### B.3 Spherical symmetry

In the generic case, i.e., outside a symmetry center, of spherical symmetry we have ,
( denotes the linear span), and the connection form can be gauged to be
Here, are (local) coordinates on and, as usual, we use the basis elements of .
is given by , whereas are the scalar field components. Equation (89) contains, as
special cases, the invariant connections found in [135]. These are gauge equivalent by gauge transformations
depending on the angular coordinates , i.e., they correspond to homomorphisms , which are not
constant on the orbits of the symmetry group.
In order to specify the general form (89) further, the first step is again to find all conjugacy classes of
homomorphisms . To do so, we can make use of Equation (83), to which end
we need the following information about SU(2) (see, e.g., [115]). 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 [136]). The map is given by
, and the remaining components of , which give us the scalar
field, are determined by subject to Equation (84), 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

which have non-trivial solutions only if , namely

The configuration variables of the system are the above fields of the -connection
form on the one hand and the two scalar-field components

on the other hand. Under a local U(1)-gauge transformation they transform as
and , which can be read off from
In order to obtain a standard symplectic structure (see Equation (92) below), we reconstruct the general
invariant connection form

An invariant densitized-triad field is analogously given by
with coefficients canonically conjugate to ( and are non-vanishing only for ).
The symplectic structure
can be derived by inserting the invariant expressions into .
Information about the topological charge can be found by expressing the volume in terms of the
reduced triad coefficients . Using

we have
We can now see that in all the sectors with the volume vanishes because then . All
these degenerate sectors have to be rejected on physical grounds and we arrive at a unique sector of
invariant connections given by the parameter .