Gravity and Strings

630 Appendix B
representation of the gamma matrices and the generators of the AdS4 group. Since s − t = 1
we expect only a Majorana representation. It appears in two forms, which are equivalent
through similarity transformations. We call them electric and magnetic and denote them by
a − or a + subscript or superscript, respectively.
The magnetic representation. It is built by the standard procedure from the Clifford algebra
associated with the SO(2, 3) metric ˆη â ˆb : first we construct five gamma matrices
satisfying
ˆγâ, ˆγ ˆb =+2ˆηâ ˆb . (B.123)
These matrices can be constructed by using the four SO(1, 3) Dirac matrices:
ˆγ+−1 = γ5 =−iγ 0 ···γ 3 , ˆγ+ a = γa, (B.124)
and, using purely imaginary Dirac matrices, we obtain a purely imaginary representation
of the SO(2, 3) Clifford algebra.
The SO(2, 3) generators in the magnetic spinorial representation are constructed
from the Clifford algebra in the usual fashion
Ɣ+ ˆM â ˆb = 1
2 ˆγ +â ˆb , (B.125)
and they automatically satisfy the so(2, 3) algebra Eq. (4.152).
SO(2, 3) spinors ˆψ+ α transform with the exponential of all these generators and are,
in particular, Lorentz spinors.
Since t = 2, we know that D = D−. Furthermore, the only B leading to consistent
Majorana spinors is B− and thus we can use only C+, which is antisymmetric. We
can take
C+ = D− =ˆγ 0 −1
+ ˆγ + = γ 0 γ5. (B.126)
It is easy to check that the charge-conjugation matrix C+ satisfies
C+ ˆγ â C −1
+ =+ˆγ â T =−ˆγ â † . (B.127)
Since the Dirac conjugation and charge-conjugation matrices are identical, B− =
I4×4, the Majorana condition ¯ˆψ + = ˆψ c + implies that Majorana spinors are purely real
spinors in this representation, ˆψ ∗ + = ˆψ+.
In this representation the ten matrices
ˆM â ˆb
αβ
(B.128)
Ɣ+
are real and symmetric. This is necessary in order to build the osp(N/4) supersymmetry
algebra. The six matrices
−1
C
αβ,
+ i ˆγ â + C−1
αβ
+
(B.129)
are real and antisymmetric and we will use them to add other (“central”) charges in
the anticommutator {Q α i , Q β j } supersymmetry algebra.
C −1
+