Gravity and Strings

Lie groups, symmetric spaces, and Yang–Mills fields 595
We can diagonalize and normalize the Killing metric using GL(n, R) transformations
so that it only has ±1, 0inthe diagonal. The zeros are associated with invariant Abelian
subalgebras and the +1s with non-compact directions.
A.2 Yang–Mills fields
A.2.1 Fields and covariant derivatives
Fields always transform in finite-dimensional representations of the symmetry group, even
if they are not unitary. 3 Ɣr(g) i j i, j = 1,...,dim(r) denotes the matrix corresponding to
group element g in the representation labeled by r. The indices which are also carried by
the fields will in general not be shown. In any representation, there are three different types
of fields according to the way they transform: contravariant fields, represented by a column
vector and transforming according to
ψ i ′ = (Ɣr(g)) i
jψ j , (A.26)
covariant fields, represented by a row vector and transforming according to
ξ ′
i = ξ j(Ɣ −1
r (g)) j i, (A.27)
and Lie-algebra-valued fields that transform under the adjoint action of the group
ϕ = ϕ I Ɣr(TI ) ,
ϕ ′ = Ɣr(g)ϕƔ −1
r (g), ⇒ ϕ′ I = ƔAdj(g) I
J ϕ J .
(A.28)
The relation among the three kinds of fields depends on the group and representation we
are considering. If the representation r is unitary and ψ is contravariant then ψ † is covariant.
If the group is defined by the property that it preserves the scalar product associated
with a metric η 〈u|v〉=u † ηv so u ′ = Ɣv(g)u and Ɣ † v (g)ηƔv(g), where Ɣv(g) is the matrix
associated with the group element g in the defining fundamental or vector representation
(these are the groups SO(n+, n−),SU(n+, n−), and Sp(n)) then, given a contravariant vector
field ψ, the row vector ψ † η transforms as a covariant vector field. It is also possible to
relate contravariant and covariant fields in the spinor representations of SO(n+, n−) groups
(see Appendix B).
Since
Ɣr(g) = exp{σ I Ɣr(TI )}≡exp{σr}, (A.29)
for infinitesimal values of the parameters (group manifold coordinates) σ I , i.e. for transformations
near the identity or infinitesimal transformations, the various fields transform as
follows:
δσ ψ = σ I Ɣr(TI ) ψ = σrψ,
δσ ξ =−ξσI Ɣr(TI ) =−ξσr,
δσ ϕ = [σr,ϕ], ⇒ δσ ϕ I = σ I Adj J ϕ J (A.30)
.
3 Their solutions correspond to states in the quantum theory and therefore must fit into unitary representations,
according to Wigner’s theorem, however.