Gravity and Strings

626 Appendix B
B.1.11 Six dimensions
With the above representation of the five-dimensional gamma matrices, we can construct a
six-dimensional representation
ˆγ â
=ˆγ â ⊗ σ 1 , â = 0, 1, 2, 3, 4, ˆγ 5
which is a Weyl representation, since the chirality matrix ˆγ 7 is
A useful formula is
= I4×4 ⊗ iσ 2 , (B.98)
ˆγ 7 = ˆγ 0 ··· ˆγ 5 = I4×4 ⊗ σ 3 . (B.99)
ˆγ â1···ân (−1)
= [n/2]
(6 − n)! ˆɛâ1···ân ˆb1··· ˆb6−n ˆγ ˆγ ˆb1··· ˆb6−n 7. (B.100)
B.2 Spaces with arbitrary signatures
We now want to generalize our results on spinors and gamma matrices to d-dimensional
spaces with signatures (+ t , − s ), where t is the number of timelike dimensions and s is the
number of spacelike dimensions. The essential reference is [642] and other useful references
are [404, 828, 878], which we roughly follow.
The general setup is the same as in the signature-(1, d − 1) case: we consider the generators
of the Clifford algebra associated with the metric ηab = diag(+ t , − s ), where the indices
are a, b =−(t − 1), −(t − 2),...,0, 1,...,s, which is the metric of SO(t, s). These
are the 2 [d/2] × 2 [d/2] gamma matrices Ɣ a which satisfy the usual anticommutation relations
and out of which one can build the generators of so(t, s) in the spinorial representation
in the usual form. They are unique up to similarity transformations. The complex
2 [d/2] -component vectors in the representation space are Dirac spinors. We consider unitary
representations and, therefore, all timelike (spacelike) gamma matrices are Hermitian
(anti-Hermitian):
Ɣ a † =+Ɣ a , a ≤ 0, Ɣ a † =−Ɣ a , a > 0. (B.101)
Arepresentation can be constructed by “Wick-rotating” the signature-(1, d − 1) matrices,
multiplying them by factors of i if necessary. Given a d-even representation, one can
construct the chirality matrix Q = Ɣd+1,
Ɣd+1 = ϕ(s, t)Ɣ −(t−1) Ɣ −(t−2) ···Ɣ −1 Ɣ 0 Ɣ 1 ···Ɣ s , ϕ(s, t) =−e πi
4 (s−t) , (B.102)
which is unitary and Hermitian and anticommutes with all the Ɣ a s. Using it, we can construct
a representation of the (d + 1)-dimensional gamma matrices: if the signature is
(t, s + 1) we define
Ɣ s+1 =−iƔd+1, (B.103)
and, if the signature is (t + 1, s),wesimply define
Ɣ −t = Ɣd+1. (B.104)