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A.1 Partial backgrounds

To describe a theory of connections, we need to fix a principal fiber bundle P(Σ, G, π) over the analytic base manifold Σ with compact structure group G. Let S < Aut (P ) be a Lie symmetry subgroup of bundle automorphisms acting on the principal fiber bundle P. Using the bundle projection π : P → Σ we get a symmetry operation of S on Σ. For simplicity we will assume that all orbits of S are of the same type. If necessary we will have to decompose the base manifold into several orbit bundles Σ (F) ⊂ Σ, where ∼ F = Sx is the isotropy subgroup of S consisting of elements fixing a point x of the orbit bundle Σ(F) (isotropy subgroups for different points in Σ (F) are not identical but conjugate to each other). This amounts to a special treatment of possible symmetry axes or centers.

By restricting ourselves to one fixed orbit bundle we fix an isotropy subgroup F ≤ S up to conjugacy, and we require that the action of S on Σ is such that the orbits are given by S (x) ∼= S∕F for all x ∈ Σ. This will be the case if S is compact but also in most other cases of physical interest. Moreover, we will have to assume later on that the coset space S∕F is reductive [202Jump To The Next Citation Point203Jump To The Next Citation Point], i.e., that ℒS can be written as a direct sum ℒS = ℒF ⊕ ℒF ⊥ with Ad (ℒF ) ⊂ ℒF F ⊥ ⊥. If S is semi-simple, ℒF ⊥ is the orthogonal complement of ℒF with respect to the Cartan–Killing metric on ℒS. Further examples are provided by freely acting symmetry groups, in which case we have F = {1 }, and semi-direct products of the form S = N ⋊ F, where ℒF ⊥ = ℒN. The latter cases are relevant for homogeneous and isotropic cosmological models.

The base manifold can be decomposed as ∼ Σ = Σ∕S × S∕F, where ∼ Σ ∕S = B ⊂ Σ is the base manifold of the orbit bundle and can be realized as a sub-manifold B of Σ via a section in this bundle. As already noted in the main text, the action of a symmetry group on space introduces a partial background into the model. In particular, full diffeomorphism invariance is not preserved but reduced to diffeomorphisms only on the reduced manifold B. To see what kind of partial background we have in a model, it is helpful to contrast the mathematical definition of symmetry actions with the physical picture.

To specify an action of a group on a manifold, one has to give, for each group element, a map between space points satisfying certain conditions. Mathematically, each point is uniquely determined by labels, usually by coordinates in a chosen (local) coordinate system. The group action can then be written down in terms of maps of the coordinate charts, and there are compatibility conditions for maps expressed in different charts to ensure that the ensuing map on the manifold is coordinate independent. If we have active diffeomorphism invariance, however, individual points in space are not well defined. This leads to the common view that geometrical observables, such as the area of a surface, are, for physical purposes, not actually defined by integrating over a sub-manifold simply in parameter form, but over subsets of space defined by the values of matter fields [253251]. Since matter fields are subject to diffeomorphisms, just as the metric, area defined in such a manner is diffeomorphism invariant.

Similarly, orbits of the group action are not to be regarded as fixed sub-manifolds, but as being deformed by diffeomorphisms. Fixing a class of orbits filling the space manifold Σ corresponds to selecting a special coordinate system adapted to the symmetry. For instance, in a spherically-symmetric situation one usually chooses spherical coordinates (r,ϑ,ϕ ), where r > 0 labels the orbits and ϑ and ϕ are angular coordinates and can be identified with some parameters of the symmetry group SO(3). In a Euclidean space the orbits can be embedded as spheres S2 of constant curvature. Applying a diffeomorphism, however, will deform the spheres and they are in general only topological 2 S. Physically, the orbits can be specified as level surfaces of matter fields, similar to specifying space points. This concept allows us to distinguish, in a diffeomorphism-invariant manner, between curves (such as edges of spin networks) that are tangential and curves that are transversal to the group orbits.

It is, however, not possible to label single points in a given orbit in such a physical manner, simply because we could not introduce the necessary matter fields without destroying the symmetry. Thus we have to use the action of the symmetry group, which provides us with additional structure, to label the points, e.g., by using the angular coordinates in the example above. A similar role is played by the embedding of the reduced manifold B into Σ by choosing a section of the orbit bundle, which provides a base point for each orbit (a north pole in the example of spherical symmetry). This amounts to a partial fixing of the diffeomorphism invariance by allowing only diffeomorphisms that respect the additional structure. The reduced diffeomorphism constraint will then, in general, require only invariance with respect to diffeomorphisms of the manifold B.

In a reduced model, a partial fixing of the diffeomorphism invariance does not cause problems because all fields are constant along the orbits anyway. However, if we study symmetric states as generalized states of the full theory, as in Section 7, we inevitably have to partially break the diffeomorphism invariance. The distributional evaluation of symmetric states and the dual action of basic operators thus depends on the partial background provided by the symmetry.

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