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, AC-flat∕G¯M,⋆⊂A¯M∕G¯M,⋆. If cellular decomposition C2 is finer than cellular decomposition C1, there is a coarse graining map πC2→C1:AC2-flat∕G¯M,⋆→AC1-flat∕G¯M,⋆. We prove that the triple (AC2-flat∕G¯M,⋆,πC2→C1,AC1-flat∕G¯M,⋆) is a principal fiber bundle with a preferred global section given by the natural inclusion map iC1→C2:AC1-flat∕G¯M,⋆→AC2-flat∕G¯M,⋆. Since the spaces AC-flat∕G¯M,⋆ are partially ordered (by inclusion) and this order is directed in the direction of refinement, we can define a continuum limit, C→M. We prove that, in an appropriate sense, limC→MAC-flat∕G¯M,⋆=A¯M∕G¯M,⋆. We also define a construction of measures in A¯M∕G¯M,⋆ as the continuum limit (not a projective limit) of effective measures.