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

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathcal{R}} $$\end{document}. We also show that the stabilizer stratification coincides with the rank stratification of a cosmooth generalized distribution D0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal{D}}_0 $$\end{document} and describe it explicitly using the Whitney stratification of a four-dimensional semi-algebraic set P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{P} $$\end{document}. The stabilizer groups along the strata are isomorphic with SU(2), SU(3), G2 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{D} $$\end{document} which turns out to be integrable in the case of compactifications down to AdS3.


Introduction
General compactifications of M-theory on eight-manifolds provide a rich class of geometries which are of physical interest due to their relation to F-theory [1][2][3]. They can serve to test ideas such as exceptional generalized geometry [4][5][6][7][8][9][10], since eight is the first dimension for which the problem of "dual gravitons" [11][12][13][14][15][16] appears. Given these aspects, it is rather surprising that current understanding of such backgrounds is quite limited. The notable exception is the class of compactifications down to 3-dimensional Minkowski space, which were studied intensively following the seminal work of [17] (for the N = 1 case) and [18] (for the N = 2 case). Such backgrounds are obtained by constraining the internal part of the supersymmetry generators to be Majorana-Weyl rather than merely Majorana. As expected from no-go theorems (first used within this setting in [19]), such Minkowski compactifications cannot support a flux at the classical level. 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. This happens because the internal parts of the supersymmetry generators are no longer required to be Majorana-Weyl. This seemingly innocuous extension leads to a surprisingly intricate geometry, as already apparent in the case of N = 1 unbroken supersymmetry [19,22], which can be described using the theory of JHEP11(2015)007 singular foliations [23,24]. By comparison, little is known 1 about N = 2 compactifications down to AdS 3 . In this paper, we consider certain aspects of the geometry of N = 2 eightdimensional backgrounds by working directly in eight dimensions. Namely, we solve the question of classifying the stratified reductions of structure group which arise on the internal eight-manifold M , showing that the full picture is considerably richer than has been previously presumed. Pointwise positions of internal supersymmetry generators as well as their stabilizer groups are described by stratifications of the internal space M : the first by the chirality stratification and the second by the stabilizer stratification. Unlike the case N = 1, the two stratifications need not agree. We find that these stratifications can be described explicitly using the preimages through certain maps b : M → R 3 and B : M → R 4 of the connected refinements of the canonical Whitney stratifications [28,29] of semi-algebraic [30][31][32] subsets R ⊂ R 3 and P ⊂ R 4 , where R is obtained from P by projection on the threedimensional space corresponding to the first three coordinates of R 4 . The maps b and B are constructed from bilinears in the internal supersymmetry generators, while the semialgebraic set P can be described explicitly using algebraic constraints implied by the Fierz identities. This gives a geometric picture of such backgrounds which shows how they can be approached using the theory of stratified manifolds. We classify the stabilizer groups for each stratum, thus giving a complete description of the "stratified G-structure" which arises in such backgrounds. In particular, we find that a generic eight-manifold M of this type contains an open stratum on which the structure group reduces to SU (2). In a certain sense, this stratum is the "largest", but it was not considered previously. We also classify the amount of supersymmetry preserved by an M2-brane transverse to M along each stratum.
Since the classification results mentioned above are independent of the precise form of the supersymmetry equations, they hold more generally than the case of compactifications down to AdS 3 . To highlight this, we develop the formalism required to describe the "topological part" of the conditions for supersymmetry, characterizing those finite-dimensional spaces of globally-defined Majorana spinors which can be spanned by solutions of constrained generalized Killing equations (so-called "virtual CGK spaces"). We show that such spaces must obey a local non-degeneracy condition which puts them in bijection with trivial sub-bundles of the bundle of Majorana spinors, endowed with a trivial flat connection. This formulation clarifies some aspects of the mathematical description of so-called "off-shell supersymmetric" backgrounds.
The paper is organized as follows. Section 1 gives the general description of virtual CGK spaces K and of the chirality and stabilizer stratifications which they induce on M and shows how this framework arises in the case of compactifications down to AdS 3 . We also treat the case N = 1 as a warm-up, pointing out its differences with the case N = 2. The rest of the paper is devoted to the detailed study of the latter case. Section 2 discusses the scalar and one-form bilinears which can be constructed using a basis of K when dim K = 2 and introduces two cosmooth generalized distributions D and D 0 (where D 0 ⊂ D) which 1 Such N = 2 backgrounds were considered in [25] using a nine-dimensional formalism and were also discussed in [26] with similar methods, but without carefully studying the corresponding geometry of the eight-manifold. Certain N = 1 compactifications down to three-dimensional Minkowski space but with torsion-full SU(4) structure were studied in [27, section 3].

