Field Theory on a Non-commutative Plane: a Non-Perturbative Study

Abstract

The 2d gauge theory on the lattice is equivalent to the twisted Eguchi-Kawai model, which we simulated at N ranging from 25 to 515. We observe a clear large N scaling for the 1- and 2-point function of Wilson loops, as well as the 2-point function of Polyakov lines. The 2-point functions agree with a universal wave function renormalization. The large N double scaling limit corresponds to the continuum limit of non-commutative gauge theory, so the observed large N scaling demonstrates the non-perturbative renormalizability of this non-commutative field theory. The area law for the Wilson loops holds at small physical area as in commutative 2d planar gauge theory, but at large areas we find an oscillating behavior instead. In that regime the phase of the Wilson loop grows linearly with the area. This agrees with the Aharonov-Bohm effect in the presence of a constant magnetic field, identified with the inverse non-commutativity parameter. Next we investigate the 3d λϕ^4 model with two non-commutative coordinates and explore its phase diagram. Our results agree with a conjecture by Gubser and Sondhi in d=4, who predicted that the ordered regime splits into a uniform phase and a phase dominated by stripe patterns. We further present results for the correlators and the dispersion relation. In non-commutative field theory the Lorentz invariance is explicitly broken, which leads to a deformation of the dispersion relation. In one loop perturbation theory this deformation involves an additional infrared divergent term. Our data agree with this perturbative result.

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Cited by in corpus (7)

• Phase diagram and dispersion relation of the non-commutative λϕ^{4} model in d=3
• A non-perturbative study of 4d U(1) non-commutative gauge theory -- the fate of one-loop instability
• Probing the fuzzy sphere regularisation in simulations of the 3d λϕ^4 model
• The Continuum Phase Diagram of the 2d Non-Commutative lambda phi**4 Model
• On the relation between non-commutative field theories at theta = infinity and large N matrix field theories
• Probability distribution of the index in gauge theory on 2d non-commutative geometry
• Area-preserving diffeomorphisms in gauge theory on a non-commutative plane: a lattice study