Gravity and Strings

Gamma matrices and spinors 625
and, using the explicit form of the three-dimensional gamma matrices,
γ5 = I2×2 ⊗ σ 2 . (B.90)
It is obviously Hermitian and imaginary (and therefore antisymmetric). In this representation
we can use as charge-conjugation matrix C− = iγ 0 , which is real and antisymmetric.
The Majorana condition says that Majorana spinors are purely real spinors, ψ = ψ ∗ .
There is another possible choice; namely C+ = γ5γ 0 , which is also real and antisymmetric
and would impose the following condition on Majorana spinors: ψ =−iγ5ψ ∗ . This is
inconsistent (just take the complex conjugate of this relation) and so with it we can define
only symplectic-Majorana spinors.
We can also build a Weyl representation that is complex:
γ a = γ a
(3) ⊗ σ 2 , a = 0, 1, 2, γ 3 = I2×2 ⊗ iσ 1 . (B.91)
With the above Majorana representation of three-dimensional gamma matrices, we find
γ5 = I2×2 ⊗ σ 3 . (B.92)
There are no Majorana–Weyl fermions in four dimensions and there are no Majorana–
Weyl representations of the gamma matrices. The Weyl and Majorana representations given
here are related by the similarity transformation (which is valid also for γ5)
We also have the identity
γ a M = Sγ a W S−1 , S = I2×2 ⊗ (I2×2 − iσ 1 ), (B.93)
γ a1···an = (−1) [n/2] i
(4 − n)! ɛa1···anb1···b4−n γb1···b4−n γ5. (B.94)
Using this identity the d = 4 Fierz identities for anticommuting spinors take the form
( ¯λMχ)( ¯ψ Nϕ) =− 1
4 ( ¯λMNϕ)( ¯ψχ) − 1
4 ( ¯λMγ a Nϕ)( ¯ψγaχ)
+ 1
8 ( ¯λMγ ab Nϕ)( ¯ψγabχ)+ 1
4 ( ¯λMγ a γ5Nϕ)( ¯ψγaγ5χ)
− 1
4 ( ¯λMγ5Nϕ)( ¯ψγ5χ). (B.95)
B.1.10 Five dimensions
There are no Majorana representations in d = 5, but only pairs of (complex) symplectic-
Majorana spinors that can be combined into a single unconstrained Dirac spinor.
Using any representation of the four-dimensional gamma matrices γ a , a = 0, 1, 2, 3, we
can construct a five-dimensional representation (which is necessarily complex, even if the
four-dimensional gamma matrices are purely imaginary)
ˆγ a = γ a , a = 0, 1, 2, 3, ˆγ 4 =−iγ5 = γ0γ1γ2γ3. (B.96)
Then the product of all the five-dimensional gammas is
ˆγ 0 ··· ˆγ 4 =+1. (B.97)