Zero modes of the Dirac operator in three dimensions

TL;DR

This paper explores the conditions under which zero modes of the Dirac operator exist in three dimensions, linking their presence to topological properties when finite Chern--Simons action is required.

Contribution

It establishes a connection between zero modes of the Dirac operator and topological properties under the finite Chern--Simons action constraint in three dimensions.

Findings

Zero modes are related to specific topological properties.

Existence of zero modes depends on the finite Chern--Simons action.

Topological conditions determine zero mode existence.

Abstract

We investigate zero modes of the Dirac operator coupled to an Abelian gauge field in three dimensions. We find that the existence of a certain class of zero modes is related to a specific topological property precisely when the requirement of finite Chern--Simons action is imposed.