The index of the overlap Dirac operator on a discretized 2d non-commutative torus

TL;DR

This paper explores the properties of the index of the overlap Dirac operator on a discretized 2d non-commutative torus, revealing that the probability of non-zero index vanishes in the continuum limit, contrasting with commutative cases.

Contribution

It provides the first detailed calculation of the index of the overlap Dirac operator in a non-commutative 2d torus setting, connecting it with topological charge and classical solutions.

Findings

The index approximates integer values for small action, matching the topological charge.

The index is a multiple of the lattice size N under certain conditions.

The probability of a non-zero index diminishes in the continuum limit.

Abstract

The index, which is given in terms of the number of zero modes of the Dirac operator with definite chirality, plays a central role in various topological aspects of gauge theories. We investigate its properties in non-commutative geometry. As a simple example, we consider the U(1) gauge theory on a discretized 2d non-commutative torus, in which general classical solutions are known. For such backgrounds we calculate the index of the overlap Dirac operator satisfying the Ginsparg-Wilson relation. When the action is small, the topological charge defined by a naive discretization takes approximately integer values, and it agrees with the index as suggested by the index theorem. Under the same condition, the value of the index turns out to be a multiple of N, the size of the 2d lattice. By interpolating the classical solutions, we construct explicit configurations, for which the index is of order 1, but the action becomes of order N. Our results suggest that the probability of obtaining a non-zero index vanishes in the continuum limit, unlike the corresponding results in the commutative space.