Probability distribution of the index in gauge theory on 2d non-commutative geometry

TL;DR

This paper explores how non-commutative geometry influences the topological properties of 2D gauge theory, revealing a dominance of trivial topology in the continuum limit, contrasting with commutative cases.

Contribution

It introduces a non-perturbative approach to study topological distributions in non-commutative gauge theory using Monte Carlo simulations and analyzes the effects of non-commutativity on topological sectors.

Findings

Distribution is asymmetric under nu -> -nu due to parity violation.

In the continuum limit, trivial topological sector dominates.

Contrasts with Gaussian distribution in commutative lattice simulations.

Abstract

We investigate the effects of non-commutative geometry on the topological aspects of gauge theory using a non-perturbative formulation based on the twisted reduced model. The configuration space is decomposed into topological sectors labeled by the index nu of the overlap Dirac operator satisfying the Ginsparg-Wilson relation. We study the probability distribution of nu by Monte Carlo simulation of the U(1) gauge theory on 2d non-commutative space with periodic boundary conditions. In general the distribution is asymmetric under nu -> -nu, reflecting the parity violation due to non-commutative geometry. In the continuum and infinite-volume limits, however, the distribution turns out to be dominated by the topologically trivial sector. This conclusion is consistent with the instanton calculus in the continuum theory. However, it is in striking contrast to the known results in the commutative case obtained from lattice simulation, where the distribution is Gaussian in a finite volume, but the width diverges in the infinite-volume limit. We also calculate the average action in each topological sector, and provide deeper understanding of the observed phenomenon.