In the earlier physics literature, invariant connections and other fields have indeed been determined by
trial and error [135], but the same problem has been solved in the mathematical literature [202, 203, 116
]
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, independent 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 for 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 sub-bundle
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 [116
]:
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 [115
]
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