Fully Integrable Pfaffian Systems

2015

Abstract:We study stratified G-structures in N = 2 compactifications of M-theory on eight-manifolds M using the uplift to the auxiliary nine-manifoldM = M × S 1 . We show that the cosmooth generalized distributionD onM which arises in this formalism may have pointwise transverse or non-transverse intersection with the pull-back of the tangent bundle of M , a fact which is responsible for the subtle relation between the spinor stabilizers arising on M andM and for the complicated stratified G-structure on M which we uncovered in previous work. We give a direct explanation of the latter in terms of the former and relate explicitly the defining forms of the SU(2) structure which exists on the generic locus U of M to the defining forms of the SU(3) structure which exists on an open subsetÛ ofM , thus providing a dictionary between the eight-and nine-dimensional formalisms.

Section: Jhep11(2015)174mentioning

confidence: 91%

Internal circle uplifts, transversality and stratified G-structures

2015

“…Furthermore, Pfaff's condition is no longer equivalent with Pfaff integrability, unlike the case when D is regular. Conditions for Cartan integrability of cosmooth generalized distributions were given in [73]. Almost all cosmooth generalized distributions arising in practice fail to be globally Cartan integrable.…”

Section: Jhep03(2015)116mentioning

confidence: 99%

“…For such singular distributions, the Stefan-Sussmann integrability theorem states (similarly to the Frobenius theorem) that D is integrable iff it is locally involutive with respect to the Poisson bracket. 19 • The integrability conditions for a non-regular cosmooth distribution (equivalently, for a non-regular smooth codistribution) are much more complicated [73] than those given by Stefan and Sussmann for smooth distributions.…”

Section: Jhep03(2015)116mentioning

confidence: 99%

Singular foliations for M-theory compactification

2015

Abstract:We use the theory of singular foliations to study N = 1 compactifications of eleven-dimensional supergravity on eight-manifolds M down to AdS 3 spaces, allowing for the possibility that the internal part ξ of the supersymmetry generator is chiral on some locus W which does not coincide with M . We show that the complement M \ W must be a dense open subset of M and that M admits a singular foliationF endowed with a longitudinal G 2 structure and defined by a closed one-form ω, whose geometry is determined by the supersymmetry conditions. The singular leaves are those leaves which meet W. When ω is a Morse form, the chiral locus is a finite set of points, consisting of isolated zero-dimensional leaves and of conical singularities of seven-dimensional leaves. In that case, we describe the topology ofF using results from Novikov theory. We also show how this description fits in with previous formulas which were extracted by exploiting the Spin(7) ± structures which exist on the complement of W.

“…As explained in appendix D of [24], their integrability theory (see [57]) is in some sense "orthogonal" to that of smooth generalized distributions [58][59][60][61]. When integrable, a cosmooth generalized distribution integrates to a Haefliger structure (a.k.a.…”

Section: Jhep11(2015)007mentioning

confidence: 99%

The landscape of G-structures in eight-manifold compactifications of M-theory

2015

We consider spaces of "virtual" constrained generalized Killing spinors, i.e. spaces of Majorana spinors which correspond to "off-shell" s-extended supersymmetry in compactifications of eleven-dimensional supergravity based on eight-manifolds M . Such spaces naturally induce two stratifications of M , called the chirality and stabilizer stratification. For the case s = 2, we describe the former using the canonical Whitney stratification of a three-dimensional semi-algebraic set R. We also show that the stabilizer stratification coincides with the rank stratification of a cosmooth generalized distribution D 0 and describe it explicitly using the Whitney stratification of a four-dimensional semi-algebraic set P. The stabilizer groups along the strata are isomorphic with SU(2), SU(3), G 2 or SU(4), where SU (2) corresponds to the open stratum, which is generically non-empty. We also determine the rank stratification of a larger generalized distribution D which turns out to be integrable in the case of compactifications down to AdS 3 .