In order to specify the general form (74) further, the first step is again to find all conjugacy classes of homomorphisms . To do so we can make use of Equation (68) to which end we need the following information about SU(2) (see, e.g., ): The standard maximal torus of SU(2) is given by
and the Weyl group of SU(2) is the permutation group of two elements, , its generator acting on by .
All homomorphisms in are given by
for any , and we have to divide out the action of the Weyl group leaving only the maps , , as representatives of all conjugacy classes of homomorphisms. We see that spherically symmetric gravity has a topological charge taking values in (but only if degenerate configurations are allowed, as we will see below).
We will represent as the subgroup of the symmetry group , and use the homomorphisms out of each conjugacy class. This leads to a reduced structure group for and (; this is the sector of manifestly invariant connections of ). The map is given by , and the remaining components of , which give us the scalar field, are determined by subject to Equation (69), which here can be written as
Using and we obtain
where , is an arbitrary element of . Since and are arbitrary, this is equivalent to the two equations
A general ansatz
with arbitrary parameters yields
The configuration variables of the system are the above fields of the -connection form on the one hand and the two scalar field components
In order to obtain a standard symplectic structure (see Equation (77) below), we reconstruct the general invariant connection form
Information about the topological charge can be found by expressing the volume in terms of the reduced triad coefficients : Using
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