Finite N Matrix Models of Noncommutative Gauge Theory

TL;DR

This work provides a constructive, nonperturbative definition of noncommutative gauge theory at finite $N$ by formulating a twisted Eguchi–Kawai–type unitary matrix model with quotient constraints. Through a lattice basis and a lattice star-product, it yields a manifestly star-gauge invariant NC lattice gauge theory that reduces to Wilson lattice gauge theory for specific parameter choices and admits a continuum limit with finite volume $L$ and finite noncommutativity scale $\lambda$. The construction makes explicit Morita duality to a NC Yang–Mills theory on a dual torus and enables the definition of both closed and open Wilson-type observables that are star-gauge invariant on a finite lattice. It provides a concrete framework for nonperturbative studies of NC gauge theories, including potential numerical simulations, while connecting to D-brane physics and the B-field description in string theory.

Abstract

We describe a unitary matrix model which is constructed from discrete analogs of the usual projective modules over the noncommutative torus and use it to construct a lattice version of noncommutative gauge theory. The model is a discretization of the noncommutative gauge theories that arise from toroidal compactification of Matrix theory and it includes a recent proposal for a non-perturbative definition of noncommutative Yang-Mills theory in terms of twisted reduced models. The model is interpreted as a manifestly star-gauge invariant lattice formulation of noncommutative gauge theory, which reduces to ordinary Wilson lattice gauge theory for particular choices of parameters. It possesses a continuum limit which maintains both finite spacetime volume and finite noncommutativity scale. We show how the matrix model may be used for studying the properties of noncommutative gauge theory.