Gravity and Strings

632 Appendix B
Although these two representations are built in different ways, they are equivalent: they
are related by a complete chiral–dual-type change of basis in spinor space: 12
Ɣ+
ˆψ+ = S ˆψ−,
ˆM â ˆb = SƔ− ˆM â ˆb S−1 ,
C+ = (S −1 ) T C−S −1 ,
S = 1 √ 2 (1 + iγ5).
(B.139)
Observe that, given that C± is always real and antisymmetric and squares to minus the
identity, the condition (from now on we suppress + and − subindices)
CƔ ˆM â ˆb
C −1
=−Ɣ ˆM â ˆb T
(B.140)
is equivalent to the statement that the 4 × 4 matrices Ɣ ˆM â
ˆb αβ are at the same time a
spinorial representation of the algebra so(2, 3) and a fundamental representation of the
algebra sp(4, R). Using Cαβ as a metric to raise and lower indices, 13 one can construct real,
symmetric representations of sp(4, R) [443] mâ ˆb αβ , where
Ɣ ˆM â ˆb
C −1 αβ
. (B.141)
These objects satisfy the identity
m â ˆb αβ =
m â ˆb αβ
mĉ ˆd αβ = 2δ[â ˆb]
[ĉ ˆd] , (B.142)
which simply states that these matrices are an orthonormal basis in the ten-dimensional
space of 4 × 4real symmetric matrices with the trace of the standard product of matrices
as scalar product and, therefore, for any symmetric matrix Oαβ,
Oαβ = 1
2 mâ ˆb γδ mâ ˆb αβ Oγδ, ⇒ m â ˆb γδ mâ ˆb αβ = 2δ(γ δ) (αβ), (B.143)
and, by definition of m â ˆb ,weobtain the identity
m â ˆb αβ γδ −1
mâ ˆb = C αγ −1
C βδ −1
+ C αδ −1
C βγ , (B.144)
which is crucial for the consistency of the osp(4/N) superalgebra.
The matrices mâ ˆb αβ can also be used to convert objects in the adjoint of so(2, 3) into
objects in the fundamental of sp(4, R), which are somewhat easier to deal with.
Let us now consider the six real, antisymmetric matrices nâ αβ ,
nâ αβ = 1
â −1 √ i ˆγ C
2
αβ ,
n4 αβ = 1
−1 √ C
2
αβ ,
12 By this we mean a change of basis, not just a rotation of the spinors as in Eqs. (5.84).
13 Upper-left indices are contracted with adjacent lower-right indices: ξα = ξ β Cβα =−Cαβ ξ β .
(B.145)