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are naturally associated to the one-form bilinears. The rank stratification of D 0 turns out to coincide with the stabilizer stratification, thus providing a way to identify the latter. In the case of compactifications to AdS 3 , the distribution D 0 need not be integrable, but one can show that the larger distribution D integrates to a singular foliation in the sense of Haefliger (topologically, this is a Haefliger structure [33] which may be non-regular). Section 3 discusses the chirality stratification, giving its explicit description in terms of a convex three-dimensional semi-algebraic body R and a smooth map b : M → R 3 whose image is contained in R. Section 4 discusses the algebraic constraints on zero-and one-form spinor bilinears which are induced by Fierz identities, showing how they can be described using a four-dimensional semi-algebraic set P. In the same section, we discuss the geometry of P and of its boundary, its canonical Whitney stratification and the preimage of ∂R inside ∂P through the map which projects on the first three coordinates. Section 5 shows that the rank stratifications of D and D 0 (where the latter coincides with the stabilizer stratification) are different coarsenings of the B-preimage of the connected refinement of the canonical Whitney stratification of P, where B is a map from M to R 4 with image contained in P. The two coarsenings are given explicitly, leading to the classification of stabilizer groups. In the same section, we show how the chirality stratification fits into this picture, while in section 6 we conclude. The appendices contain various proofs as well as other technical details. The main results of this paper are Theorems 1, 2, 3 and 4, which can be found in subsections 3.6, 5.1 and 5.2. For ease of reference, various results are summarized in tables and figures. The notations and conventions used in the paper are explained in appendix A.

Virtual CGK spaces
The eight-manifold M can be used in various ways to construct a supersymmetric background M of eleven-dimensional supergravity [34], for example by taking M to be foliated in eight-manifolds with typical leaf M or by taking it to be a (warped) product between M and some non-compact 3-manifold N endowed with a metric of Minkowski signature. In such backgrounds, supersymmetry generators can be constructed starting from globallydefined solutions ξ ∈ Γ(M, S) of equations of the type:

