On the chirality of a discrete Dirac-K\"ahler equation

TL;DR

This paper develops a discrete version of the Dirac-Kähler equation that preserves chiral properties, using a combinatorial double complex and discrete exterior calculus, and analyzes its chiral invariance and behavior under mass terms.

Contribution

It introduces a novel discrete Dirac-Kähler equation with a carefully constructed discrete Hodge star operator that captures chirality.

Findings

Discrete chiral invariance in the massless case is demonstrated.

The discrete Dirac-Kähler operator flips chirality in the massive case.

A combinatorial double complex is used to define discrete exterior calculus operations.

Abstract

We discuss a discrete analogue of the Dirac-K\"{a}hler equation in which chiral properties of the continual counterpart are captured. We pay special attention to a discrete Hodge star operator. To build one a combinatorial construction of double complex is used. We describe discrete exterior calculus operations on a double comlex and obtain the discrete Dirac-K\"{a}hler equation using these tools. Self-dual and anti-self-dual discrete inhomogeneous forms are presented. The chiral invariance of the massless discrete Dirac-K\"{a}hler equation is shown. Moreover, in the massive case we prove that a discrete Dirac-K\"{a}hler operator flips the chirality.