Gravity and Strings

608 Appendix A
A.4.1 H-covariant derivatives
The H-covariant derivative of any object that transforms contravariantly, φ ′ = Ɣr(h)φ, or
covariantly, ψ ′ = ψƔr(h −1 ) (for instance, u(x) itself), in the representation r of H is
Dµφ ≡ ∂µφ − ϑµ i Ɣr(Mi)φ, Dµψ ≡ ∂µψ + ψϑµ i Ɣr(Mi). (A.122)
The curvature Fi is
F i = dϑ i − 1
2ϑ j ∧ ϑ k f jk i
(A.123)
and it is covariantly constant with respect to the full (Lorentz-plus-gauge) covariant derivative
if we use the torsional spin connection Eq. (A.118) and also in the symmetric case.
This statement is clearly equivalent to the covariant constancy of the Lorentz curvature.
This implies that, in any reductive coset space G/H, there is a solution of the Yang–Mills
equations of motion for the group H in the curved geometry associated with the (torsionful)
spin connection defined above. This result is implicitly or explicitly used in many places.
The simplest example is provided by the coset manifold SU(2)/U(1), which gives a round
2-sphere. The U(1) connection solves the Maxwell equations and corresponds to the Dirac
monopole (see the calculation of the Robinson–Bertotti superalgebra on page 386 and Appendix
C.1).
The gauge field ϑi is invariant under the combination of the diffeomorphisms generated
by the Killing vectors k(I ) and gauge transformations generated by WI i , i.e.
is the standard Lie derivative.
−Lk(I ) ϑi = DµWI i , (A.124)
where Lk(I )
The H-covariant Lie derivative with respect to the Killing vectors17 k(I ) of contravariant
(φ)orcovariant (ψ) objects in the representation r of H is
Lk(I ) φ ≡ Lk(I ) φ + WI i Ɣr(Mi)φ, Lk(I ) ψ ≡ Lk(I ) ψ − ψWI i Ɣr(Mi). (A.125)
This Lie derivative satisfies, among other properties
[Lk(I ) , Lk(J) ] = L[k(I ),k(J)], (A.126)
Lk(I ) ea = 0, (A.127)
Lk(I ) u = Lk(I ) u − uWI i Mi = TI u, (A.128)
where the last property follows from Eqs. (A.115) and (A.112). The connection 1-forms
ϑ i are not covariant or contravariant objects and this definition does not apply to them.
The best one can do for them is to combine the standard Lie derivative and a compensating
gauge transformation. The resulting operator acting on ϑ i is identically zero, due to
Eq. (A.124).
17 H-covariant Lie derivatives can be defined with respect to any vector, but the property Eq. (A.126) holds
only for Killing vectors. The spinorial Lie derivative [632, 633, 655] and the more general Lie–Lorentz
and Lie–Maxwell derivatives that appear in calculations of supersymmetry algebras [390, 748] discussed
in Section 13.2.1 can actually be seen as particular examples of this more general operator (see e.g. [460])
and, actually, are identical objects when they are acting on Killing spinors of maximally supersymmetric
spacetimes [25].