-theory on Spin(7) manifolds, fluxes and 3D, =1 supergravity

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: 70%

Internal circle uplifts, transversality and stratified G-structures

2015

“…E Some topological properties of singular foliations defined by a Morse one-form 55 E.1 Some topological invariants of M 55 E.2 Estimate for the number of splitting saddle points 55 E. 3 Estimates for c and N min 56 E. 4 Criteria for existence and number of homologically independent compact leaves 56 E.5 Generic forms 56 E.6 Exact forms 57 E. 7 Behavior under exact perturbations 57 1 Introduction N = 1 flux compactifications of eleven-dimensional supergravity on eight-manifolds M down to AdS 3 spaces [1,2] provide a vast extension of the better studied class of compactifications down to 3-dimensional Minkowski space [3][4][5], having the advantage that they are already consistent at the classical level [2]. 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].…”

Section: Jhep03(2015)116mentioning

confidence: 99%

“…• Cases 1 and 2 correspond to the classical limit of the compactifications studied in [3][4][5]. Case 3 was studied in [2,8].…”

Section: Remarksmentioning

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.

“…The Ansatz for the supersymmetry generator is: 20) where ξ i ∈ Γ(M, S) are Majorana spinors of spin 1/2 on the internal space (M, g) and ζ i are Majorana spinors on (N, g 3 ) which satisfy the Killing equation with positive Killing constant. 3 Assuming that ζ i are Killing spinor on the AdS 3 space (N, g 3 ), the supersymmetry condition is satisfied if ξ i satisfies (1.1), where:…”

Section: The Case Of Compactifications To Adsmentioning

confidence: 99%

“…However, they can support small fluxes at the quantum level, which are suppressed by inverse powers of the size of the compactification manifold. Since such fluxes are difficult to control beyond leading order [20,21], a natural idea is to consider instead compactifications down to AdS 3 spaces. As pointed out in [19], compactifications of M-theory down to AdS 3 do support classical fluxes, which are therefore not suppressed.…”

Section: Introductionmentioning

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 .