Gravity and Strings

476 From eleven to four dimensions
O(n) × O(n) on the left,
V ′ = NV. (16.132)
It is natural to use indices V = V A
−1
and V = V A where A, B = 1,...,2n. On
the other hand, the change of basis from L to the diagonal metric η is
V ≡ VR −1 . (16.133)
The combinations VF and V∂V −1 which are invariant under global O(n, n) transformations
appear naturally in the reduction of the fermionic supersymmetry transformation
rules. We are interested only in the purely bosonic transformation rules (with all
fermions set to zero). The reduction is made in two steps: first one reduces all the tensor
fields, then one decomposes ten-dimensional 32-component spinors into d-dimensional
spinors with extra internal indices and ten-dimensional gamma matrices into tensor products
of d-dimensional gamma matrices and matrices associated with the internal symmetries.
The second step depends strongly on n and, thus, we are going to perform only
the first step, with the ultimate goal of finding the right truncation of the matter vector
fields.
It is convenient to split the 2n-dimensional indices A, B into A1, B1 running from 1 to
n and A2, B2 that take values between n + 1 and 2n. Furthermore, in order to indicate the
correct contractions with the gamma matrices, we have defined the 2n “vector”
The result of the reduction is, then,
δˆɛ
ˆψ (+)
a
ˆƔ A = ( ˆƔ d+1 ,..., ˆƔ d+p , ˆƔ d+1 ,..., ˆƔ d+p ). (16.134)
=∇(+)
a
− 1
δˆɛ ˆψ (+) A2 =−
ˆɛ (+) +
δˆɛ(ˆλ (−) − ˆƔ i ˆψ (+)
i ) = ∂φ − 1
H
12
√ 2
4 ˆƔA1 V A1 F a ˆɛ (+)
4 ˆƔA1 V A1 ∂aV B1 ˆƔ B1 ˆɛ (+) ,
√
2
8 V A2
F ˆɛ (+) − 1
2V A2
∂V A1 ˆƔ A1 ˆɛ (+) ,
ˆɛ (+) +
√ 2
8 ˆƔA1 V A1 F ˆɛ (+) .
(16.135)
will split into several d-dimensional gravitinos (four in d = 4, since
have 16 real components for each a, which cor-
will split into
It is clear that ˆψ (+)
a
the ten-dimensional chiral spinors ˆψ (+)
a
responds to N = 4, d = 4 SUEGRA) and the combination ˆλ (−) − ˆƔ i ˆψ (+)
i
several d-dimensional dilatinos (again four in d = 4) since they transform into the dilaton
under supersymmetry. The n spinors ˆψ (+) A2 transform into vectors and so they will
split into d-dimensional gauginos (4n in d = 4) of the n vector supermultiplets. The supersymmetry
parameter splits into as many d-dimensional supersymmetry parameters as the
gravitino, giving the number N of independent supersymmetry transformations that can be
performed (N = 4ind = 4).