On the relation between non-commutative field theories at   =   and largeNmatrix field theories

Abstract

It is well-known that non-commutative (NC) field theories at θ = ∞ are "equivalent" to large N matrix field theories to all orders in perturbation theory, due to the dominance of planar diagrams. By formulating a NC field theory on the lattice nonperturbatively and mapping it onto a twisted reduced model, we point out that the above equivalence does not hold if the translational symmetry of the NC field theory is broken spontaneously. As an example we discuss NC scalar field theory, where such a spontaneous symmetry breakdown has been confirmed by Monte Carlo simulations.

Lectures on Matrix Field Theory

“…See [135,136] for a pedagogical introduction to the subject of the functional renormalization group. The Wilson renormalization group recursion formula was also used in [130][131][132][133][134] to study matrix scalar models which, as it turns out, are of great relevance to the limit θ −→ ∞ of noncommutative scalar field theory [137].…”

Area-preserving diffeomorphisms in gauge theory on a non-commutative plane: a lattice study

“…and[19] 2. We remark, however, that this restoration does not need to hold generally: it can fail non-perturbatively for instance in the case of spontaneous symmetry breaking[21].…”

Wilson Loops in 2D Noncommutative Euclidean Gauge Theory: 1. Perturbative Expansion

“…In contrast to this the Wilson loop averages in the 2D noncommutative gauge theory (1.4) exhibit a nontrivial dependence on θ . References [27,28,29,30,31,32,33,34,35,36,37] are devoted to the analysis of 2D noncommutative gauge theory.…”