Admissibility Condition and Nontrivial Indices on a Noncommutative Torus

TL;DR

This paper numerically investigates the index of the Ginsparg-Wilson Dirac operator on a noncommutative torus, introducing an admissibility condition to identify configurations with nontrivial indices, demonstrating the potential for defining such indices in noncommutative geometry.

Contribution

It formulates an admissibility condition and provides the first numerical evidence of nontrivial indices on a noncommutative torus using the Ginsparg-Wilson relation.

Findings

Identified gauge configurations with nontrivial indices

Explicit example of index 1 configuration

First numerical evidence of nontrivial indices in this setting

Abstract

We study the index of the Ginsparg-Wilson Dirac operator on a noncommutative torus numerically. To do this, we first formulate an admissibility condition which suppresses the fluctuation of gauge fields sufficiently small. Assuming this condition, we generate gauge configurations randomly, and find various configurations with nontrivial indices. We show one example of configurations with index 1 explicitly. This result provides the first evidence that nontrivial indices can be naturally defined on the noncommutative torus by utilizing the Ginsparg-Wilson relation and the admissibility condition.