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Definition. A virtual CGK space K is called B-compatible if there exists a B-compatible connection D on S and a global endomorphism Q ∈ Γ(M, End(S)) such that K = K(D, Q).
First remarks. The physics literature of flux compactifications sometimes makes a distinction between: 2 (a) The topological condition for supersymmetry, namely that the given background admits a number s of independent and globally-defined spinors ξ 1 , . . . , ξ s of the desired type; (b) The algebro-differential conditions for supersymmetry, namely that the spinors at (a) satisfy an equation of the form (1.1).
Recall that Γ(M, S) has a canonical structure of module over C ∞ (M, R). Since the latter is an R-algebra, this also endows Γ(M, S) with a structure of (infinite-dimensional) vector space over R.
Definition. The globally-defined spinors ξ 1 , . . . , ξ s are called weakly linearly independent if they are linearly independent over the field R of real numbers, i.e. linearly independent as elements of the infinite-dimensional R-vector space Γ(M, S) of smooth globally-defined sections of S. They are called strongly linearly independent if they are linearly independent over C ∞ (M, R), i.e. linearly independent as elements of the C ∞ (M, R)-module Γ(M, S). Weak linear independence of ξ 1 , . . . , ξ s means that the relation: where c 1 , . . . , c s are real constants, implies c 1 = . . . = c s = 0. Strong linear independence means that the relation: where c 1 , . . . , c s ∈ C ∞ (M, R) are smooth real-valued functions defined on M , implies c 1 (p) = . . . = c s (p) = 0 for all p ∈ M . Since constant real-valued functions are smooth, it is clear that strong linear independence implies weak linear independence. It is also clear that strong linear independence amounts to the condition that ξ 1 (p), . . . , ξ s (p) are linearly independent inside the vector space S p for all p ∈ M . As we show below, condition (b) implies that the independence condition at (a) should be understood as strong linear independence. The supersymmetry equations (1.1) do not specify precise choices of globally-defined spinors but only a subspace K of Γ(M, S), namely the space K(D, Q) of all globally-defined solutions of (1.1). Hence we need a formulation of the strong linear independence condition which does not rely on choosing a basis for K. Since this is a pointwise condition, it JHEP11(2015)007 can be formulated in a frame-free manner using the evaluation map. This leads to the notion of locally non-degenerate subspaces of Γ(M, S). As we show below, a subspace K of Γ(M, S) is a virtual CGK space iff it obeys this non-degeneracy condition. When K is Bcompatible, the freedom to change an orthonormal basis of K is related to the R-symmetry of supersymmetric effective actions built using such backgrounds.
Remark. The fact that some subspace K ⊂ Γ(M, S) is a virtual CGK space does not mean that K consists of internal parts of supersymmetry generators for any specific background of eleven-dimensional supergravity built on M . To know whether this is the case, one has to study which pairs (D, Q) can arise in a given class of backgrounds. The notion of virtual CGK space encodes the "topological part" of the supersymmetry conditions, which is much weaker than the full supersymmetry conditions in a given background or class of backgrounds.

Locally non-degenerate subspaces of Γ(M, S)
For any p ∈ M , let ev p : Γ(M, S) → S p be the evaluation map at p: Notice that ev p is R-linear and surjective. Any subspace K ⊂ Γ(M, S) induces a generalized linear sub-bundle ev * (K) def.
= p∈M ev p (K) of S, which is smooth in the sense of [37].
The local non-degeneracy condition means that any element ξ ∈ K is either the zero section of S or a section of S which does not vanish anywhere on M . A locally nondegenerate subspace K of Γ(M, S) has finite dimension s def.
= dim K ≤ rkS = 16. In this case, it is easy to see that ev * (K) is an ordinary sub-bundle of S which is topologically trivial, because any basis ξ 1 , . . . , ξ s of K obviously forms a frame of K. Let Grn s (M, S) denote the set of locally non-degenerate s-dimensional subspaces of Γ(M, S); notice that Grn s (M, S) can be viewed as an infinite-dimensional manifold. Let Trivf s (M, S) denote the set of pairs (K, D), where K is a trivial (in the sense of globally trivializable) smooth rank s sub-bundle of S and D is a trivial flat connection on K.
Remark. Given a trivial rank s sub-bundle K of S and a point p ∈ M , trivial flat connections on K can be identified (using parallel transport) with bundle isomorphisms ϕ p : K ∼ → M × S p , so Trivf s (M, S) can be identified with the set of all pairs (K, ϕ p ). Notice that this identification depends on the choice of p ∈ M and hence it is natural only if we work with pointed manifolds (M, p). A natural description which does not require the choice of a base point is given below.
Proposition. There exists a natural bijection Φ s : Grn s (M, S) ∼ −→ Trivf s (M, S), whose inverse is given by Φ −1 s (K, D) = Γ flat (K, D), where: is the space of all D-flat sections of K.

