# On global aspects of duality invariant theories: M2-brane vs DFT

## TL;DR

This paper compares the global aspects of duality invariance in supermembrane theory and Double Field Theory, highlighting similarities and potential connections through their treatment of torus bundles and monodromies.

## Contribution

It analyzes the global structure of duality invariant theories, revealing parallels between M2-brane torus bundles and Double Field Theory's doubled torus fibrations.

## Key findings

- M2-brane bundles classified by $SL(2,Z)$ monodromies
- Double Field Theory formulated with $O(D,D,Z)$ monodromies
- Potential links between the global structures of both theories

## Abstract

Supermembrane compactified on a $M_9\times T^2$ target space is globally described by the inequivalent classes of torus bundles over torus. These torus bundles have monodromy in $SL(2,Z)$ when they correspond to the nontrivial central charge sector and they are trivial otherwise. The first ones contain eight inequivalent classes of M2-brane bundles which at low energies, are in correspondence with the eight type II gauged supergravities in $9D$. The relation among them is completely determined by the global action of T-duality which interchanges topological invariants of the two tori. The M2-brane torus bundles are invariant under $SL(2,Z)\times SL(2,Z) \times Z_2$. From the effective point of view, there is another dual invariant theory, called Double Field Theory which describe invariant actions under $O(D,D)$. Globally it is formulated in terms of doubled $2D$ torus fibrations over the spacetime with a monodromy given by $O(D,D,Z)$. In this note we discuss T-duality global aspects considered in both theories and we emphasize certain similarities between both approaches which could give some hints towards a deeper relationship between them.