Classical gauge theory in Riem

In the geometrodynamical setting of general relativity in Lagrangian form, the objects of study are the Riemannian metrics (and their time derivatives) over a given 3-manifold M. It is our aim in this paper to study some geometrical aspects of the space $\mathcal{M}:=$ Riem(M) of all metrics over M. For instance, the Hamiltonian constraints by themselves do not generate a group, and thus its action on Riem(M) cannot be viewed in a geometrical gauge setting. It is possible to do so for the momentum constraints however. Furthermore, in view of the recent results representing GR as a dual theory, invariant under foliation preserving 3–diffeomorphisms and 3D conformal transformations, but not under refoliations, we are justified in considering the gauge structure pertaining only to the groups $\mathcal{D}$ of diffeomorphisms of M, and $\mathcal{C}$ , of conformal diffeomorphisms on M. For these infinite-dimensional symmetry groups, $\mathcal{M}$ has a natural principal fiber bundle structure, which renders the gravitational field amenable to the full range of gauge-theoretic treatment. The aim of the paper is to use the geometrical structure present in the configuration space of general relativity to build gauge connection forms. The interpretation of the gauge connection form for the 3-diffeomorphism group is that it yields parallel translation of coordinates. For the conformal group, it yields parallel translation of scale. We focus on the concept of a gauge connection forms for these structures and construct explicit formulae for supermetric-induced gauge connections. To apply the formalism, we compute general properties for a specific connection bearing strong resemblance to the one naturally induced by the deWitt supermetric, showing it has desirable relationalist properties.