Differential geometry on the space of connections via graphs and projective limits
The authors consider a certain completion \(\overline {{\mathcal A}/{\mathcal G}}\) of the space \({\mathcal A}/{\mathcal G}\) of gauge equivalent connections, characterized as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs. Using this characterization, differential geometry is developed through algebraic methods; thus, in \(\overline {{\mathcal A}/{\mathcal G}}\), are introduced: differential forms, exterior derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and associated heat kernels and heat kernel measures. The described approach does not require a background metric and provides a geometrical framework well suited for diffeomorphism invariant theories such as quantum relativity.
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