Towards the solution of noncommutative $Y{M}_2$: Morita equivalence and large N-limit

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.

21 pages, JHEP3 preprint tex-format

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• The Meaning of Infrared Singularities in Noncommutative Gauge Theories
• Morita Duality and Large-N Limits
• Gauge Theory On The Fuzzy Torus
• Solitons on Noncommutative Torus as Elliptic Algebras and Elliptic Models

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