Gravity and Strings

494 The type-IIB superstring and type-II T duality
magnetic dual of the mass parameter. 4 In turn, the IIA Ĉ (9) implies the existence of
aIIBĈ (10) associated with the D9-brane. On combining these results with S duality,
we arrive at the possible existence of the S dual of Ĉ (10) , ˆB (10) ,whose on-shell
supersymmetry transformation is given in Eq. (17.41). A new T duality implies the
existence of another ˆB (10) in the IIA theory. These potentials play an interesting role,
as we are going to see in Section 17.5 [123, 580]. T-duality rules for B (10) and B (6)
have been given in [375].
17.3.1 The type-II T-duality Buscher rules
We are now ready to relate the N = 2A, B, d = 10 fields. For the sake of completeness we
summarize the relation between ten- and nine-dimensional fields for each theory first.
Summary of the type-IIA reduction
NSNS fields:
gµν =ˆgµν −ˆgµx ˆgνx/ ˆgxx,
RR fields:
ˆgµν = gµν − k 2 A (1) µ A (1) ν,
ˆBµν = Bµν − A (1) [µ A (2) ν],
ˆφ = φ + 1
ln k, 2
ˆgµx =−k2A (1) µ,
ˆBµx = A (2) µ,
ˆgxx =−k 2 ,
Bµν = ˆBµν +ˆg[µ|x| ˆBν]x/ ˆgxx,
φ = ˆφ − 1
ln |ˆgxx|,
4
A (1) µ =ˆgµx/ ˆgxx,
A (2) µ = ˆBµx,
k =|ˆgxx| 1 2 .
Ĉ (2n−1) µ1···µ2n−1 = C (2n−1) µ1···µ2n−1 + (2n − 1)A(1) [µ1 C (2n−2) µ2···µ2n−1],
Ĉ (2n+1) µ1···µ2n x = C (2n) µ1···µ2n ,
C (2n−1) µ1···µ2n−1 = Ĉ (2n−1) µ1···µ2n−1 − (2n − 1) ˆg[µ1|x| Ĉ (2n−1) µ2···µ2n−1]x/ ˆgxx,
(17.32)
C (2n) µ1···µ2n = Ĉ (2n+1) µ1···µ2n x. (17.33)
Summary of the type-IIB reduction
NSNS fields:
ˆjµν = gµν − k −2 A (2) µ A (2) ν,
ˆBµν = Bµν + A (1) [µ A (2) ν],
ˆϕ = φ − 1
ln k, 2
ˆjµy =−k−2 A (2) µ,
ˆBµy = A (1) µ,
ˆjyy =−k −2 ,
gµν =ˆjµν −ˆjµy ˆjνy/ ˆjyy,
Bµν = ˆBµν +ˆj[µ|y| ˆBν]y/ ˆjyy,
φ =ˆϕ − 1
ln |ˆjyy|,
4
A (1) µ = ˆBµy,
A (2) µ =ˆjµy/ ˆjyy,
k =|ˆjyy| − 1 2 .
(17.34)
4 As explained on page 342, a non-vanishing value of the potential Ĉ (9) is related to the GDR associated with
the shifts of Ĉ (0) that we discussed before, which are in turn related to the mass parameter of Romans’
theory. Clearly, the whole picture is consistent.