Differential calculus and gauge theory on finite sets

TL;DR

This paper develops a differential calculus and gauge theory framework on finite sets, especially cyclic groups, connecting to Connes' two-point model and q-calculus, with potential applications in physics.

Contribution

It introduces a novel formulation of differential calculus and gauge theory on finite sets with group structures, unifying and extending previous models like Connes' two-point approach.

Findings

Recovered Connes' two-point model within this framework

Established reductions to lower-dimensional calculi

Linked the complete reduction to q-calculus on periodic lattices

Abstract

We develop differential calculus and gauge theory on a finite set G. An elegant formulation is obtained when G is supplied with a group structure and in particular for a cyclic group. Connes' two-point model (which is an essential ingredient of his reformulation of the standard model of elementary particle physics) is recovered in our approach. Reductions of the universal differential calculus to `lower-dimensional' differential calculi are considered. The `complete reduction' leads to a differential calculus on a periodic lattice which is related to q-calculus.