Ginsparg-Wilson Fermions in Odd Dimensions

TL;DR

This paper explores the application of the Ginsparg-Wilson relation to lattice Dirac operators in odd dimensions, addressing parity invariance, anomalies, and proposing a lattice Chern-Simons term.

Contribution

It extends the Ginsparg-Wilson framework to odd dimensions, clarifies the origin of parity anomalies, and introduces a consistent lattice Chern-Simons term.

Findings

Parity invariance under generalized transformations

Parity anomaly arises from fermion measure

A natural lattice Chern-Simons term is proposed

Abstract

The Ginsparg-Wilson relation, if written in a suitable form, can be used as a condition for lattice Dirac operators of massless fermions also in odd dimensions. The fermion action with such a Dirac operator is invariant under a generalized parity transformation, which reduces to the ordinary parity transformation in the (naive) continuum limit. The fermion measure, however, transforms non-trivially under the generalized parity transformation, and hence the parity anomaly arises solely from the fermion measure. The analogy to the lattice construction of chiral gauge theories in even dimensions is clarified by considering a dimensional reduction. We also propose a natural definition of a lattice Chern-Simons term, which is consistent with odd dimensional Ginsparg-Wilson fermions.