · 3!
T ˆM −1 ˆH + 1
4 · 5! ˆF 2 − 1
27 · 33 1
|ˆjE| ɛ ˆD ˆH Tη ˆ
H . (17.22)
Observe that the factor ˆg 2 B /(16πG(10) NB )isS-duality-invariant because it does not depend
on ˆgB (see Eq. (19.26)). Thus it is in the Einstein frame that the full action is invariant
under S duality and masses measured in this frame are S-duality-invariant. As usual, this
is not the metric in which we should measure masses (at least masses that we want to
compare with the string spectrum) because, if the string metric is asymptotically flat, the
Einstein metric is not. We should use the modified Einstein frame. This metric will also
be S-duality-invariant, but the action will have the prefactor 1/(16πG (10)
NB ) which is not
invariant and, thus, masses measured in it will not be invariant.
It is easy to find how the stringy fields ˆH, ˆG (3) , and Ĉ (4) transform under SL(2, R):
ˆH ′
= δ + γ Ĉ (0)
ˆH + γ ˆG (3) ,
ˆG (3) ′ 1
=
|γ ˆτ + δ| 2
δ + γ Ĉ (0)
ˆG (3) − γ e−2 ˆϕ
ˆH ,
Ĉ (4) ′ = Ĉ (4) − 3 Ĉ (2) ˆB (17.23)
(2)
αγ βγ Ĉ
βγ δβ ˆB
.
ˆτ transforms as above and we stress that the string metric does transform under SL(2, R):
ˆj ′ =|γ ˆτ + δ|ˆj. (17.24)
Some of the SL(2, R) transformations of N = 2B, d = 10 SUEGRA involve an inversion
of the dilaton, and, hence, of the string coupling constant ˆgB, just as we discussed in the
case of N = 4, d = 4 SUEGRA, the effective theory of the heterotic string (Sections 12.2
and 16.5.5). These are, therefore, non-perturbative transformations from the string-theory