Gravity and Strings

Appendix B
Gamma matrices and spinors
In this appendix we explain our conventions for gamma matrices in diverse dimensions and
their relations (via dimensional reduction). We start by reviewing basic facts about spinors
and gamma matrices in diverse dimensions. At the end we review spinors and gamma
matrices in spaces of arbitrary dimensions and signatures.
B.1 Generalities
Let us first review some facts about gamma matrices. 1 Gamma matrices are the generators
of the d-dimensional Clifford algebra associated with the metric ηab = diag
(+−···−), a, b = 0,...,d − 1and, therefore, satisfy the anticommutation relations
{Ɣa,Ɣb} =+2ηab. (B.1)
Any other element of the Clifford algebra can be constructed as a linear combination of the
gamma matrices and their products.
Clifford algebras are relevant in physics due to the fact that a representation of the
d-dimensional Clifford algebra for the above metric ηab can be used to construct a representation
of the d-dimensional Lorentz algebra so(1, d − 1),
[Mab, Mcd] =−ηacMbd − ηbd Mac + ηad Mbc + ηbcMad, (B.2)
that we denote by Ɣs by taking antisymmetric products of two gamma matrices:
(We use the notation
Ɣs(Mab) = 1
2 Ɣab, Ɣab ≡ Ɣ[aƔb]. (B.3)
Ɣ a1···an = Ɣ [a1 Ɣ a2 ···Ɣ an]
for the antisymmetrized (with weight unity) product of n gamma matrices.)
1 We will follow [913] but using our mostly minus-signature metric. See also [221, 923, 947].
611
(B.4)