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Present address: Departamento de Fisica, Universidad de Antofagasta, Aptdo 02800, Chile.

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We obtain the complete classification of the inequivalent classes of M2-brane symplectic torus bundles with monodromy in

M-theory is a theory candidate for unification of all the interactions in nature that contains supermembrane theory—also called M2-brane theory—as one of its building blocks. Any quantum consistent definition of M-theory will require its understanding. Supermembranes are (

There are other types of supergravity theories, like gauged/massive supergravities. Gauged supergravities can be obtained from string/supergravity theories in a number of ways: by compactifying on manifolds with nontrivial holonomy

In contrast, the determination of the M-theory action origin of the gauged/massive supergravity deformations has become much more elusive. The M-theory uplift of SS nine-dimensional reduction was conjectured to be related to torus bundles with monodromy in

Another topic that has received a lot of attention from the community are the

Global aspects of the

The paper is structured as follows: In Sec.

In this section we will review the supermembrane theory compactified on an

Let us review the construction of the symplectic M2-brane torus bundle in this first part of the section. The Hamiltonian of a supermembrane theory with central charges formulated in the LCG on a target space

The Hamiltonian is subject to the APD group residual constraints (connected to the identity

The theory is invariant under two different

The second one is associated with an invariance of the mass operator involving

Now we formulate the previous embedding description in terms of a symplectic torus bundle with monodromy in

The supermembrane symplectic torus bundles are characterized by two types of coinvariants relevant to the characterization of the supermembrane bundle. The class of coinvariants associated with the fiber

Associated with the monodromy subgroup

A coinvariant class in the winding sector is given by

The dependence of the bundle on the winding charges

Disclaimer: in this section we analyze the action of

The

The

Suppose now that instead of mapping

The duality transformation on the symplectic torus bundle has an action not only on the charges but also on the geometrical moduli. We define dimensionless variables

Let us recall

In this section we are going to establish the precise correspondence between the type IIA side of the supermembrane bundle with parabolic, elliptic, hyperbolic, and trombone monodromies. We will focus on the

The

The M2-brane torus bundles with parabolic monodromies has two inequivalent nontrivial monodromies classes,

We will analyze the two cases separately, and we will see that under

The elliptic monodromy group

It can be shown that Eq.

In contrast, Eq.

Let us now analyze the case of hyperbolic monodromy. There are infinite Abelian monodromy groups of hyperbolic matrices constructed in terms of

Let us now compare the dual monodromies of elliptic and hyperbolic 2-torus bundles with respect to their conjugate classes. We consider the generic monodromy case

Notice that for the parabolic case (

Clearly the elliptic and hyperbolic coinvariant classes are mapped under

Trombone symmetry produces supergravities that do not have a Lagrangian but are uniquely defined through the equations of motion. The reason is that the trombone symmetry is not a symmetry of the action since it scales the Langrangian, but it is a symmetry of the equations of motion. At quantum level, however, there exists a well-defined action since it is possible to define an invariant hamiltonian. In type II supergravity in 9D, the global symmetries are

Let us consider a nonlinear representation of the group

This transformation generates the complete lattice of charges for a given vacuum (that is, the asymptotic value of the scalar moduli).

The

We notice that in the expression of the Hamiltonian of the trombone torus bundle with coinvariant

The gauging is obtained by means of the nonlinear representation

The precise relations between the M2-brane bundle with monodromy in

We find eight independent inequivalent classes of supermembrane torus bundles with monodromy linearly and nonlinearly realized in

The

Trombone symmetry of the supermembrane torus bundle is a nonlinear realization of the

J. M. P. is grateful to the Scientific Programme Stringy Geometry, Mainz Institute for Theoretical Physics (MITP) for its hospitality and its partial support during part of the realization of this work. M. P. G. d. M. is grateful to I. Cavero-Pelaez for the helpful comments on the manuscript. M. P. G. d. M. has been supported by Convenio Marco UES ANT1398 PROJECT and ANT1555 PROJECT, Plan plurianual de fortalecimiento Institucional, MECESUP, Chile. A. R. and M. P. G. d. M. are partially supported by Project No. FONDECYT 1161192 (Chile). M. P. G. d. M. and A. R. participated as external nodes of the EU-COST Action No. MP1210 “The String Theory Universe” during the realization of this work.

Let us define the map associated with the coinvariant classes on the KK sector to the coinvariant classes on the winding sector. We take one element

If

In this appendix we obtain the most general transformation of

One can distinguish among three different cases, according to the value of the trace

The parabolic case

The elliptic case corresponds to have

For the hyperbolic case we will analyze separately three different cases restricted to assume that

In first place consider (

If (

The last case to consider occurs when

In conclusion there always exists a parabolic transformation