Gravity and Strings

Gamma matrices and spinors 629
Table B.3. Possible spinors in d = t + s dimensions with signatures
(+ t , − s ).Mstands for Majorana, pM for pseudo-Majorana, SM for
symplectic-Majorana, pSM for pseudo-symplectic-Majorana, MW for
Majorana–Weyl, and pMW for pseudo-Majorana–Weyl; ∗ meaning
that d has to be even. In addition to this, Weyl spinors are possible
for any even d.
s − t
1 2 3 4 5 6 7 8
pSM pSM pSM
pM pM pM
M M M
SM SM SM
pMW∗ MW∗ which, using the relation among D, C,andB, can be simplified to
which is satisfied for s − t = 0 mod 4.
Q ∗ = BQB −1 , (B.119)
(Pseudo-)symplectic-Majorana spinors. When BB ∗ =−1 the Majorana reality condition
cannot be consistently imposed. However, then one can introduce an even number
of Dirac spinors labeled by i = 1,...,2n and impose the reality condition
where is real and satisfies
¯ψ i = ψi c ≡ ijψ jc , (B.120)
ij jk =−δik. (B.121)
This condition can be rewritten in the more transparent form
ψ i ∗ = ijBψ j , (B.122)
which is consistent if BB ∗ =−1. The cases in which these spinors can be defined are
represented in Table B.3.
Now we are going to use these results in several examples of interest.
B.2.1 AdS4 gamma matrices and spinors
The spinor representations of SO(2, 3) (which we also refer to as AdS4) have the same
dimension (four) as those of SO(1, 3). The corresponding gamma matrices, which we write
with hats, are 4 × 4 matrices and any representation of them includes a representation of the
SO(1, 3) (unhatted) gamma matrices. Furthermore, it is clear that AdS4 spinors transform
as Lorentz spinors under the Lorentz subgroup. Our goal now will be to construct an explicit