Gravity and Strings

496 The type-IIB superstring and type-II T duality
compatible with fermions, which, combined with the standard KK Ansatz Eqs. (11.33),
give Eqs. (15.30). In our case we have already reduced the N = 2A fermions and supersymmetry
transformation rules using the standard KK Ansatz (with Aµ renamed A (1) µ) and
it just turns out that we can obtain agreement with those results only by using the lower
sign in Eqs. (15.30) (with Bµ renamed A (2) µ)[505, 506]. To be more explicit, the Ansatz
that we must use in the reduction of the N = 2B, d = 10 theory is
ê ˆµ â
=
⎛
⎝ eµ a −k −1 A (2) µ
0 −k −1
⎞
⎠,
êâ ˆµ
=
⎛
⎝ ea µ −A (2) a
0 −k
⎞
⎠. (17.38)
The sign is irrelevant in the reduction of the bosonic sector, as we stressed, and, thus, it
does not change the type-II Buscher rules we just derived.
The N = 2B, d = 10 spinors are pairs of Majorana–Weyl spinors with only 16 real nonvanishing
components out of the 32 in the chiral basis that we are using with ˆƔ11 = I16×16 ⊗
σ 3 . The indices that label each pair of fermions are usually not explicitly shown. The Pauli
matrices that appear in the supersymmetry rules Eqs. (17.10) act only on those indices and
both survive in the nine-dimensional theory. In the decomposition of the ten-dimensional
gamma matrices that we have used in the type-IIA case new Pauli matrices appear but they
do not act on those indices; rather, they act on the chiral (upper and lower) components of
the 32-component spinors. These Pauli matrices do not survive the reduction.
Taking all these facts into account, the ten-dimensional 32-component fermions, which,
including the supersymmetry parameter, are ˆζ i
ˆµ , ˆχ i , and ˆε i and the nine-dimensional,
16-component fermions ψi µ ,λi ,ρi , and ɛi ,are related by
ˆζ i y =
0
−k−1ρi
, ˆζ i µ =
0
ψi µ − k−1 A (2) µρi ˆχ i = σ 2
0
λi − ρi
, ˆε i
0
=
ɛi
,
,
(17.39)
and, using these relations, we obtain complete agreement with the nine-dimensional supersymmetry
transformation rules that we derived from the N = 2A theory, Eqs. (16.88)–
(16.90).
Using Eqs. (16.86) and (17.39), we can derive Buscher’s rules for the fermions. They
are not too interesting except for the supersymmetry parameters, since they can be used for
Killing spinors, if they are independent of the compact coordinates, which is not always the
case, as we discussed in Section 16.6. Recalling that the gamma matrix which points into
the direction into which we T dualize is ˆƔ 9 = I16×16 ⊗ iσ 1 ,itisimmediately possible to
derive the two T-duality rules:
ˆɛ =ˆε2 − i ˆƔ 9ˆε 1 ,
ˆε 1 =− i
ˆƔ 9 1 + ˆƔ11 ˆɛ, ˆε
2 2 = 1
1 − ˆƔ11 ˆɛ.
2
(17.40)