U-duality in quantum M2-branes and gauged supergravities

In this paper, we study the relation of the M2-brane with fluxes and monodromy in SL(2, ℤ), which has a quantum discrete supersymmetric spectrum with finite multiplicity and type IIB gauged supergravities in nine dimensions. SL(2, ℤ) is the group of isotopy classes of the area-preserving diffeomorphisms. The global description of these M2-branes we are considering is formulated on twisted torus bundles, and they are classified in terms of H²(Σ, ℤ_ρ), or equivalently, by their coinvariants for a given monodromy subgroup. We find the ‘gauge’ symmetries between equivalent M2-branes on torus bundles with monodromy that lead to ℝ, SO(2), or SO(1, 1), the symmetry groups of type IIB gauged supergravities in 9d. We obtain an explicit relation between the equivalent classes of M2-brane bundles and the mass parameters that classify the gaugings of type IIB supergravities in 9d. We also find that the symmetries between inequivalent M2-branes on twisted torus bundles for a given monodromy are related to ℤ, ℤ₃, ℤ₅, ℤ₉, or ℤ_2n − 7 for n ≥ 5, the U-duality symmetry group, a subgroup of SL(2, ℤ). In distinction, in the case without monodromy, related to type II maximal supergravity at low energies, its U-duality group corresponds to the full SL(2, ℤ).