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Proof. Let Π 1 (M ) be the first homotopy groupoid of M and A(K) be the isomorphism groupoid of K (the groupoid whose objects are the points of M and whose Hom-set from p to q is the set of linear isomorphisms from K p to K q ). The map which assigns the pair (p, q) to curves starting at p and ending at q induces a functor E : Π 1 (M ) → M ×M , where M × M is the trivial groupoid whose objects are the points of M . Given K ∈ Grn s (M, S), the rank s bundle K def.
= ev * (K) is trivial, as pointed out above. The corestriction: of ev p | K to its image is bijective for all p ∈ M . Given p, q ∈ M , consider the bijection: This satisfies: U qr • U pq = U pr and U pp = id Kp , ∀p, q, r ∈ M and hence defines a functor U : M × M → A(K) whose image is a trivial sub-groupoid of A(K) (the Hom-sets of the image being singleton sets). There exists a unique flat connection D on K whose holonomy functor Hol D (the functor which associates to every morphism of the groupoid Π 1 (M ) the parallel transport of D along curves belonging to that homotopy class) coincides with the composition U • E : Π 1 (M ) → A(K). This flat connection is trivial since the image of Hol D = U • E (which coincides with the image of U) is a trivial groupoid. This construction gives a natural map Φ s : Grn s (M, S) → Trivf s (M, S) given by Φ s (K) = (K, D). Relation (1.3) implies that any ξ ∈ K satisfies: which implies Dξ = 0. Hence K is contained in the space Γ flat (K, D). Since dim Γ flat (K, D) = rkK = s = dim K, we must have K = Γ flat (K, D). This shows that K is uniquely determined by (K, D) and hence that Φ s is injective. Consider now a pair (K, D) ∈ Trivf s (M, S) and set K def.
= Γ flat (K, D). We have dim K = rkK = s. The map ev p | K is injective with image equal to K p . Thus K is locally non-degenerate and K = ev * (K). Since D is a trivial flat connection, its parallel transport along curves from p to q depends only on p and q, being given by (1.3). Thus (K, D) = Φ s (K), which shows that Φ s is surjective. (1.5) Any B-compatible locally nondegenerate subspace K is endowed with a Euclidean met- is the constant function equal to one on M . For simplicity, we will not distinguish notationally between B 0 and the C ∞ (M, R)-valued bilinear form B| K⊗K = B 0 1 M . Condition (1.5)

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is an invariant way of saying that K admits a basis ξ 1 , . . . , ξ s having the property that the scalar products B p (ξ i (p), ξ j (p)) are independent of the point p ∈ M for all i, j = 1 . . . s. Using the Gram-Schmidt algorithm for B 0 , it is easy to see that this amounts to the condition that K admits a basis which is everywhere orthonormal in the sense B p (ξ i (p), ξ j (p)) = δ ij for all i, j = Φ s (K). Condition (1.5) is equivalent with: where the map e p was defined in (1.2). Since e p : K → K p is bijective for all p, the relation e q = U pq • e p (which follows from (1.3)) shows that (1.6) is equivalent with the condition: which amounts to the requirement that U pq be an isometry from (K p , B p | Kp ) to (K q , B q | Kq ) for all p, q ∈ M . In turn, this is equivalent with the requirement that the trivial flat connection D be B-compatible.
Let K ∈ Grn s (M, S) and (K, D) def.
= Φ s (K). The following statement is obvious in view of the above: Proposition. Let ξ 1 , . . . , ξ s ∈ K and Ξ def.
2. When K is B-compatible, Ξ is an orthonormal basis of K iff it is an everywhereorthonormal D-flat global frame of K.

Relation to virtual CGK spaces
Let K(D, Q) denote the space of solutions to (1.1) and s def.
Proposition. K(D, Q) is a locally non-degenerate subspace of Γ(M, S).
Proof. For ease of notation, let K def.
= K(D, Q). Let P pq (M ) denote the set of curves in M starting at p and ending at q. For any γ ∈ P pq (M ), let: denote the parallel transport of D along γ. Since the connection D need not be flat, the isomorphisms U pq (γ) may depend on γ and not only on its homotopy class. For any ξ ∈ K, the first equation in (1.1) implies: (1.8)
Proposition. The bundle K is D-invariant, thus: Furthermore, the restriction of D to K is a trivial flat connection on K which coincides with D: Proof. Defining e p as in (1.2), relation (1.8) implies: showing that U pq (γ)(K p ) = K q for all p, q ∈ M and γ ∈ P pq (M ). This means that D preserves the bundle K, i.e. relation (1.9) holds. Corestricting U pq to its codomain, (1.11) gives the parallel transport of the connection D 0 induced by D on the sub-bundle K:

