M-theory FDA, Twisted Tori and Chevalley Cohomology

TL;DR

This paper analyzes FDA algebras from twisted tori compactifications of M-theory using Chevalley cohomology, revealing conditions under which non-trivial fluxes produce non-trivial FDA structures.

Contribution

It develops a general cohomological formalism to classify FDA algebras from M-theory compactifications, clarifying when fluxes lead to non-trivial structures.

Findings

Trivial FDA.s arise when fluxes are cohomologically trivial.

Non-trivial FDA.s require flux cohomology classes satisfying additional algebraic conditions.

A formalism based on a double elliptic complex enables analysis of various twisted torus compactifications.

Abstract

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 FDA.s, all new p--form generators being contractible when the 4--form flux is cohomologically trivial. Non trivial D=4 FDA.s 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.