Geometry of C-flat connections, coarse graining and the continuum limit

TL;DR

This paper introduces a background-metric-independent framework for effective gauge fields using cellular decompositions, establishing a continuum limit that recovers the space of generalized connections and constructing measures as limits of effective measures.

Contribution

It defines a new notion of effective gauge fields via cellular decompositions and proves the continuum limit of these spaces converges to the space of generalized connections.

Findings

The space of effective gauge fields forms a principal fiber bundle with a natural global section.

The continuum limit of cellular decompositions yields the space of generalized connections.

A method for constructing measures as limits of effective measures is proposed.

Abstract

A notion of effective gauge fields which does not involve a background metric is introduced. The role of scale is played by cellular decompositions of the base manifold. Once a cellular decomposition is chosen, the corresponding space of effective gauge fields is the space of flat connections with singularities on its codimension two skeleton, ${\cal A}_{C-flat} \subset \bar{\cal A}_M$. If cellular decomposition $C_2$ is finer than cellular decomposition $C_1$, there is a coarse graining map $\pi_{C_2 \to C_1}: {\cal A}_{C_2-flat} \to {\cal A}_{C_1-flat}$. We prove that the triple $({\cal A}_{C_2-flat}, \pi_{C_2 \to C_1}, {\cal A}_{C_1-flat})$ is a principal fiber bundle with a preferred global section given by the natural inclusion map $i_{C_1 \to C_2}: {\cal A}_{C_1-flat} \to {\cal A}_{C_2-flat}$. Since the spaces ${\cal A}_{C-flat}$ are partially ordered (by inclusion) and this order is directed in the direction of refinement, we can define a continuum limit, $C \to M$. We prove that, in an appropriate sense, $\lim_{C \to M} {\cal A}_{C-flat} = \bar{\cal A}_M$. We also define a construction of measures in $\bar{\cal A}_M$ as the continuum limit (not a projective limit) of effective measures.