U-duality in quantum M2-branes and gauged supergravities

A bstract 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 H 2 (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 , ℤ 3 , ℤ 5 , ℤ 9 , or ℤ 2 n − 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 ).