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A.3 Classification of invariant connections

Now let w be an S-invariant connection on the symmetric bundle P classified by ([c],Q), i.e., s*w = w for any s (- S. After restriction, w induces a connection ~w on the reduced bundle Q. Because of S-invariance of w the reduced connection w~ is a one-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 p*v (- s*Tp(p)B, where s is the embedding of B into S. 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 w by f (- F applied to v is by definition f *wp(v) = wf(p)(df (v)) = wf(p)(v). Now using the fact that f acts as gauge transformation in the fibers and observing the definition of cp and the adjoint transformation of w, we obtain wf(p)(v) = Adcp(f)-1wp(v). By assumption the connection w is S-invariant implying f *wp(v) = Adcp(f)-1wp(v) = wp(v) for all f (- F. This shows that wp(v) (- LZG(cp(F )), and w can be restricted to a connection on the bundle Qc with structure group Z c.

Furthermore, using w we can construct the linear map ~ /\p : LS --> LG, X '--> wp(X) for any p (- P. Here, X~ is the vector field on P given by * ~X(h) := d(exp(tX) h)/dt|t=0 for any X (- LS and h (- C1(P, R). For X (- LF the vector field ~X is a vertical vector field, and we have /\p(X) = dcp(X), where dc : LF --> LG is the derivative of the homomorphism defined above. This component of /\ is therefore already given by the classifying structure of the principal fiber bundle. Using a suitable gauge, c can be held constant along B. The remaining components /\p |LF _L yield information about the invariant connection w. They are subject to the condition

/\p(Adf (X)) = Adc (f)(/\p(X)) for f (- F,X (- LS, (69) p
which follows from the transformation of w under the adjoint representation and which provides a set of equations determining the form of the components /\.

Keeping only the information characterizing w we have, besides ~w, the scalar field ~f : Q --> LG ox LF * _L, which is determined by /\p|LF _L and can be regarded as having dim LF_ L components of LG-valued scalar fields. The reduced connection and the scalar field suffice to characterize an invariant connection [82]:

Theorem 2 (Generalized Wang Theorem) Let P(S, G) be an S-symmetric principal fiber bundle classified by ([c],Q) according to Theorem 1, and let w be an S-invariant connection on P.

Then the connection w is uniquely classified by a reduced connection ~w on Q and a scalar field ~f : Q × LF_ L --> LG obeying Equation (69View Equation).

In general, ~ f transforms under some representation of the reduced structure group Zc: Its values lie in the subspace of LG determined by Equation (69View Equation) and form a representation space for all group elements of G (which act on /\) whose action preserves the subspace. These are by definition precisely elements of the reduced group.

The connection w can be reconstructed from its classifying structure (~w, ~f) as follows: According to the decomposition ~ S = B × S/F we have

w = w~+ wS/F, (70)
where w S/F is given by /\ o i*hMC in a gauge depending on the (local) embedding i: S/F --> S. Here hMC is the Maurer-Cartan form on S taking values in LS. Through /\, w depends on c and f~.

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