Finite N matrix models of noncommutative gauge 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.

Symmetry breaking in twisted Eguchi–Kawai models

“…There are other motivations for such a study: first, it allows us to do simulations with much larger N than with the traditional method, since the number of external degrees of freedom (volume) is reduced to its minimum; and second, it provides an alternative method to approach large N physics, with which we can compare (and hopefully confirm) the numerous results already obtained via the conventional method. Finally we point to the importance of reduced models, in particular the twisted version, in matrix models of noncommutative field theory [10] and M-theory [11].…”

Instantons, fluxons, and open gauge string theory

“…An explicit realization of two-dimensional noncommutative gauge theory as a matrix model has also been constructed in [14,15] by exploiting the general relation [16] between its lattice regularization and the twisted Eguchi-Kawai (TEK) model [17]- [19]. According to the general paradigm of reduced models, ordinary QCD 2 at large N should be recovered from the TEK model as well.…”

Flux tube solutions in noncommutative gauge theories

“…Here, however, we have a coupled interacting system. An alternative approach to noncommutative theories is to acknowledge that they refer to operator-valued fields and work directly with operators on the quantum phase space (X, Y ) which are functions of z and t [19,28,29]. In this approach, star products become operator products and integration over the (X, Y ) plane becomes trace:…”

“…We will look for solutions of (19,20) that represent static magnetic flux tubes in the z-direction. We therefore take ∂ z A i = 0 and also make the gauge choice A z = 0.…”