Lattice gauge fields and discrete noncommutative Yang-Mills theory

We present a lattice formulation of noncommutative Yang-Mills theory inarbitrary even dimensionality. The UV/IR mixing characteristic ofnoncommutative field theories is demonstrated at a completely nonperturbativelevel. We prove a discrete Morita equivalence between ordinary Yang-Millstheory with multi-valued gauge fields and noncommutative Yang-Mills theory withperiodic gauge fields. Using this equivalence, we show that genericnoncommutative gauge theories in the continuum can be regularizednonperturbatively by means of {\it ordinary} lattice gauge theory with 't~Hooftflux. In the case of irrational noncommutativity parameters, the rank of thegauge group of the commutative lattice theory must be sent to infinity in thecontinuum limit. As a special case, the construction includes the recentdescription of noncommutative Yang-Mills theories using twisted large $N$reduced models. We study the coupling of noncommutative gauge fields to matterfields in the fundamental representation of the gauge group using the latticeformalism. The large mass expansion is used to describe the physical meaning ofWilson loops in noncommutative gauge theories. We also demonstrate Moritaequivalence in the presence of fundamental matter fields and use this propertyto comment on the calculation of the beta-function in noncommutative quantumelectrodynamics.Comment: 48 pages LaTeX2e, no figure