Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory

TL;DR

This paper develops a nonperturbative lattice formulation of noncommutative Yang-Mills theory, demonstrating UV/IR mixing, Morita equivalence, and regularization methods, with applications to matter coupling and beta-function calculations.

Contribution

It introduces a lattice approach to noncommutative Yang-Mills theory, establishing Morita equivalence with ordinary gauge theories and providing a framework for nonperturbative regularization.

Findings

UV/IR mixing demonstrated nonperturbatively

Morita equivalence between noncommutative and ordinary gauge theories proven

Regularization of noncommutative theories via lattice gauge theory with 't Hooft flux

Abstract

We present a lattice formulation of noncommutative Yang-Mills theory in arbitrary even dimensionality. The UV/IR mixing characteristic of noncommutative field theories is demonstrated at a completely nonperturbative level. We prove a discrete Morita equivalence between ordinary Yang-Mills theory with multi-valued gauge fields and noncommutative Yang-Mills theory with periodic gauge fields. Using this equivalence, we show that generic noncommutative gauge theories in the continuum can be regularized nonperturbatively by means of {\it ordinary} lattice gauge theory with 't~Hooft flux. In the case of irrational noncommutativity parameters, the rank of the gauge group of the commutative lattice theory must be sent to infinity in the continuum limit. As a special case, the construction includes the recent description of noncommutative Yang-Mills theories using twisted large $N$ reduced models. We study the coupling of noncommutative gauge fields to matter fields in the fundamental representation of the gauge group using the lattice formalism. The large mass expansion is used to describe the physical meaning of Wilson loops in noncommutative gauge theories. We also demonstrate Morita equivalence in the presence of fundamental matter fields and use this property to comment on the calculation of the beta-function in noncommutative quantum electrodynamics.