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Noncommutative fermions and Morita equivalence

We study the Morita equivalence for fermion theories on noncommutative two-tori. For rational values of the $θ$ parameter (in appropriate units) we show the equivalence between an abelian noncommutative fermion theory and a nonabelian theory of twisted fermions on ordinary space. We study the chiral anomaly and compute the determinant of the Dirac operator in the dual theories showing that the Morita equivalence also holds at this level.