Comments on the Morita Equivalence

TL;DR

This paper provides a straightforward derivation of Morita equivalence between noncommutative and ordinary Yang-Mills theories on a torus, highlighting the correspondence of gauge-invariant observables like Wilson loops.

Contribution

It offers a simple derivation of Morita equivalence and details the mapping of observables, including Polyakov loops and open Wilson loops, between the theories.

Findings

Morita equivalence relates noncommutative and ordinary Yang-Mills theories on a torus.

Polyakov loops in ordinary YM map to open noncommutative Wilson loops.

The paper clarifies the correspondence of gauge-invariant observables under Morita transformation.

Abstract

It is known that noncommutative Yang-Mills theory with periodical boundary conditions on torus at the rational value of the noncommutativity parameter is Morita equivalent to the ordinary Yang-Mills theory with twisted boundary conditions on dual torus. We present simple derivation of this fact. We describe one-to-one correspondence between and gauge invariant observables in these two theories. In particular, we show that under Morita map Polyakov loops in the ordinary YM theory go to the open noncommutative Wilson loops discovered by Ishibashi, Iso, Kawai and Kutazawa.