DIFFERENTIAL GEOMETRY ON THE SPACE OF CONNECTIONS VIA GRAPHS AND PROJECTIVE-LIMITS

In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion (A/G) over b ar of the space A/G of gauge equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory. (A/G) over bar is a very large space and serves as a ''universal home'' for measures in theories in which the Wilson loop observables are well def ined. In this paper, (A/G) over bar is considered as the projective li mit of a projective family of compact Hausdorff manifolds, labelled by graphs (which can be regarded as ''floating lattices'' in the physics terminology). Using this characterization, differential geometry is d eveloped through algebraic methods. In particular, we are able to intr oduce the following notions on (A/G) over bar: differential forms, ext erior derivatives, volume forms, vector fields and Lie brackets betwee n them, divergence of a vector field with respect to a volume form, La placians and associated heat kernels and heat kernel measures. Thus, a lthough (A/G) over bar is very large, it is small enough to be mathema tically interesting and physically useful. A key feature of this appro ach is that it does not require a background metric. The geometrical f ramework is therefore well suited for diffeomorphism invariant theorie s such as quantum general relativity.