The M-theory origin of the IIB gauged supergravities in nine dimensions, classified according to the inequivalent classes of monodromy, is shown to exactly corresponds to the global description of the supermembrane with central charges. The global description is a realization of the sculpting mechanism of gauging (arXiv:1107.3255) and it is associated to particular deformation of fibrations. The supermembrane with central charges may be formulated in terms of sections on symplectic torus bundles with SL(2,Z) monodromy. This global formulation corresponds to the gauging of the abelian subgroups of SL(2,Z) associated to monodromies acting on the target torus. We show the existence of the trombone symmetry in the supermembrane formulated as a non-linear realization of the SL(2,Z) symmetry and construct its gauging in terms of the supermembrane formulated on an inequivalent class of symplectic torus fibration. The supermembrane also exhibits invariance under T-duality and we find the explicit T-duality transformation. It has a natural interpretation in terms of the cohomology of the base manifold and the homology of the target torus. We conjecture that this construction also holds for the IIA origin of gauged supergravities in 9D such that the supermembrane becomes the origin of all type II supergravities in 9D. The geometric structure of the symplectic torus bundle goes beyond the classification on conjugated classes of SL(2,Z). It depends on the elements of the coinvariant group associated to the monodromy group. The possible values of the (p,q) charges on a given symplectic torus bundle are restricted to the corresponding equivalence class defining the element of the coinvariant group.Comment: 41 pages, Latex. Typos corrected, references added, appendix added. Sections enlarged with more examples and clarifying explanations. Minor corrections in section 8. Results unchange

We describe a compactified Supermembrane, or M2-brane, with 2-form fluxes generated by constant three-forms that are turned on a 2-torus of the target space M 9 × T 2 . We compare this theory with the one describing a 11D M2-brane formulated on M 9 ×T 2 target space subject to an irreducible wrapping condition. We show that the flux generated by the bosonic 3-form under consideration is in a one to one correspondence to the irreducible wrapping condition. After a canonical transformation both Hamiltonians are exactly the same up to a constant shift in one particular case. Consequently both of them, share the same spectral properties. We conclude that the Hamiltonian of the M2-brane with 2-form target space fluxes on a torus has a purely discrete spectrum with eigenvalues of finite multiplicity and it can be considered to describe a new sector of the microscopic degrees of freedom of M-theory. We also show that the total membrane momentum in the direction associated to the flux condition adquires a quantized contribution in correspondence to the flux units that have been turned on.

We give the explicit formulation of the 11D supermembrane as a symplectic torus bundle with non trivial monodromy and non vanishing Euler class. This construction allows a classification of all supermembranes showing explicitly the discrete SL(2, Z) symmetries associated to dualities. It hints as the origin in M-theory of the gauging of the effective theories associated to string theories.

In this paper we obtain the complete classification of the inequivalent classes of M2-brane symplectic torus bundles with monodromy in SL(2, Z) and their precise U-duality relations among them. There are eight inequivalent classes of bundles whose monodromy groups, at low energies, are in correspondence with the gauging groups of the eight type II gauged supergravities in nine dimensions. Four of those have been previously found and they correspond to the 'type IIB side'. In this paper we provide the explicit realization of the remaining four classes associated to the 'type IIA side'. The precise M2-brane U-duality relations between the eight inequivalent classes of bundles have allowed to identified the remaining four ones. We conjecture that the classes of gaugings -classifying the eight types of II gauged supergravity in nine dimensions-are determined by the inequivalent coinvariant classes associated to the base and the fiber of the supermembrane bundles and their duals.