Morita equivalence of multidimensional noncommutative tori

One can describe an n-dimensional noncommutative torus by means of an antisymmetric n×n matrix θ. We construct an action of the group SO(n, n|Z) on the space of n× n antisymmetric matrices and show that, generically, matrices belonging to the same orbit of this group give Morita equivalent tori. Some applications to physics are sketched. By definition [R5], an n-dimensional noncommutative torus is an associative algebra with involution having unitary generators U1, ..., Un obeying the relations UkUj = e(θkj)UjUk, (1) where e(t) = e and θ is an antisymmetric matrix. The same name is used for different completions of this algebra. In particular, we can consider the noncommutative torus as a C-algebra Aθ (the universal C -algebra generated by n unitary operators satisfying (1) ). Noncommutative tori are important in many problems of mathematics and physics. It was shown recently that they are essential in consideration of compactifications of M(atrix) theory ([CDS]; for further development see [T]). The results of the present paper also have application to physics. If two algebras A and  are Morita equivalent (see the definition below), then for every A-module R one can construct an Â-module R̂ in such a way that the correspondence R → R̂ is an equivalence of categories of A-modules and Â-modules. M. A. R. supported in part by NSF grant DMS 96-13833; A. S. supported in part by NSF grant DMS 95-00704; both authors supported in part by NSF Grant No. PHY94-07194 during visits to the Institute for Theoretical Physics, Santa Barbara. A.S. acknowledges also the hospitality of MIT, IAS and Rutgers University