Gravity and Strings

602 Appendix A
As usual, the curvature is another Lie-algebra-valued field defined through the commutator
of two covariant derivatives in any representation:
ˆDµ, ˆDν ˆψ =−ˆRµν ˆψ, (A.79)
where
ˆRµν = 1
2 ˆRµν â
ˆb
Ɣ ˆM â ˆb ,
transforms as ˆσ and satisfies the Bianchi identities
ˆRµν â ˆb
= 2∂[µ ˆων] â ˆb
− 2 ˆω[µ| âĉ ˆb
ˆω|ν]ĉ , (A.80)
ˆD[µ ˆRνρ] = 0. (A.81)
SO(3) and three-dimensional real Lie algebras. For the particular case n = 3 the adjoint
representation coincides with the vector representation. Then we have two different notations
(with one and with two antisymmetric indices) for the same representation. The
relation between the two of them is
and the Tis satisfy the algebra
Ti = 1
2 ɛijkTjk, Tij = ɛijkTk, (A.82)
[Ti, Tj] =−ɛijkη kl Tl, ⇒ fij l =−ɛijkη kl , ⇒ Kij =−2ηij. (A.83)
For ηkl = δkl (for the group SO(3)) anexplicit representation is given in Eq. (A.94). The
only other possibility, η = diag (++−), isSO(2, 1). They are identical as complex algebras
(it suffices to multiply T1 and T2 by i), but not as real algebras.
All the possible real three-dimensional Lie algebras can be written in terms of a matrix
Qkl .
[Ti, Tj] =−ɛijkQ kl Tl, Q (lk) ɛkijQ ij = 0. (A.84)
If we make the separation Qlk = Q (kl) − ɛkliai, the constraint on Q is just Q (kl) al = 0. Q (kl)
can be diagonalized and its eigenvectors ai can be found, and all the three-dimensional Lie
algebras (nine in total) can be classified. This is the Bianchi classification (see e.g. [640]).
The only semisimple ones are those of SO(3) and SO(2, 1).
A.3 Riemannian geometry of group manifolds
We can define Riemannian metrics on Lie groups. The most interesting ones are those
invariant under the left- and right-translation diffeomorphisms (bi-invariant metrics). If
BIJ are the components of a non-singular metric in the basis of left-invariant vector fields
{eI }, then
ds 2 = BIJe I ⊗ e J , (A.85)
where the eI are a basis of left-invariant 1-forms constructed for instance as in Eq. (A.12),
is automatically a metric invariant under the left-translation diffeomorphisms and has an
isometry group G. Under right translations g → gh
e I → ƔAdj(h −1 ) I J e J , ⇒ BIJ → ƔAdj(h −1 ) K I ƔAdj(h −1 ) L J BKL. (A.86)