Gravity and Strings

606 Appendix A
which is a function of g and x:
gu(x) = u(x ′ )h. (A.105)
Now we construct the left-invariant Maurer–Cartan 1-form and expand it in horizontal, e a ,
and vertical, ϑ i , components:
V ≡−u −1 du = e a Pa + ϑ i Mi. (A.106)
The horizontal components e a can be used as Vielbeins for G/H and, given any metric
Bab on k, wecan construct a Riemannian metric
ds 2 = Babe a ⊗ e b . (A.107)
To find under what conditions this metric will be (left-)G-invariant, we have to look at
the transformation of the Maurer–Cartan 1-forms under left multiplication by a constant
element g ∈ G, u(x ′ ) = gu(x)h −1 :
e a (x ′ ) = (he(x)h −1 ) a = ƔAdj(h) a be b (x),
ϑ i (x ′ ) = (hϑ(x)h −1 ) i + (h −1 dh) i + (he(x)h −1 ) i .
(A.108)
The last term in the second equation is zero in the reductive case and the ϑ i s transform
as a connection. Furthermore, the restriction of ƔAdj(h) to k is a representation of h. The
Riemannian metric will be invariant under the left action of G if
fi(a c Bb)c = 0, (A.109)
which is guaranteed if we can set Bab = Kab, the projection on k of the (non-singular)
Killing metric. There are many important cases in which G is not semisimple but there is
a non-degenerate invariant metric. For instance, we can describe Minkowski space as the
quotient of the Poincaré group (which is not semisimple because it contains the Abelian
invariant subgroup of translations) by the Lorentz subgroup. The Minkowski metric is a
non-degenerate invariant metric for this coset. Another example is provided by the Hppwave
spacetimes constructed in Section 10.1.1.
The resulting Riemannian metric contains G in its isometry group (generically G ×
N(H)/H) and must admit n Killing vector fields k(I ). The Killing vectors k(I ) and the Hcompensator
WI i are defined through the infinitesimal version of gu(x) = u(x ′ )h with
Using these equations in
we obtain
g = 1 + σ I TI ,
h = 1 − σ I WI i Mi,
x µ ′ = x µ + σ I k(I ) µ .
(A.110)
u(x + δx) = u(x) + σ I k(I )u, (A.111)
TI u = k(I )u − uWI i Mi. (A.112)