M-theory FDA, twisted tori and Chevalley cohomology

The FDA algebras emerging from twisted tori compactifications of M-theory with fluxes are discussed within the general classification scheme provided by Sullivan's theorems and by Chevalley cohomology. It is shown that the generalized Maurer-Cartan equations which have appeared in the literature, in spite of their complicated appearance, once suitably decoded within cohomology, lead to trivial FDAs, all new p-form generators being contractible when the 4-form flux is cohomologically trivial. Non-trivial D=4 FDAs can emerge from non-trivial fluxes but only if the cohomology class of the flux satisfies an additional algebraic condition which appears not to be satisfied in general and has to be studied for each algebra separately. As an illustration an exhaustive study of Chevalley cohomology for the simplest class of SS algebras is presented but a general formalism is developed, based on the structure of a double elliptic complex, which, besides providing the presented results, makes possible the quick analysis of compactification on any other twisted torus.