Difference Discrete Connection and Curvature on Cubic Lattice

TL;DR

This paper develops a formal framework for defining difference discrete connections and curvature on cubic lattices, linking discrete differential calculus with lattice gauge theory and integrable systems.

Contribution

It introduces equivalent definitions of discrete connections and curvature on lattices, extending to random lattices and connecting with existing lattice gauge theories.

Findings

Defined difference discrete connection and curvature on cubic lattices

Extended definitions to random lattices

Applied framework to discrete integrable systems

Abstract

In a way similar to the continuous case formally, we define in different but equivalent manners the difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space. We deal with the difference operators as the discrete counterparts of the derivatives based upon the differential calculus on the lattice. One of the definitions can be extended to the case over the random lattice. We also discuss the relation between our approach and the lattice gauge theory and apply to the discrete integrable systems.