Superspace plays the role of the configuration space for general relativity in the traditional metric representation. The associated cotangent bundle, when properly defined, is the phase space for the Hamiltonian formulation of the theory. As a Hamiltonian formulation is the starting point for the quantization of any mechanical or field system, the role of superspace and the need to understand its mathematical structure cannot be overemphasized. A secondary role of superspace is that of providing “variables for the wave function” in a functional Schrödinger representation for quantum gravity. However, it should be noted at this point that even in the quantization of the simplest field theories – such as scalar fields – it is necessary to suitably enlarge this configuration space and allow for distributional, non-smooth objects to arrive at a consistent model (see, for example, [12]). How – and if – this can be done in the geometrodynamical setting is an interesting, if hard, question. This is directly related to the Wheeler–DeWitt approach to the quantization of GR [79].

The precise definition of the geometry of a three manifold requires some discussion (see [87, 98] and references therein for a nice introduction to the subject). Here we will content ourselves with a quick review of the most important issues. It is important to remark at this point that the non-generic character of geometries with non-trivial isometry groups has a very clear reflection in superspace: they correspond to singularities.

The geometry of a four-dimensional manifold in relativists’ parlance refers to equivalence classes of suitably smooth Lorentzian metrics defined on it. Two metrics are declared equivalent if they are connected by a diffeomorphism. Though one might naively think that this is just a mathematically-sensible requirement, in fact, it is quite natural from a physical point of view. The reason is that ultimately the geometry must be probed by physical means. This, in turn, demands an operational definition of the (possibly idealized) physical processes allowing us to explore – actually measure – it. This is in the spirit of special and general relativity, where the definition of physical magnitudes such as lengths, distances, velocities and the like requires the introduction of concrete procedures to measure them by using basic tools such as clocks, rulers and light rays. Every transformation of the manifold (and the objects defined on it) that does not affect the operational definition of the measuring processes will be physically unobservable. Diffeomorphisms are such transformations. Notice that this prevents us from identifying physical events with points in the spacetime manifold as a diffeomorphism can take a given event from one point of the manifold to another (see [163]).

The precise definition and description of the space of geometries requires the introduction of mathematical objects and structures at different levels:

- The three-dimensional smooth manifold with the required differential structure.
- A class of smooth Riemannian metrics on this manifold that should be considered (smooth in the sense of being ). We will denote this as .
- A suitable topology and differential structure on this space.
- An equivalence relation between different smooth metrics provided by the action of (a class of) smooth () diffeomorphisms on .

After doing this one has to study the quotient . Naturally, superspace will inherit some background properties from those carried by the different elements needed to properly define it. The resulting space has a rich structure and interesting properties that we will very quickly comment on here (the interested reader is referred to [98] and the extensive bibliography cited there).

An important issue is related to the appearance of singularities in this quotient space associated with the fact that in many instances the spatial manifold allows for the existence of invariant metrics under non-trivial symmetry groups (leading to a non-free action of the diffeomorphisms). This turns out to be a problem that can be dealt with in the sense that the singularities are minimally resolved (see [87]). It is important to mention at this point that the symmetry reductions that we will be considering here consist precisely in restrictions to families of symmetric metrics that, consequently, sit at the singularities of the full superspace. This fact, however, does not necessarily imply that the reduced systems are pathological. In fact some of them are quite well behaved as we will show in Section 4.

Finally, we point out that both the space of Riemannian metrics and the quotient space mentioned above are endowed with natural topologies that actually turn them into very well-behaved topological spaces (for instance, they are metrizable – and hence paracompact –, second countable and connected). The space of metrics can be described as a principal bundle with basis and structure group given by (that is, the proper subgroup of those diffeomorphisms of that fix a preferred point and the tangent space at this point). Finally, a family of ultralocal metrics is naturally defined in superspace [98]. Some of these properties are inherited by the spaces of symmetric geometries that we consider here.

Other approaches to the quantization of GR, and in particular loop quantum gravity, rely on spaces of connections rather than in spaces of metrics. Hence, in order to study symmetry reductions in these frameworks, one should discuss the properties of such “connection superspaces” and then consider the definition of symmetric connections and how they fit into these spaces. The technical treatment of the spaces of Yang–Mills connections modulo gauge transformations has been developed in the late seventies by Singer and other authors [198, 166]. These results have been used by Ashtekar, Lewandowski [11] and others to give a description of the spaces of connections modulo gauge (encompassing diffeomorphisms) and their extension to symmetry reductions have been explored by Bojowald [44] and collaborators as a first step towards the study of symmetry reductions in LQG. These will be mentioned in Section 5.

Living Rev. Relativity 13, (2010), 6
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