On Pure Lattice Chern-Simons Gauge Theories

TL;DR

This paper analyzes the lattice formulation of Abelian Chern-Simons gauge theories, revealing additional zero modes that challenge gauge invariance and proposing a mechanism similar to Wilson fermions to eliminate these zeros.

Contribution

It identifies extra zero eigenvalues in lattice Abelian Chern-Simons models and proposes a Wilson-like term to remove them, improving lattice gauge theory formulations.

Findings

Extra zero eigenvalues are present inside the Brillouin zone.

Adding a Maxwell term opens a gap, removing the extra zeros.

The zero mode proliferation is analogous to the Nielsen-Ninomiya theorem.

Abstract

We revisit the lattice formulation of the Abelian Chern-Simons model defined on an infinite Euclidean lattice. We point out that any gauge invariant, local and parity odd Abelian quadratic form exhibits, in addition to the zero eigenvalue associated with the gauge invariance and to the physical zero mode at p=0 due to traslational invariance, a set of extra zero eigenvalues inside the Brillouin zone. For the Abelian Chern-Simons theory, which is linear in the derivative, this proliferation of zero modes is reminiscent of the Nielsen-Ninomiya no-go theorem for fermions. A gauge invariant, local and parity even term such as the Maxwell action leads to the elimination of the extra zeros by opening a gap with a mechanism similar to that leading to Wilson fermions on the lattice.