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A.3 Classification of invariant connections

Now let ω be an S-invariant connection on the symmetric bundle P classified by ([λ],Q), i.e., s∗ω = ω for any s ∈ S. After restriction, ω induces a connection &tidle;ω on the reduced bundle Q. Because of the S-invariance of ω, the reduced connection &tidle;ω is a 1-form on Q with values in the Lie algebra of the reduced structure group. To see this, fix a point p ∈ P and a vector v in TpP, such that π ∗v ∈ σ∗Tπ(p)B, where σ is the embedding of B into Σ. Such a vector, which does not have components along symmetry orbits, is fixed by the action of the isotropy group: df (v) = v. The pull back of ω by f ∈ F applied to v is by definition f ⋆ωp(v) = ωf(p)(df (v)) = ωf(p)(v). Now using the fact that f acts as gauge transformation in the fibers and observing the definition of λp and the adjoint transformation of ω, we obtain ωf (p)(v) = Ad λp(f)−1ωp(v). By assumption, the connection ω is S-invariant implying f ⋆ωp(v) = Ad λp(f)−1ωp(v) = ωp(v) for all f ∈ F. This shows that ωp (v ) ∈ ℒZG (λp(F)), and ω can be restricted to a connection on the bundle Q λ with structure group Z λ.

Furthermore, using ω we can construct the linear map &tidle; Λp : ℒS → ℒG, X ↦→ ωp(X ) for any p ∈ P. Here, X&tidle; is the vector field on P given by ⋆ &tidle;X (h) := d(exp(tX ) h)∕dt|t=0 for any X ∈ ℒS and h ∈ C1(P, ℝ). For X ∈ ℒF the vector field &tidle;X is a vertical vector field, and we have Λp (X ) = dλp(X ), where dλ : ℒF → ℒG is the derivative of the homomorphism defined above. This component of Λ is therefore already given by the classifying structure of the principal fiber bundle. Using a suitable gauge, λ can be held constant along B. The remaining components Λp |ℒF⊥ yield information about the invariant connection ω. They are subject to the condition

Λp(Adf (X )) = Ad λ (f)(Λp (X )) for f ∈ F,X ∈ ℒS , (84 ) p
which follows from the transformation of ω under the adjoint representation and which provides a set of equations determining the form of the components Λ.

Keeping only the information characterizing ω we have, besides &tidle;ω, the scalar field &tidle;φ : Q → ℒG ⊗ ℒF ⋆ ⊥, which is determined by Λp|ℒF⊥ and can be regarded as having dim ℒF ⊥ components of ℒG-valued scalar fields. The reduced connection and the scalar field suffice to characterize an invariant connection [116]:

Theorem 2 (Generalized Wang Theorem) Let P(Σ, G ) be an S-symmetric principal fiber bundle classified by ([λ],Q ) according to Theorem 1, and let ω be an S-invariant connection on P. Then the connection ω is uniquely classified by a reduced connection &tidle;ω on Q and a scalar field &tidle;φ: Q × ℒF ⊥ → ℒG obeying Equation (84View Equation).

In general, &tidle;φ transforms under some representation of the reduced structure group Z λ; its values lie in the subspace of ℒG determined by Equation (84View Equation) and form a representation space for all group elements of G (which act on Λ), whose action preserves the subspace. These are, by definition, precisely elements of the reduced group.

The connection ω can be reconstructed from its classifying structure (&tidle;ω, &tidle;φ) as follows. According to the decomposition Σ ∼ B × S ∕F = we have

ω = &tidle;ω + ωS ∕F , (85 )
where ωS∕F is given by Λ ∘ ι⋆θMC in a gauge depending on the (local) embedding ι: S ∕F `→ S. Here θMC is the Maurer–Cartan form on S taking values in ℒS. Through Λ, ω depends on λ and &tidle; φ.


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