Foliated eight-manifolds for M-theory compactification

Abstract

We characterize compact eight-manifolds M which arise as internal spaces in N=1 flux compactifications of M-theory down to AdS3 using the theory of foliations, for the case when the internal part of the supersymmetry generator is everywhere non-chiral. We prove that specifying such a supersymmetric background is equivalent with giving a codimension one foliation of M which carries a leafwise G2 structure, such that the O'Neill-Gray tensors, non-adapted part of the normal connection and torsion classes of the G2 structure are given in terms of the supergravity four-form field strength by explicit formulas which we derive. We discuss the topology of such foliations, showing that the C star algebra of the foliation is a noncommutative torus of dimension given by the irrationality rank of a certain cohomology class constructed from the four-form field strength, which must satisfy the Latour obstruction. We also give a criterion in terms of this class for when such foliations are fibrations over the circle. When the criterion is not satisfied, each leaf of the foliation is dense in M.Comment: 63 pages, 2 figure

Singular foliations for M-theory compactification

“…The Morse case is generic in the sense that such 1-forms constitute an open and dense subset of the set of all closed one-forms belonging to the cohomology class f. In the Morse case, the singular foliationF can be described using the foliation graph [16][17][18] associated to the corresponding decomposition of M (see [18][19][20] and [21]- [29]), which provides a combinatorial way to encode some important aspects of the foliation's topology -up to neglecting the information contained in the so-called minimal components of the decomposition, components which should possess an as yet unexplored non-commutative geometric description. This provides a far-reaching extension of the picture found in [8] for the everywhere non-chiral case U = M , a case which corresponds to the situation when the foliation graph is reduced to either a circle (when F has compact leaves, being a fibration over S 1 ) or to a single so-called exceptional vertex (when F has non-compact dense leaves, being a minimal foliation). In the minimal case of the backgrounds considered [8], the exceptional vertex corresponds to a noncommutative torus which encodes the noncommutative geometry [30,31] of the leaf space.…”

“…In this paper, we extend the results of [8] to the general case when the internal part ξ of the supersymmetry generator is allowed to become chiral on some locus W ⊂ M . Assuming that W = M , i.e.…”

“…They form a useful testing ground for various proposals aimed at providing unified descriptions of flux backgrounds [6] and may be relevant to recent attempts to gain a better understanding of F-theory [7]. When the internal part ξ of the supersymmetry generator is everywhere non-chiral, such backgrounds can be studied [8] using foliations endowed with longitudinal G 2 structures, an approach which permits a geometric description of the supersymmetry conditions while providing powerful tools for studying the topology of such backgrounds.…”

“…The geometric information along the non-chiral locus U is encoded [8] by a regular foliation F which carries a longitudinal G 2 structure and whose geometry is determined by the supersymmetry conditions in terms of the supergravity four-form field strength. When ∅ = W M , we show that F extends to a singular foliationF of the whole manifold M by adding leaves which are allowed to have singularities at points belonging to W. This singular foliation "integrates" a cosmooth 1 [11][12][13][14] singular distribution D (a.k.a.…”

“…The singular foliationF carries a longitudinal G 2 structure, which is allowed to degenerate at the singular points of singular leaves. On the non-chiral locus U , the problem can be studied using the approach of [8] or the approach advocated in [1], which makes use of two Spin(7) ± structures. We show explicitly how one can translate between these two approaches and prove that the results of [8] agree with those of [1] along this locus.…”