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Supermembrane compactified on a $M_9\times T^2$ target space is globallydescribed by the inequivalent classes of torus bundles over torus. These torusbundles have monodromy in $SL(2,Z)$ when they correspond to the nontrivialcentral charge sector and they are trivial otherwise. The first ones containeight inequivalent classes of M2-brane bundles which at low energies, are incorrespondence with the eight type II gauged supergravities in $9D$. Therelation among them is completely determined by the global action of T-dualitywhich interchanges topological invariants of the two tori. The M2-brane torusbundles are invariant under $SL(2,Z)\times SL(2,Z) \times Z_2$. From theeffective point of view, there is another dual invariant theory, called DoubleField Theory which describe invariant actions under $O(D,D)$. Globally it isformulated in terms of doubled $2D$ torus fibrations over the spacetime with amonodromy given by $O(D,D,Z)$. In this note we discuss T-duality global aspectsconsidered in both theories and we emphasize certain similarities between bothapproaches which could give some hints towards a deeper relationship betweenthem.

On global aspects of duality invariant theories: M2-brane vs DFT