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, ${\mathcal{A}}_{C-flat}\subset {\bar{\mathcal{A}}}_M$. If cellular decomposition $C_2$ is finer than cellular decomposition $C_1$, there is a coarse graining map ${\pi }_{C_2\to C_1}:{\mathcal{A}}_{C_2-flat}\to {\mathcal{A}}_{C_1-flat}$. We prove that the triple $({\mathcal{A}}_{C_2-flat},{\pi }_{C_2\to C_1},{\mathcal{A}}_{C_1-flat})$ is a principal fiber bundle with a preferred global section given by the natural inclusion map $i_{C_1\to C_2}:{\mathcal{A}}_{C_1-flat}\to {\mathcal{A}}_{C_2-flat}$. Since the spaces ${\mathcal{A}}_{C-flat}$ 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, ${\lim }_{C\to M}{\mathcal{A}}_{C-flat}={\bar{\mathcal{A}}}_M$. We also define a construction of measures in ${\bar{\mathcal{A}}}_M$ as the continuum limit (not a projective limit) of effective measures.

29 pages, 2 figures

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