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

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 $π_{C_2 \to C_1}: {\cal A}_{C_2-flat} \to {\cal A}_{C_1-flat}$. We prove that the triple $({\cal A}_{C_2-flat}, π_{C_2 \to C_1}, {\cal A}_{C_1-flat})$ 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.