Discrete differential calculus graphs, topologies and gauge theory

摘要: Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra a set can be regarded as ‘‘reduction’’ ‘‘universal algebra’’ and this allows systematic exploration algebras given set. Associated with (di)graph where two vertices are connected by at most (antiparallel) arrows. The interpretation such graph ‘‘Hasse diagram’’ determining (locally finite) topology then establishes contact recent work other authors which discretizations topological spaces corresponding field theories were considered retain their global structure. It shown that theories, particular gauge formulated close analogy continuum case. framework presented generalizes ordinary lattice theory recovered from an oriented (hypercubic) graph. also includes, e.g., two‐point space used Connes Lott (and others) models elementary particle physics. formalism suggests latter approximation manifold thus opens way to relate ‘‘internal’’ (a la et al.) dimensionally reduced fields. Furthermore, ‘‘symmetric lattice’’ studied (in certain limit) turns out related ‘‘noncommutative calculus’’ manifolds.