Abstract
We show that the D = 11 supermembrane theory (M2-brane) compactified on a M9× T2 target space, with constant fluxes C± naturally incorporates the geometrical structure of a twisted torus. We extend the M2-brane theory to a formulation on a twisted torus bundle. It is consistently fibered over the world volume of the M2-brane. It can also be interpreted as a torus bundle with a nontrivial U(1) connection associated to the fluxes. The structure group G is the area preserving diffeomorphisms. The torus bundle is defined in terms of the monodromy associated to the isotopy classes of symplectomorphisms with π0(G) = SL(2, Z), and classified by the coinvariants of the subgroups of SL(2, Z). The spectrum of the theory is purely discrete since the constant flux induces a central charge on the supersymmetric algebra and a modification on the Hamiltonian which renders the spectrum discrete with finite multiplicity. The theory is invariant under symplectomorphisms connected and non connected to the identity, a result relevant to guarantee the U-dual invariance of the theory. The Hamiltonian of the theory exhibits interesting new U(1) gauge and global symmetries on the worldvolume induced by the symplectomorphim transformations. We construct explicitly the supersymmetric algebra with nontrivial central charges. We show that the zero modes decouple from the nonzero ones. The nonzero mode algebra corresponds to a massive superalgebra that preserves either 1/2 or 1/4 of the original supersymmetry depending on the state considered.
Highlights
Twisted torus seen as fiber bundles is associated to nontrivial torus bundles with monodromy over torus
We show that the D = 11 supermembrane theory (M2-brane) compactified on a M9 × T 2 target space, with constant fluxes C± naturally incorporates the geometrical structure of a twisted torus
The M2-brane formulated on a twisted torus bundle considers the twisted torus TW3 to contain two of its dimensions associated to the 2-torus target space and the third one to the fiber of the nontrivial U(1) principal bundle associated to a central charge on the worldvolume
Summary
We characterize the symmetries of the theory by a study on the area preserving diffeomorphisms APD which in two dimensions are equivalent to 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 They will be relevant in the definition of symplectomorphims connected to the identity Symp0(Σ) and not connected to the identity, SympG(Σ), respectively. Associated to Ai we will introduce a one-form connection on a trivial U(1) principal bundle which carries the dynamical degrees of freedom In this second case the complete transformation acts only on the exact part of the embedding map A leaving invariant the harmonic sector. This transformation is associated to a symplectic connection A that carries the dynamical degrees of freedom.
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