Classification of M2-brane 2-torus bundles, U -duality invariance, and type II gauged supergravities

Abstract

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.

Fluxes, twisted tori, monodromy and U(1) supermembranes

“…A way to obtain nine dimensional type II gauged supergravities as an Effective Field Theory (EFT) at low energies has been done from F-theory compactified on a 3-twisted torus ( [20]) or from M-theory [27]. In this paper we will show that the supermembrane with constant fluxes C ± compactified on M 9 × T 2 can be understood as a supermembrane compactified on a twisted tori, providing new hints of its connection [28,29] with type II gauged supergravities in nine dimensions [12,17].…”

“…The formulation of the supermembrane in terms of the geometrical objects of a twisted torus bundle will then follow directly. In such case, the classification of inequivalent classes of M2-brane torus bundles already known [28,29] and related to gauge supergravities [12,17], can be enriched with the associated monopole structure [31]. This construction can be of relevance for applications of the M2-brane phenomenology.…”

“…The Hamiltonian of the supermembrane with central charges formulated in the LCG besides its invariance under symplectomorphisms is invariant under two discrete symmetries associated to the monodromy of the fiber denoted formerly by the authors [29] as S U duality, 1 and of the base respectively, both contained in SL(2, Z), the group of isotopy classes of symplectomorphims. One of the discrete group symmetries SL(2, Z) is associated to the change in the basis of harmonic one-forms of the worldvolume torus of the M2brane.…”

“…The resulting transformation on dX i is [29]. This result is important since the supermembrane is then well defined on the class of T 2 flat torus modulo the full group of symplectomorphisms.…”

“…Focusing on the M2-brane bundle description and considering the 2-torus target space as its fiber, with structure group the symplectomorphisms preserving its canonical symplectic two-form, and the worldvolume as its base manifold, there is a formulation of the M2-brane on a torus bundle with monodromy in SL(2, Z) as realized in [29]. In this paper we show that the nontrivial U(1) principal bundle over the base manifold and the 2-torus fiber determine a 3-twisted torus bundle T 3 W or equivalently a T 2 with a connection U(1) that is consistently fibered over the worldvolume base.…”