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50 D. Bullock, C. Frohman, J. Kania-Bartoszy´nska Part 1. Lattice Gauge Field Theory Herein we develop, from a self contained and axiomatic approach, the machinery of gauge field theory on an abstract, oriented, ciliated graph. For basic background on Hopf algebras we rely on Sweedler [18] and Kassel [13]. The discussion here is restricted to the definitions and basic results confirming that the theory is consistent and computationally viable. For origins of the ideas we refer the reader to [1, 3, 4, 5, 8, 10].

The positive number α1 above can be chosen as arbitrarily small. 17), but now we assume that the expression in front of the exponential 32 T. Balaban above is sufficiently small. β − 4 +α1 ≤ 1, β − 8 η γ−α1 |σ| ≤ 1. 27) Then the expansion can be bounded by β − 8 , which is obviously arbitrarily small for β large enough. 17). Let us remark that there is nothing special about the powers of β above. 26) in many other ways among the three factors. We formulate the obtained localization results in the lemma.

52) with ε = const. 58). 54). Let us recall that we have the additional factor βk 2 < 2β − 2 η 2 d−2 multiplying these terms, so altogether we get the right scaling factor η 2 +γ−α1 multiplied by const. β −1+α1 . The last expression is arbitrarily small for β large enough, we 1 1 can even assume that const. β − 2 +α1 ≤ 1 and we are left simply with β − 2 . 58). 61) dτ (j) ˜ + τ C (k) 21 ψ ), g). M (x; ψk(j) (ψ (k) (θ) τ2 1 We construct a localization expansion for the operator Hk C (k) 2 in the usual way, and 1 ξ −α |ψ |.