Can nonlocal Dirac operators be topologically proper ?

TL;DR

This paper investigates the limits of topologically-proper lattice Dirac operators, demonstrating that nonlocal operators can still satisfy the Atiyah-Singer index theorem and possess exact zero modes in nontrivial gauge backgrounds.

Contribution

It shows that nonlocal lattice Dirac operators can be topologically proper and satisfy the Atiyah-Singer index theorem, expanding understanding of lattice gauge theories.

Findings

Nonlocal Dirac operators can have exact zero modes.

Topologically-proper lattice Dirac operators tend to nonlocal operators.

Nonlocal operators can satisfy the Atiyah-Singer index theorem.

Abstract

By examining the analyticity of a sequence of topologically-proper lattice Dirac operators, we show that they tend to a nonlocal Dirac operator. This implies that a nonlocal lattice Dirac operator can have exact zero modes satisfying the Atiyah-Singer index theorem, in gauge backgrounds with nonzero topological charge.