We investigate in the Matrix theory framework, the subgroup of dualities of the DLCQ of M-theory compactified on three-tori, which corresponds to T-duality in the auxiliary Type II string theory. We show how these dualities are realized in the supersymmetric Yang-Mills gauge theories on dual noncommutative three-tori.

We present a cohomological method for obtaining the non-Abelian Seiberg-Witten map for any gauge group and to any order in θ. By introducing a ghost field, we are able to express the equations defining the Seiberg-Witten map through a coboundary operator, so that they can be solved by constructing a corresponding homotopy operator. * email address: dmbrace@lbl.gov † email address: BLCerchiai@lbl.gov ‡

We show that the Born-Infeld theory with n complex abelian gauge fields written in an auxiliary field formulation has a U (n, n) duality group. We conjecture the form of the Lagrangian obtained by eliminating the auxiliary fields and then introduce a new reality structure leading to a Born-Infeld theory with n real gauge fields and an Sp(2n, IR) duality symmetry. The real and complex constructions are extended to arbitrary even dimensions. The maximal noncompact duality group is U (n, n) for complex fields. For real fields the duality group is Sp(2n, IR) if half of the dimension of space-time is even and O(n, n) if it is odd. We also discuss duality under the maximal compact subgroup, which is the self-duality group of the theory obtained by fixing the expectation value of a scalar field. Supersymmetric versions of self-dual theories in four dimensions are also discussed.