Towards the solution of noncommutative $YM_2$: Morita equivalence and large N-limit

TL;DR

This paper explores solving noncommutative U(1) gauge theories on the plane by using Morita equivalence to relate them to known U(N) theories, employing large N limits and rational approximations to analyze their partition functions and soliton configurations.

Contribution

It introduces a method to solve noncommutative U(1) theories via Morita equivalence and large N limits, connecting them to known U(N) solutions and non-commutative solitons.

Findings

Constructed rational approximants for noncommutative theories.

Computed partition functions and Wilson line correlators in the large N limit.

Identified configurations corresponding to non-commutative solitons.

Abstract

In this paper we shall investigate the possibility of solving U(1) theories on the non-commutative (NC) plane for arbitrary values of $\theta$ by exploiting Morita equivalence. This duality maps the NC U(1) on the two-torus with a rational parameter $\theta$ to the standard U(N) theory in the presence of a 't Hooft flux, whose solution is completely known. Thus, assuming a smooth dependence on $\theta$, we are able to construct a series rational approximants of the original theory, which is finally reached by taking the large $N-$limit at fixed 't Hooft flux. As we shall see, this procedure hides some subletities since the approach of $N$ to infinity is linked to the shrinking of the commutative two-torus to zero-size. The volume of NC torus instead diverges and it provides a natural cut-off for some intermediate steps of our computation. In this limit, we shall compute both the partition function and the correlator of two Wilson lines. A remarkable fact is that the configurations, providing a finite action in this limit, are in correspondence with the non-commutative solitons (fluxons) found independently by Polychronakos and by Gross and Nekrasov, through a direct computation on the plane.