Morita duality and large- N limits

Abstract

We study some dynamical aspects of gauge theories on noncommutative tori. We show that Morita duality, combined with the hypothesis of analyticity as a function of the noncommutativity parameter Θ, gives information about singular large- N limits of ordinary U( N) gauge theories, where the large-rank limit is correlated with the shrinking of a two-torus to zero size. We study some nonperturbative tests of the smoothness hypothesis with respect to Θ in theories with and without supersymmetry. In the supersymmetric case this is done by adapting Witten's index to the present situation, and in the nonsupersymmetric case by studying the dependence of energy levels on the instanton angle. We find that regularizations which restore supersymmetry at high energies seem to preserve Θ-smoothness whereas nonsupersymmetric asymptotically free theories seem to violate it. As a final application we use Morita duality to study a recent proposal of Susskind to use a noncommutative Chern–Simons gauge theory as an effective description of the fractional Hall effect. In particular we obtain an elegant derivation of Wen's topological order.