Differential Geometry on the Space of Connections via Graphs and Projective Limits

TL;DR

This paper develops a differential geometric framework on the quantum configuration space of gauge theories, using projective limits of graphs, enabling analysis without background metrics, relevant for quantum gravity.

Contribution

It introduces a novel algebraic differential geometry on the space of connections as a projective limit, applicable to background-independent quantum theories.

Findings

Defined differential forms and derivatives on the space of connections

Constructed volume forms, vector fields, and Lie brackets in this setting

Developed Laplacians and heat kernels without a background metric

Abstract

In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion, $\agb$, of the space $\ag$ 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. $\agb$ is a very large space and serves as a ``universal home'' for measures in theories in which the Wilson loop observables are well-defined. In this paper, $\agb$ is considered as the projective limit 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 developed through algebraic methods. In particular, we are able to introduce the following notions on $\agb$: 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. Thus, although $\agb$ is very large, it is small enough to be mathematically interesting and physically useful. A key feature of this approach is that it does not require a background metric. The geometrical framework is therefore well-suited for diffeomorphism invariant theories such as quantum general relativity.