In mathematical terms, a gauge field (with gauge group , say) is a connection in a principal bundle over spacetime . A gauge field is symmetric with respect to the action of a symmetry group of , if it is described by an -invariant connection on . Hence, finding the symmetric gauge fields involves the task of classifying the principal bundles , which admit the symmetry group , acting by bundle automorphisms. This program was carried out by Brodbeck and Straumann for arbitrary gauge and symmetry groups , (see also [34, 39]), generalizing earlier work of Harnad et al. , Jadczyk  and Künzle .
The gauge fields constructed in the above way are invariant under the action of up to gauge transformations. This is also the starting point of the alternative approach to the problem, due to Forgács and Manton . It implies that a gauge potential is symmetric with respect to the action of a Killing field , say, if there exists a Lie algebra valued function , such that
Let us now consider a stationary spacetime with (asymptotically) time-like Killing field . A stationary gauge potential can be parameterized in terms of a one-form orthogonal to , in the sense that , and a Lie algebra valued potential ,9 By virtue of the above decomposition, the field strength becomes , where is the Yang–Mills field strength for and . Using the expression (6.5) for the vacuum action, one easily finds that the EYM action,
The main difference between the Abelian and the non-Abelian case concerns the variational equation for , that is, the Yang–Mills equation for : For non-Abelian gauge groups, is no longer an exact two-form, and the gauge covariant derivative of introduces source terms in the corresponding Yang–Mills equation:
As an application, we note that the effective action (6.16) is particularly suited for analyzing stationary perturbations of static (), purely magnetic () configurations , such as the Bartnik–McKinnon solitons  and the corresponding black-hole solutions [310, 208, 24]. The two crucial observations in this context are [35, 312]:
The second observation follows from the fact that the magnetic Yang–Mills equation (6.18) and the Einstein equations for and become background equations, since they contain no linear terms in and . Therefore, the purely electric, non-static perturbations are governed by the twist equation (6.17) and the electric Yang–Mills equation (obtained from variations of with respect to ).
Using Eq. (6.17) to introduce the twist potential , the fluctuation equations for the first-order quantities and assume the form of a self-adjoint system . Considering perturbations of spherically-symmetric configurations, one can expand and in terms of isospin harmonics. In this way one obtains a Sturm–Liouville problem, the solutions of which reveal the features mentioned in the last paragraph of Section 5.5 .
Living Rev. Relativity 15, (2012), 7
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