Towards the solution of noncommutative YM₂: Morita equivalence and large N-limit

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In this paper we shall investigate the possibility of solving U(1) theories on the non-commutative (NC) plane for arbitrary values of θ by exploiting Morita equivalence. This duality maps the NC U(1) on the two-torus with a rational parameter θ 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 θ, 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.