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A.2 Classification of symmetric principal fiber bundles

Fields that are invariant under the action of a symmetry group S on space Σ 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
s∗A = g (s )− 1Ag(s) + g(s)−1dg(s) (81 )
with a local gauge transformation g(s) for each s ∈ S. These gauge transformations are not arbitrary, since two symmetry transformations s1 and s2 applied one after another have to imply a gauge transformation with g(s2s1) related to g (s1) and g(s2). However, this does not simply amount to a group homomorphism property and allowed maps g: S → G are not easily determined by group theory. Thus, even though, for a known map g, one simply has to solve a system of linear equations for A, finding appropriate maps g 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 physics literature, invariant connections and other fields have indeed been determined by trial and error [135Jump To The Next Citation Point], but the same problem has been solved in the mathematical literature [202203116Jump To The Next Citation Point] 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 A and g(s) 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 A on Σ are invariant up to gauge, is equivalent to a connection 1-form ω defined on the full fiber bundle P, which satisfies the simple invariance conditions s∗ω = ω for all s ∈ S. This is indeed simpler to analyze, since we now have a set of linear equations for ω alone. However, even though hidden in the notation, the map g: S → G is still present. The invariance conditions for ω defined on P are well defined only if we know a lift from the original action of S on the base manifold Σ to the full bundle P. As with maps g: S → G, 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 S from Σ to P. Given a point x ∈ Σ, the action of the isotropy subgroup F yields a map −1 −1 F : π (x) → π (x ) of the fiber over x, which commutes with the right action of G on the bundle. To each point p ∈ π−1(x) we can assign a group homomorphism λp: F → G defined by f (p) =: p ⋅ λp(f ) for all f ∈ F. To verify this we first note that commutativity of the action of S < Aut (P ) with right multiplication of G on P implies that we have the conjugate homomorphism λp ′ = Adg −1 ∘ λp for a different point ′ p = p ⋅ g in the same fiber:

′ ′ p ⋅ λp′(f) = f (p ⋅ g) = f (p ) ⋅ g = (p ⋅ λp(f)) ⋅ g = p ⋅ Adg −1λp(f) .

This yields

(f1 ∘ f2)(p ) = f1(p ⋅ λp(f2)) = (p ⋅ λp(f2)) ⋅ Ad λp(f2)−1λp(f1) = p ⋅ (λp(f1) ⋅ λp(f2))

demonstrating the homomorphism property. We thus obtain a map λ: P × F → G, (p,f) ↦→ λp(f) obeying the relation λ = Ad −1 ∘ λ p⋅g g p.

Given a fixed homomorphism λ : F → G, we can build the principal fiber sub-bundle

Q λ(B, Zλ, πQ) := {p ∈ P|B : λp = λ} (82 )
over the base manifold B, which as structure group has the centralizer
Zλ := ZG (λ(F )) = {g ∈ G : gf = fg for all f ∈ λ(F )}

of λ(F ) in G. P|B is the restricted fiber bundle over B. A conjugate homomorphism λ ′ = Adg −1 ∘ λ simply leads to an isomorphic fiber bundle.

The structure elements [λ] and Q classify symmetric principal fiber bundles according to the following theorem [116Jump To The Next Citation Point]:

Theorem 1 An S-symmetric principal fiber bundle P (Σ, G,π ) with isotropy subgroup F ≤ S of the action of S on Σ is uniquely characterized by a conjugacy class [λ ] of homomorphisms λ: F → G together with a reduced bundle Q (Σ∕S, ZG (λ(F )),πQ ).

Given two groups, F and G, we can make use of the relation [115Jump To The Next Citation Point]

Hom (F, G)∕Ad ∼= Hom (F,T (G ))∕W (G ) (83 )
in order to determine all conjugacy classes of homomorphisms λ: F → G. Here, T (G) is a maximal torus and W (G) the Weyl group of G. 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 [4396].
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