From Large N Matrices to the Noncommutative Torus

TL;DR

This paper rigorously demonstrates how the noncommutative two-torus can be approximated by finite-dimensional matrix models, clarifying the continuum limit and Morita equivalence, with applications to noncommutative gauge theories.

Contribution

It provides a rigorous framework for approximating the noncommutative torus with matrix geometries, elucidating the continuum limit and Morita equivalence in noncommutative gauge theories.

Findings

Finite-dimensional matrix models approximate the noncommutative torus.

The continuum limit of matrix models is precisely characterized.

Applications to noncommutative Yang-Mills theory are discussed.

Abstract

We describe how and to what extent the noncommutative two-torus can be approximated by a tower of finite-dimensional matrix geometries. The approximation is carried out for both irrational and rational deformation parameters by embedding the algebra of the noncommutative torus into an approximately finite algebra. The construction is a rigorous derivation of the recent discretizations of noncommutative gauge theories using finite dimensional matrix models, and it shows precisely how the continuum limits of these models must be taken. We clarify various aspects of Morita equivalence using this formalism and describe some applications to noncommutative Yang-Mills theory.