Discrete differential calculus, graphs, topologies and gauge theory

TL;DR

This paper develops a framework for differential calculus on discrete sets using noncommutative geometry, enabling the formulation of gauge theories and topological models on discrete structures, bridging continuum and discrete approaches.

Contribution

It introduces a systematic method to derive differential algebras on discrete sets and connects them with graph-based topologies, extending lattice gauge theories and relating discrete models to noncommutative geometry.

Findings

Discrete differential calculus can be systematically constructed from universal differential algebra.

Gauge theories can be formulated on discrete sets similarly to continuum theories.

The framework generalizes lattice theories and relates discrete models to noncommutative geometry.

Abstract

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