Noncommutative Geometry of Lattice and Staggered Fermions

TL;DR

This paper develops a noncommutative geometric framework for lattice and staggered fermions, connecting differential structures to Dirac operators that match known staggered Dirac operators in specific dimensions.

Contribution

It introduces a noncommutative differential structure on lattices and relates it explicitly to the Dirac-Connes operator, providing a geometric interpretation of staggered fermions.

Findings

Differential structure modeled as noncommutative exterior algebra.

Dirac-Connes operator corresponds to staggered Dirac operator in 1, 2, and 4 dimensions.

Framework unifies lattice differential calculus with noncommutative geometry.

Abstract

Differential structure of a d-dimensional lattice, which is essentially a noncommutative exterior algebra, is defined using reductions in first order and second order of universal differential calculus in the context of noncommutative geometry (NCG) developed by Dimakis et al. This differential structure can be realized adopting a Dirac-Connes operator proposed by us recently within Connes' NCG. With matrix representations being specified, our Dirac-Connes operator corresponds to staggered Dirac operator, in the case that dimension of the lattice equals to 1, 2 and 4.