LATTICE GAUGE-FIELDS AND NONCOMMUTATIVE GEOMETRY

Conventional approaches to lattice gauge theories do not properly cons ider the topology of spacetime or of its fields, In this paper, we dev elop a formulation which tries to remedy this defect. It starts from a cubical decomposition of the supporting manifold (compactified space- time or spatial slice) interpreting it as a finite topological approxi mation in the sense of Sorkin. This finite space is entirely described by the algebra of cochains with the cup product. The methods of Conne s and Lott are then used to develop gauge theories on this algebra and to derive Wilson's actions for the gauge and Dirac fields therefrom w hich can now be given geometrical meaning. We also describe very natur al candidates for the QCD theta-term and Chern-Simons action suggested by this algebraic formulation, Some of these formulations are simpler than currently available alternatives. The paper treats both the func tional integral and Hamiltonian approaches.