### A.2 Classification of symmetric principal fiber bundles

Fields that are invariant under the action of a symmetry group on space are defined by a set of linear equations for invariant field components. Nevertheless, finding invariant fields in gauge theories is not always straightforward since, in general, fields need to be invariant only up to gauge transformations which depend on the symmetry transformation. An invariant connection, for instance, satisfies the equation
with a local gauge transformation for each . These gauge transformations are not arbitrary since two symmetry transformations and applied one after another have to imply a gauge transformation with related to and . However, this does not simply amount to a homomorphism property and allowed maps are not easily determined by group theory. Thus, even though for a known map one simply has to solve a system of linear equations for , finding appropriate maps can be difficult. In most cases, the equations would not have any non-vanishing solution at all, which would certainly be insufficient for interesting reduced field theories.

In the earlier physical literature, invariant connections and other fields have indeed been determined by trial and error [91], but the same problem has been solved in the mathematical literature [13813982] in impressive generality. This uses the language of principal fiber bundles which already provides powerful techniques. Moreover, the problem of solving one system of equations for and at the same time is split into two separate problems, which allows a more systematic approach. The first step is to realize that a connection whose local 1-forms on are invariant up to gauge is equivalent to a connection 1-form defined on the full fiber bundle , which satisfies the simple invariance conditions for all . This is indeed simpler to analyze since we now have a set of linear equations for alone. However, even though hidden in the notation, the map is still present. The invariance conditions for defined on are well-defined only if we know a lift from the original action of on the base manifold to the full bundle . As with maps , there are several inequivalent choices for the lift which have to be determined. The advantage of this procedure is that this can be done by studying symmetric principal fiber bundles, i.e., principal fiber bundles carrying the action of a symmetry group, independently of the behavior of connections. In a second step, one can then ask what form invariant connections on a given symmetric principal fiber bundle have.

We now discuss the first step of determining lifts of the symmetry action of from to . Given a point , the action of the isotropy subgroup yields a map of the fiber over , which commutes with the right action of on the bundle. To each point we can assign a group homomorphism defined by for all . To verify this we first note that commutativity of the action of with right multiplication of on implies that we have the conjugate homomorphism for a different point in the same fiber:

This yields

demonstrating the homomorphism property. We thus obtain a map obeying the relation .

Given a fixed homomorphism , we can build the principal fiber subbundle

over the base manifold , which as structure group has the centralizer

of in . is the restricted fiber bundle over . A conjugate homomorphism simply leads to an isomorphic fiber bundle.

The structure elements and classify symmetric principal fiber bundles according to the following theorem [82]:

Theorem 1 An -symmetric principal fiber bundle with isotropy subgroup of the action of on is uniquely characterized by a conjugacy class of homomorphisms together with a reduced bundle .

Given two groups and we can make use of the relation [81]

in order to determine all conjugacy classes of homomorphisms . Here, is a maximal torus and the Weyl group of . Different conjugacy classes correspond to different sectors of the theory, which can be interpreted as having different topological charge. In spherically symmetric electromagnetism, for instance, this is just magnetic charge [3566].