Gravity and Strings

Gamma matrices and spinors 623
We will see explicitly that it is possible to have ˆƔ11 defined as above in terms of the ten
gamma matrices and at the same time having precisely that form. The sign of ˆƔ11 is chosen
in order to have that relation with positive sign and to have as well
which leads to
ˆƔ11 = 1
10! ˆɛâ1···â10 ˆƔ â1···â10 =
Ɣ11 ˆƔ â1···ân = (−1) [(10−n)/2]+1
(10 − n)!
1
10! |ˆg| ˆɛ ˆµ1··· ˆµ10 ˆƔ ˆµ1··· ˆµ10 , (B.76)
ˆɛ â1···ân ˆb1··· ˆb10−n ˆƔ ˆb1··· ˆb10−n
. (B.77)
In the Majorana–Weyl representation each 32-component real Majorana spinor ˆψ can be
constructed from one positive-chirality and one negative-chirality 16-component spinor:
ˆψ
ˆψ =
(+)
. (B.78)
ˆψ (−)
B.1.5 Nine dimensions
We have chosen a Majorana–Weyl representation for the ten-dimensional gamma matrices.
They can be constructed from a purely real representation of the nine-dimensional ones:
where Ɣ 8 satisfies
ˆƔ a = Ɣ a ⊗ σ 2 , a = 0,...,8, ˆƔ 9 = I16×16 ⊗ iσ 1 , (B.79)
Ɣ 8 = Ɣ 0 ···Ɣ 7 . (B.80)
As usual, it will be proportional to the eight-dimensional chiral matrix Ɣ(8) 9 (see below).
One can explicitly check that, with these definitions, the ten-dimensional representation of
the gamma matrices is indeed chiral and ˆƔ11 = I16×16 ⊗ σ 3 .
B.1.6 Eight dimensions
The purely nine-dimensional gamma matrices we are using can be constructed in the standard
way from a purely real eight-dimensional representation (which is not chiral):
The chirality matrix is defined by
Ɣ a = Ɣ a (8) , a = 0,...,7, Ɣ8 = Ɣ 0 ···Ɣ 7 . (B.81)
Ɣ(8) 9 = iƔ 8 = iƔ 0 ···Ɣ 7 . (B.82)
We will not be able to decompose this representation in terms of a seven-dimensional
representation. There are no purely real or imaginary (in Lorentzian signature) representations
of the gamma matrices in seven dimensions. Thus, we cannot decompose
Ɣ a (8) = Ɣa (7) ⊗ A2×2 with the same factor matrix A for all a = 0,...,6. Thus, it is impossible
to use this representation to perform a dimensional reduction from d = 8tod = 7, 6, 5
dimensions because we would break Lorentz invariance.