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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,is a principal fiber bundle with a preferred global section given by the natural inclusion mapSince the spaces A C-flat /Ḡ 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,We also define a construction of measures inĀ M /Ḡ M,⋆ as the continuum limit (not a projective limit) of effective measures.

Abstract. Given a pair (Γ, ρ) of a Fuchsian group of the first kind, and a unitary representation ρ of Γ of arbitrary rank, the problem of construction of vector-valued Poincaré series of weight 2 is considered. Implications in the theory of parabolic bundles are discussed. When the genus of the group is zero, it is shown how an explicit basis for the space of these functions can be constructed.

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Parallel transport as dictated by a gauge field determines a collection of local reference systems. Comparing local reference systems in overlapping regions leads to an ensemble of algebras of relational kinematical observables for gauge theories including general relativity. Using an auxiliary cellular decomposition, we propose a discretization of the gauge field based on a decimation of the mentioned ensemble of kinematical observables. The outcome is a discrete ensemble of local subalgebras of "macroscopic observables" characterizing a measuring scale. A set of evaluations of those macroscopic observables is called an extended lattice gauge field because it determines a G-bundle over M (and over submanifolds of M that inherit a cellular decomposition) together with a lattice gauge field over an embedded lattice. A physical observable in our algebra of macroscopic observables is constructed. An initial study of aspects of regularization and coarse graining, which are special to this description of gauge fields over a combinatorial base, is presented. The physical relevance of this extension of ordinary lattice gauge fields is discussed in the context of quantum gravity. *