Noncommutative gauge theories on $ \mathbb{R}_{\theta}^2 $ as matrix models

Abstract

We study a class of noncommutative gauge theory models on 2-dimensional Moyal space from the viewpoint of matrix models and explore some related properties. Expanding the action around symmetric vacua generates non local matrix models with polynomial interaction terms. For a particular vacuum, we can invert the kinetic operator which is related to a Jacobi operator. The resulting propagator can be expressed in terms of Chebyschev polynomials of second kind. We show that non vanishing correlations exist at large separations. General considerations on the kinetic operators stemming from the other class of symmetric vacua, indicate that only one class of symmetric vacua should lead to fast decaying propagators. The quantum stability of the vacuum is briefly discussed.

Noncommutative gauge theories on ℝ λ 3 $$ {\mathrm{\mathbb{R}}}_{\uplambda}^3 $$ : perturbatively finite models

“…In this paper, we show that natural noncommutative gauge theory models on R 3 λ can support gauge invariant harmonic terms, unlike the case of Moyal spaces. This stems from the existence of a relationship between the center of R 3 λ and the components of the gauge invariant 1-form canonical connection which arises in the derivation-based differential 1 This technical obstruction can be circumvented on R 2 θ for particular vacuum configurations [38].…”

“…the natural objects related to the canonical connection. Then, a suitable BRST gauge-fixing in the spirit of [38,56,57] gives rise to a family of matrix models with quartic interactions and kinetic operators (having compact resolvent). Their perturbative behavior is then examined.…”

“…Here, we will assume that the relevant field variable is A µ , akin to a matrix model formulation of gauge theories on R 3 λ , thus proceeding in the spirit of [38]. Natural requirement for the gauge-invariant functional are:…”

“…The equation of motion for (2.37) given by 38) one infers that Φ ρ = 0 is the absolute minimum of (2.37). 5 In the section 3, we will show that one class of gauge-invariant models pertaining to the families (2.37), (2.33) yields after gauge-fixing to a finite theory at all orders in perturbation.…”

“…The corresponding details are collected in the appendix A. The gauge-invariance of S Ω (2.37) can be translated into invariance under a nilpotent BRST operation δ 0 defined by the following structure equations [38] …”