Chiral fermion operators on the lattice

TL;DR

This paper explores a broad class of lattice Dirac operators satisfying generalized chiral symmetry and $ ext{γ}_5$-hermiticity, providing a spectral framework and discussing their role in chiral gauge theories.

Contribution

It introduces a general construction of lattice Dirac operators based on spectral representations and a unitary operator, extending the understanding of chiral fermions on the lattice.

Findings

Spectral representations are effective for analyzing Dirac operators.

A basic unitary operator is central to the construction of these operators.

Weaker conditions still yield proper Weyl fermions and chiral gauge theories.

Abstract

We only require generalized chiral symmetry and $\gamma_5$-hermiticity, which leads to a large class of Dirac operators describing massless fermions on the lattice, and use this framework to give an overview of developments in this field. Spectral representations turn out to be a powerful tool for obtaining detailed properties of the operators and a general construction of them. A basic unitary operator is seen to play a central r\^ole in this context. We discuss a number of special cases of the operators and elaborate on various aspects of index relations. We also show that our weaker conditions lead still properly to Weyl fermions and to chiral gauge theories.