The Fuzzy Disc

TL;DR

This paper develops a finite matrix model approximation to the algebra of functions on a disc using noncommutative geometry, enabling nonperturbative studies of field theories on the disc with promising accuracy.

Contribution

It introduces a new matrix model for the disc based on noncommutative geometry, including a Laplacian and correlation functions, with good agreement to exact results.

Findings

Eigenvalues of the Laplacian match exact disc results.

Two-point correlation functions agree well with exact solutions.

Model captures edge states similar to previous matrix models.

Abstract

We introduce a finite dimensional matrix model approximation to the algebra of functions on a disc based on noncommutative geometry. The algebra is a subalgebra of the one characterizing the noncommutative plane with a * product and depends on two parameters N and theta. It is composed of functions which decay exponentially outside a disc. In the limit in which the size of the matrices goes to infinity and the noncommutativity parameter goes to zero the disc becomes sharper. We introduce a Laplacian defined on the whole algebra and calculate its eigenvalues. We also calculate the two--points correlation function for a free massless theory (Green's function). In both cases the agreement with the exact result on the disc is very good already for relatively small matrices. This opens up the possibility for the study of field theories on the disc with nonperturbative methods. The model contains edge states, a fact studied in a similar matrix model independently introduced by Balachandran, Gupta and Kurkcuoglu.