Gravity and Strings

Lie groups, symmetric spaces, and Yang–Mills fields 607
Acting with u −1 on the left and using the definitions of the adjoint action and the Maurer–
Cartan 1-forms, we obtain
TJ ƔAdj(u −1 ) J I =−k(I ) a Pa − (k(I ) µ ϑ i µ + WI i )Mi, (A.113)
which, projected onto the horizontal and vertical subspaces, gives
k(I ) a =−ƔAdj(u −1 (x)) a I , (A.114)
WI i =−k(I ) µ ϑ i µ − ƔAdj(u −1 (x)) i I . (A.115)
The Killing vectors associated with the right isometry group N(H)/H are just the vectors
ea dual to the horizontal Maurer–Cartan 1-forms in the directions of N(H)/H.
These formulae simplify considerably the calculation of Killing vectors, if we construct
the space with the above recipe. As in group manifolds, the spin connection 16 can easily be
found: on comparing the Maurer–Cartan equations
with the structure equation (1.143), we obtain
if we do not allow for torsion, or
de a − ϑ i ∧ e b fib a − 1
2 eb ∧ e c fbc a = 0 (A.116)
ω a b = ϑ i fib a + 1
2 ec fcb a , (A.117)
ω a b = ϑ i fib a , T a =− 1
2 ec ∧ e b fcb a . (A.118)
It is straightforward to compute the curvature using the Maurer–Cartan equations:
dϑ i − 1
2 ϑ j ∧ ϑ k f jk i − 1
2 ea ∧ e b fab i = 0. (A.119)
In the symmetric case ( fcb a = 0) and in the reductive case with the torsionful connection
Eq. (A.118)
R a b = [dϑ i − 1
2ϑ j ∧ ϑ k f jk i ] fib a = 1
2ec ∧ e d fcd i fib a
(A.120)
(using the Maurer–Cartan equations) and is covariantly constant. In the reductive (nonsymmetric)
case
R a b = 1
2 ec ∧ e d fcd i fib a + 1
2 fcd e feb a − 1
2 fce a fdb e . (A.121)
The reductive case (symmetric or not) is particularly interesting because, as we have
said, according to Eqs. (A.108) the vertical 1-forms ϑ i transform as a connection for the
group H. The above formulae Eqs. (A.117) and (A.118) relate this gauge connection to the
spin connection (and torsion). Sometimes this is expressed by saying that the gauge group
has been embedded into the tangent-space group. These relations are used very often in the
construction of solutions. This suggests the following definitions.
16 We assume here that Bab is diagonal with only +1s and −1s on the diagonal.