Gravity and Strings

488 The type-IIB superstring and type-II T duality
Finally, the dual of the NSNS 2-form is a 6-form whose field strength can be written in
the form
⋆ ˆG (2n+3) ∧ Ĉ (2n) . (17.9)
ˆH (7) = d ˆB (6) + 1
n=3
2
n=0
17.1.2 The type-IIB supersymmetry rules
The supersymmetry transformation rules of N = 2B, d = 10 supergravity, generalized to
include the magnetic RR potentials and field strengths plus Ĉ (10) , are, suppressing the i, j =
1, 2 SO(2) indices in fermions and Pauli matrices [117],
where
δˆεê ˆµ â =−i ¯ ˆε ˆƔ â ˆζ ˆµ,
δˆε ˆζ ˆµ =∇ˆµˆε − 1
8 ˆH ˆµσ3ˆε + 1 ˆϕ e
16
δˆε ˆB ˆµˆν =−2i ¯ ˆεσ 3 ˆƔ[ ˆµ ˆζˆν],
δˆε Ĉ (2n−2) ˆµ1··· ˆµ2n−2 = i(2n − 2)e−ˆϕ ¯ ˆεPn ˆƔ[ ˆµ1··· ˆµ2n−3
δˆε ˆχ =
n=1,···,5
1
(2n − 1)! ˆG (2n−1) ˆƔ ˆµPn ˆε,
1
ˆζ ˆµ2n−2] −
2(2n − 2) ˆƔ
ˆµ2n−2] ˆχ
+ 1
2 (2n − 2)(2n − 3)Ĉ (2n−4) [ ˆµ1··· ˆµ2n−4δˆε ˆB ˆµ2n−3 ˆµ2n−4],
∂ ˆϕ − 1
12 ˆHσ 3
δˆε ˆϕ =− i
2 ¯ ˆε ˆχ,
Pn =
ˆε + 1 ˆϕ e 4
n=1,···,5
σ 1 , n even,
iσ 2 , n odd.
(n − 3)
(2n − 1)! ˆG (2n−1) Pn ˆε,
(17.10)
(17.11)
Observe that the consistency of these supersymmetry transformations demands that the
gravitinos and supersymmetry transformation parameters have the same chirality, opposite
to that of the dilatinos. Observe also that, due to the self-duality of ˆG (5) ,
ˆG (5) = ˆG
(5) 1
2 (1 + ˆƔ11). (17.12)
The ˆG (5) term in δˆε ˆζ ˆµ survives due to the negative chirality of ˆε and does not survive in
δˆε ˆχ for the same reason. This fact plays an important role in the existence of maximally
supersymmetric solutions of this theory.
17.2 Type-IIB S duality
In the original version of the ten-dimensional, chiral N = 2 supergravity [820] the theory
has a classical SU(1,1) global symmetry. The two scalars parametrize the coset space