Morita equivalence and T-duality (or B versus Theta)

T-duality in M(atrix) theory has been argued to be realized as Morita equiv alence in Yang-Mills theory on a non-commutative torus (NCSYM). Even though the two have the same structure group, they differ in their action since M orita equivalence makes crucial use of an additional modulus on the NCSYM s ide, the constant abelian magnetic background. In this paper, we reanalyze and clarify the correspondence between M(atrix) theory and NCSYM, and provi de two resolutions of this puzzle. In the first of them, the standard map i s kept and the extra modulus is ignored, but the anomalous transformation i s offset by the M(atrix) theory "rest term". In the second, the standard ma p is modified so that the duality transformations agree, and a SO(d) symmet ry is found to eliminate the spurious modulus. We argue that this is a true symmetry of supersymmetric Born-Infeld theory on a non-commutative torus, which allows to freely trade a constant magnetic background for non-commuta tivity of the base-space. We also obtain a BPS mass formula for this theory , invariant under T-duality, U-duality, and continuous SO(d) symmetry.