Gravity and Strings

486 The type-IIB superstring and type-II T duality
we will find a non-perturbative strong–weak-coupling relation between the type-I SO(32)
and the heterotic SO(32) superstring theories.
17.1 N = 2B, d = 10 supergravity in the string frame
The fields of N = 2B, d = 10 SUEGRA [571, 820, 823] associated with the massless
modes of the type-IIB superstring are given in Table 14.2. Actually, as we have mentioned
afew times, there are two N = 2B, d = 10 theories, with all fermionic chiralities reversed
and opposite self-duality properties of the RR 5-form field strength. Here we are going to
describe the theory with a RR 4-form Ĉ (4) + with self-dual 5-form field strength (suppressing
the upper index + for convenience),
ˆG (5) =+ ⋆ ˆG (5)
(17.1)
(all the RR field strengths are normalized as explained in Section 16.1.3 according to
Eqs. (16.51) and (16.52), the only difference being the use of calligraphic letters for the
NSNS fields ˆj ˆµˆν, ˆB ˆµˆν, and ˆϕ) and negative-chirality pairs of gravitinos and supersymmetry
transformation parameters ˆζ
i (−)
ˆµ and ˆε i (−) and positive-chirality dilatinos ˆχ i (+) (although
we also suppress these ± and the i = 1, 2 indices of the SO(2) global symmetry that rotates
the fermions for convenience):
ˆƔ11 ˆζ ˆµ =−ˆζ ˆµ, ˆƔ11ˆε =−ˆε, ˆƔ11 ˆχ =+ˆχ. (17.2)
Although the self-duality equation for ˆG (5) looks like a constraint, it is indeed one of the
equations of motion of the theory [820], and it arises as such in the superspace formalism
[571]. It is not hard to see that, combined with the Bianchi identity, it gives a conventionallooking
equation of motion:
∂ ˆG (5) − 10
3 ˆH ˆG (3) = 0 ⇒ ∂ ⋆ ˆG (5) − 10
3 ˆH ˆG (3) = 0. (17.3)
It is known that it is not possible to write a covariant action for the above self-duality
equation of motion [685]. Nevertheless, having an action is very useful (for instance, to
perform dimensional reductions) and we would like to write one. The main observation is
that, if we do away with the self-duality of the 5-form, we can find an action that gives
the conventional equation of motion Eq. (17.3) [111]. This equation of motion does not
imply self-duality when it is combined with the Bianchi identity, but only ∂ ˆG (5) = ∂⋆ ˆG (5) .
However, it is consistent with self-duality. This should be reflected in the following property
of the action: if we dualize the 4-form, we must end up with an identical action written in
terms of the dual 4-form. In other words, the action of the theory of the non-self-dual (NSD)
5-form must itself be “Poincaréself-dual.” In our conventions the action of such a Poincaré
self-dual NSD theory is
SNSD =
ˆg 2 B
16πG (10)
d
NB
10 ˆx
−2 ˆϕ |ˆj| e ˆR( ˆj)− 4 ∂ ˆϕ
2 1
+ ˆH 2
2 · 3!
+ 1
ˆG 2
(0)
2 + 1
ˆG
2 · 3!
(3)
2 + 1
ˆG
4 · 5!
(5)
2 − 1 1
ɛ∂Ĉ
192 |ˆj| (4) ∂Ĉ (2)
ˆB ,
(17.4)