In this paper we prove the theorem that there exists no 7--dimensional Liegroup manifold G of weak G2 holonomy. We actually prove a stronger statement, namely that there exists no7--dimensional Lie group with negative definite Ricci tensor Ric_{IJ}. This result rules out (supersymmetric and non--supersymmetric) Freund--Rubin solutions of M--theory of the form AdS_4\times G andcompactifications with non--trivial 4--form fluxes of Englert type on aninternal group manifold G. A particular class of such backgrounds which, by ourarguments are excluded as bulk supergravity compactifications corresponds tothe so called compactifications on twisted--tori, for which G has structureconstants $\tau^K{}_{IJ}$ with vanishing trace $\tau^J{}_{IJ}=0$. On the otherhand our result does not have bearing on warped compactifications of M--theoryto four dimensions and/or to compactifications in the presence of localizedsources (D--branes, orientifold planes and so forth). Henceforth our resultsingles out the latter compactifications as the preferred hunting grounds thatneed to be more systematically explored in relation with all compactificationfeatures involving twisted tori.
展开▼
机译：在本文中，我们证明了一个定理，即不存在弱G2完整性的7维李群流形G。实际上，我们证明了一个更强有力的说法，即不存在带有负确定Ricci张量Ric_ {IJ}的7维李群。该结果排除了AdS_4 \ times G形式M理论的（超对称和非超对称）Freund-Rubin解以及内部群流形G上具有Englert型非平凡4型通量的紧致。我们的论据排除了这类特殊的背景，因为我们将其排除在外，因为整体超重力压实对应于扭曲的托里上的压实，为此G具有结构常数$ \ tau ^ K {} _ {IJ} $且踪迹$ \ tau ^消失J {} _ {IJ} = 0 $。另一方面，我们的结果与M理论的四个方向的压缩致密化和/或在存在局部源（D-脑，定向平面等）的情况下不致影响致密化。从今以后，我们的结果将后者的压实作为首选的狩猎场，需要对所有涉及扭曲花托的压实特征进行更系统的探索。
展开▼
