Gravity and Strings

Lie groups, symmetric spaces, and Yang–Mills fields 599
have determinant +1, and preserve the metric ˆη â ˆb = diag(+···+−···−) (n± plus (minus)
signs), so
ˆV ′â ˆη â ˆb ˆV ′ ˆb = ˆV â ˆη â ˆb ˆV ˆb . (A.54)
This implies that SO(n+, n−) matrices satisfy the defining property
ˆη â ˆb ˆ ˆb ˆd ˆη ˆdĉ = ( ˆ −1 ) ĉ â, ˆη â ˆb ˆη ˆbĉ = δâ ĉ. (A.55)
This generalizes to arbitrary signature the n− = 0 orthogonality condition ˆ T = ˆ −1 .
If we consider also translations in the n-dimensional vector space, we obtain the group
ISO(n+, n−), which acts on contravariant vectors as follows:
ˆV ′â = ˆ â ˆb ˆV ˆb + ˆW â . (A.56)
The Poincaré group in d spacetime dimensions is ISO(1, d − 1) in this notation.
We can immediately define the action of SO(n+, n−) on covariant vectors ˆVâ:
ˆV ′
â = ˆVˆb ( ˆ −1 ) ˆb â. (A.57)
Using the defining property of SO(n+, n−) matrices Eq. (A.55), we can relate covariant
and contravariant vectors in the standard way, raising and lowering indices with ˆη:
ˆVâ =ˆηâ ˆb ˆV ˆb
, ˆV â =ˆη â ˆb ˆVˆb . (A.58)
Let us now consider infinitesimal SO(n+, n−) transformations ˆ â ˆb ∼ δâ ˆb +ˆσ â ˆb . The
defining property of SO(n+, n−) matrices in the vector representation Eq. (A.55) implies
that the infinitesimal parameters of the transformation satisfy ˆσ â ˆb [â ˆb] =ˆσ and thus the
group has n(n − 1)/2 independent generators ˆM â ˆb (one for each independent parameter),
which are conveniently labeled by an antisymmetric pair of indices â ˆb (this expresses the
fact that the adjoint representation is just the antisymmetric product of two vector represen-
tations). We can write
where
ˆσ â ˆb
Ɣv
1
= 2 ˆσ ĉ
ˆd
Ɣv ˆM
â
ĉ ˆd ˆb , (A.59)
ˆM ĉ ˆd
â ˆb =+2 ˆη[ĉ â ˆη ˆd] ˆb
(A.60)
are the SO(n+, n−) generators in the vector representation. Observe that we need to divide
by two in order to avoid counting the same generator twice. These generators are normalized
so that
Tr Ɣv ˆM â ˆb
Ɣv ˆM ĉ ˆd =−4ˆη [â ˆb][ĉ ˆd] . (A.61)
The infinitesimal transformations of contravariant and covariant vectors take the forms:
δ ˆσ ˆV â = 1
2 ˆσ ĉ
ˆd
Ɣv ˆM â
ĉ ˆd ˆb ˆV ˆb ,
δ ˆσ ˆVâ
(A.62)
= ˆVˆb
ˆM
ˆb
ĉ ˆd â .
− 1
2 ˆσ ĉ ˆd Ɣv