  In this note I review the role played by dualities in the Supermembrane
Theory compactified on a torus. Supermembrane theory realize S, T, so
U-duality, as exact symmetries of the theory. There are two well defined
sectors: with and without central charges. Both sectors have the SL(2,Z)xZ2
U-duality group in 9D as a symmetry of the theory, but the supermembrane with
central charges, also exhibits an extra SL(2,Z) symmetry of the theory
associated to the diffeomorphisms not connected with the identity changing the
homology basis. The Z2 symmetry is T-duality transformation that acts on the
supermembrane and generalize that of the string. This T-duality transformation
acts locally and also globally on the supermembrane theory. It has a natural
description in terms of the cohomology of the base manifold and the homology of
the target torus. When only string-like states are preserved, one recovers both
type II string theories in 9D. If one also consider properly limit of the
T-duality transformation characterizing the supermembrane, the usual closed
string T-duality transformation between the type IIA and type IIB mass
operators is recovered.

High Energy Physics - Theory