Twisted tori and fluxes: A no go theorem for Lie groups of weak G2 holonomy

In this paper we prove the theorem that there exists no 7-dimensional Lie group manifold G of weak G2 holonomy. We actually prove a stronger statement, namely that there exists no 7-dimensional Lie group with positive definite Ricci tensor RicIJ. This result rules out (supersymmetric and non-supersymmetric) Freund-Rubin solutions of M-theory of the form AdS4×G and compactifications with non-trivial 4-form fluxes of Englert type on an internal group manifold G. A particular class of such backgrounds which, by our arguments are excluded as bulk supergravity compactifications corresponds to the so-called compactifications on twisted tori, for which G has structure constants τKIJ with vanishing trace τJIJ=0. On the other hand our result does not have bearing on warped compactifications of M-theory to four dimensions and/or to compactifications in the presence of localized sources (D-branes, orientifold planes and so forth). Henceforth our result singles out the latter compactifications as the preferred hunting grounds that need to be more systematically explored in relation with all compactification features involving twisted tori.