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

Abstract

In this paper we prove the theorem that there exists no 7-dimensional Lie group manifold G of weak G 2 holonomy. We actually prove a stronger statement, namely that there exists no 7-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 × G and compactifications with nontrivial 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 τ K IJ with vanishing trace τ J IJ = 0. On the other hand our result does not have bearing on warped compactifications of Mtheory 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.

Gauged Supergravities

“…In [61] it was observed that if the embedding tensor is restricted to the representation (20, 2) in (4.119), upon an N = 4 truncation, the resulting theory coincides with the N = 4 gauged supergravity mentioned above, describing Type IIB superstring compactified on a T 6 /Z 2orientifold in the presence of RR and NS-NS 3-form fluxes F (3) , H (3) . One can apply a similar analysis to the study of M-theory flux-compactifications [88][89][90][91], see also [95,96]. In the toroidal compactification of eleven-dimensional supergravity to fourdimensions the relevant group with respect to which to branch the E 7(7) -representations is…”

“…Here we briefly recall the definition of a quaternionic Kähler manifold 96 M QK [163,172,271,273] and fix the notations. A quaternionic-Kähler manifold M QK of real dimension 4n H is defined as a Riemannian manifold whose holonomy group is contained inside SU(2) × USp(2n H ).…”

Supersymmetric M2-branes with Englert fluxes, and the simple group PSL(2, 7)

“…As one sees these conditions just coincide with what we named Englert equation: it suffices to set Y [3] = and identify μ 4 = −12e. In [9], together with M. Trigiante I showed that the existence of a form satisfying eqs. (1.7) is equivalent to the definition, introduced in [10], of manifolds M 7 of weak G 2 -holonomy.…”

Embedding tensor of Scherk-Schwarz flux compactifications from eleven dimensions

“…The background equation (33) implies that the four dimensional spacetime is Minkowski and that the group under consideration is "flat" [22], i.e.…”

“…In this paper, we study Scherk-Schwarz [22] 2 reductions of D = 11 supergravity with background flux [25,26,27,28,29,30,31,32,33,34,35] within the context of the formalism developed in [1]. The Scherk-Schwarz flux compactification has principally been studied from a four-dimensional gauge algebra perspective by associating background fields to particular representations in the GL (7) decomposition of the 912 representation of E 7 (7) in which the embedding tensor lives.…”