Lattice gauge fields and discrete noncommutative Yang-Mills theory

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 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.

Stripes from (non-commutative) stars

“…However, it is clearly of more interest to do this in higher dimensional theories. The lattice formalism has been developed [5] and just awaits use. Indeed, while this paper was in preparation we received a preprint [8] in which the same model was examined in 2 + 1 dimensions with similar conclusions.…”

“…However, the associated Heisenberg algebras of exponentials ofx µ (and∂ µ ) allow finite dimensional representations, as already noted by Weyl. This is the key to formulating the noncommutative theory on a finite periodic N × N lattice with lattice spacing a (see [5] for a more general formulation). The algebra (7) is replaced bŷ…”

Coherent states and N dimensional coordinate noncommutativity

“…This is not the case [10] as we are making the identification of momenta with commutators of K's and by the Jacobi identities…”

Lattice perturbation theory in noncommutative geometry and parity anomaly in 3d noncommutative QED

“…[11,12,13] (see [26,27,28] for reviews). In the literature it is common to start from a finite-N matrix model, which is then shown to be equivalent to the lattice formulation of a noncommutative field theory.…”

“…Here the space-time degrees of freedom and the internal ("color") degrees of freedom are treated on equal footing, and "star-gauge invariance" is simply described by the global U(∞) symmetry which acts on the matrix indices. The matrix model description is also useful for regularizing noncommutative field theories [9,10,11,12,13], since finite-N twisted reduced models [14] are interpreted as a lattice formulation of noncommutative field theories [11,12,13]. Such a lattice formulation provides the most reliable method to study the quantum dynamics of noncommutative field theories in a fully nonperturbative manner.…”