Gravity and Strings

Gamma matrices and spinors 627
Thus, in even dimensions the gamma matrices are independent and in odd dimensions
they are not: the product of all the gamma matrices is a power of the imaginary unit.
In even dimensions, {±Ɣ a }, {±Ɣ a † }, {±Ɣ a T }, and {±Ɣ a ∗ } generate equivalent representations.
In odd dimensions, however, due to the above-mentioned constraint, only one sign
gives an equivalent representation. The matrices of the corresponding similarity transformations
are the chirality matrix Q (Ɣd+1), the Dirac matrix D±, the charge conjugation
matrix C±, and the B± matrix:
Q Ɣ a Q −1 =−Ɣ a , D± Ɣ a D −1
± =±Ɣ a † ,
C± Ɣ a C −1
± =±Ɣ a T , B± Ɣ a B −1
± =±Ɣ a ∗ .
In even dimensions all these matrices exist and, evidently,
(B.105)
D± = D∓Q, C± = C∓Q, B± = B∓Q. (B.106)
In odd dimensions Q does not exist and only one of the C± and B± exists.
In general, D is defined (up to a phase α)by
In our conventions we find
and, thus,
and then one has the relations
D = αƔ 0 Ɣ −1 ···Ɣ −(t−1) . (B.107)
D Ɣ a D −1 = (−1) t+1 Ɣ a † , (B.108)
D = D+, for odd t, D = D−, for even t, (B.109)
C± = B T ± D, for odd t, C± = B T ∓D, for even t, (B.110)
so the existence and properties of C± are determined by the existence and properties of B±.
The main result is11 B T ± = ε±(t,
π
s)B±, ε±(t, s) = sqcos (s − t ± 1) . (B.111)
4
When ε =±1, B is symmetric or antisymmetric. When ε± = 0, B± does not exist. The
value depends on (s − t) mod 8 and it is represented in Table B.2, from [878]. Observe
that, since these matrices are assumed to be unitary, we also have
B ∗ ± B± = ε±(t, s). (B.112)
Thus, for instance, when s = t only B− exists and is symmetric, whereas for s = t + 1,
both B+ and B− exist and are, respectively, antisymmetric and symmetric.
11 The function sqcos θ is defined as the projection on the x axis of the line that forms an angle θ with
the x axis and joins the origin to a square centered on the origin and with sides of length 2. Then
sqcos(−π/4, 0,π/4) =+1, sqcos(3π/4,π,5π/4) =−1, and sqcos(π/2, 3π/2) = 0.