Gravity and Strings

622 Appendix B
is equivalent to requiring that all components of a Majorana spinor are real. Using the
property (B.67) and the definition of (anticommuting) Majorana spinors, one finds
ˆɛ ˆƔ â1···ân ˆψ = (−1) n+[n/2] ¯ ˆψ ˆƔ â1···ân ˆɛ, (B.69)
so the above bilinear is symmetric for n = 0, 3, 4, 7, and 8 and antisymmetric for n =
1, 2, 5, 6, 9, and 10.
On the other hand, taking the Hermitian conjugate 10 and using Eq. (B.63), we find
¯ˆɛ ˆƔ â1···ân
†
ˆψ = (−1) [n/2] ¯ ˆψ ˆƔ â1···ân ˆɛ, (B.70)
which implies, on comparison with Eq. (B.69), that the above bilinear is real for even n and
imaginary for odd n.
Finally, we have the useful identity
ˆƔ â1···ân (−1)
= i [n/2]+1
(11 − n)! ˆɛâ1···ân ˆb1··· ˆb11−n ˆƔ . (B.71)
ˆb1··· ˆb11−n
B.1.4 Ten dimensions
The 11-dimensional Majorana representation of gamma matrices can be constructed from
the ten-dimensional Majorana (purely imaginary) representation, according to
ˆƔ â = ˆƔ â , â = 0,...,9,
ˆƔ 10 =+i ˆƔ 0 ··· ˆƔ 9 .
(B.72)
Ten-dimensional Majorana spinors are identical to 11-dimensional spinors and the same
definitions and identities apply to them.
However, in ten dimensions we can also have Weyl spinors that satisfy many additional
identities. They are defined in terms of the chirality matrix ˆƔ11,
ˆƔ11 =−ˆƔ 0 ··· ˆƔ 9 = i ˆƔ 10 , (B.73)
so ˆƔ11 is Hermitian and satisfies ( ˆƔ11) 2 =+1. Spinors of positive, ˆψ (+) ,andnegative, ˆψ (−) ,
chiralities are defined as usual:
ˆƔ11 ˆψ (±) =±ˆψ (±) . (B.74)
Furthermore, in d = 10 we can define Majorana–Weyl fermions. It is useful to work in
a Majorana–Weyl representation of the gamma matrices in which, in addition to having
imaginary gamma matrices, the chirality matrix ˆƔ11 has the form
ˆƔ11 = I16×16 ⊗ σ 3
I16×16 0
=
. (B.75)
0 −I16×16
10 We use the convention (ab) ⋆ =+a ⋆ b ⋆ for anticommuting numbers. This is the convention used in [264,
599, 795] etc. The opposite convention is used in [946, 948] etc.