Chern-Simons Currents and Chiral Fermions on the Lattice

TL;DR

This paper analyzes how Wilson fermions on a lattice induce Chern-Simons currents, revealing topological and homotopy class changes that relate to chiral fermion models and differ from continuum behavior.

Contribution

It provides a topological interpretation of the lattice fermion propagator and shows how the Chern-Simons term varies discontinuously with fermion mass, connecting to chiral lattice fermion models.

Findings

Chern-Simons current depends on homotopy classes of momentum space maps.

Discontinuous changes in the Chern-Simons term occur at specific mass values.

Results align with the spectrum of chiral fermions bound to domain walls.

Abstract

We compute the Chern-Simons current induced by Wilson fermions on a $d=2n+1$ dimensional lattice, making use of a topological interpretation of the momentum space fermion propagator as a map from the torus to the sphere, $T^{d}\to S^{d}$. These mappings are shown to fall in different homotopy classes depending on the value of $m/r$, where $m$ is the fermion mass and $r$ is the Wilson coupling. As a result, the induced Chern-Simons term changes discontinuously at $d+1$ different values for $m$, unlike in the continuum. This behavior is exactly what is required by the peculiar spectrum found for a recently proposed model of chiral lattice fermions as zeromodes bound to a domain wall.