Chiral Fermions on the Lattice

TL;DR

This paper investigates the overlap approach to chiral gauge theories on arbitrary lattices, demonstrating how to eliminate doublers, recover known anomalies, and construct anomaly-free models, with results matching continuum theories.

Contribution

It provides a method to eliminate fermion doublers on lattices, reproduces continuum anomalies, and establishes criteria for anomaly-free chiral gauge theories.

Findings

Doublers can be eliminated in the overlap approach.

Perturbative anomalies are recovered on the lattice.

Formalism reproduces continuum anomalies in various backgrounds.

Abstract

The overlap approach to chiral gauge theories on arbitrary $D$--dimensional lattices is studied. The doubling problem and its relation to chiral anomalies for $D=2$ and 4 is examined. In each case it is shown that the doublers can be eliminated and the well known perturbative results for chiral anomalies can be recovered. We also consider the multi--flavour case and give the general criteria for the construction of anomaly free chiral gauge theories on arbitrary lattices. We calculate the second order terms in a continuum approximation to the overlap formula in $D$ dimensions and show that they coincide with the bilinear part of the effective action of $D$--dimensional Weyl fermions coupled to a background gauge field. Finally, using the same formalism we reproduce the correct Lorentz, diffeomorphism and gauge anomalies in the coupling of a Weyl fermion to 2--dimensional gravitational and Maxwell fields.