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Proposition. Let K be an s-dimensional subspace of Γ(M, S). Then the following statements are equivalent: (a) K is a virtual CGK space.
(b) K is locally non-degenerate.
Proof. The implication (a) ⇒ (b) was proved before. To prove the inverse implication, let K ∈ Grn s (M, S) and (K, D) def.
= Φ s (K). Choosing a complement K of K inside S gives a direct sum decomposition: We have K = Γ flat (K, D) ⊂ Γ(M, K) and hence Dξ = 0 for all ξ ∈ K. Let Q ∈ Γ(M, End(S)) denote the projector of S onto K parallel to K. Then K = ker Q and hence Qξ = 0 for any ξ ∈ K. Let D be any connection on K . Then the direct sum D def.
= D ⊕ D is a connection on S which satisfies Dξ = 0 for all ξ ∈ K. It follows that we have K ⊂ K(D, Q). To show the inverse inclusion, let ξ ∈ K(D, Q). Then Qξ = 0 and hence ξ ∈ Γ(M, K). The equation Dξ = 0 is thus equivalent with Dξ = 0. It follows that we have ξ ∈ Γ flat (K, D) = K and hence K(D, Q) ⊂ K.
Proposition. Let K be an s-dimensional subspace of Γ(M, S). Then the following statements are equivalent: = Φ s (K). Let K ⊥ be the B-orthocomplement of K inside S and Q the B-orthoprojector on K ⊥ . Let D be any B-compatible connection on K ⊥ and let D = D ⊕ D . The same argument as in the proof of the previous proposition shows that we have K = K(D, Q). Since K is B-compatible, the connection D is B-compatible and hence D is B-compatible as well.

The chirality stratification
Let: be the B-orthogonal projectors of S onto S ± and let (K, D) = Φ s (K) for some locally non-degenerate subspace K ⊂ Γ(M, S).
Definition. The chiral projections of K are the smooth generalized sub-bundles of S ± defined through: The chiral rank functions r ± of K are the rank functions of K ± :

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Notice that r ± are lower semicontinuous and that they satisfy: where the last inequality follows from the fact that K is a sub-bundle of the generalized bundle K + ⊕ K − .
Definition. The chiral slices of K are the following cosmooth generalized sub-bundles of K: The identity S ± = ker P ∓ implies K ± = ker(P ∓ | K ), hence we have exact sequences of generalized sub-bundles of S: which give the relations: (1.14) Definition. We say that p ∈ M is a K-special point if (r − (p), r + (p)) = (s, s). The K-special locus is the following subset of M : The open complement: will be called the non-special locus of K; its elements are the non-special points. The special locus admits a stratification induced by the chiral rank functions: Definition. The chirality stratification of M induced by K is the decomposition:

The stabilizer stratification
For any p ∈ M , consider the natural representation of the group Spin(T p M, g p ) Spin(8) on S p .
Definition. The stabilizer group of K at p is the closed subgroup of Spin(T p M, g p ) consisting of those elements which act trivially on the subspace K p ⊂ S p : = {h ∈ Spin(T p M, g p )| hu = u ∀u ∈ K p } . (1.16)

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Definition. Let K be an s-dimensional locally-nondegenerate subspace of Γ(M, S). The stabilizer stratification of M induced by K is the stratification of M given by the isomorphism type of H p . Two points p, q ∈ M belong to the same stratum of this stratification iff H p and H q are isomorphic.
Remark. Given a frame (ξ 1 , . . . , ξ s ) of K, the group H p coincides with the common stabilizer of ξ i (p): When K is B-compatible, we can formulate this as follows. Let V  = q p (H p ) is a subgroup of SO(T p M, g p ). The q p -preimage of the unit element id TpM of SO(T p M, g p ) is a two-point set which consists of the unit element of Spin(T p M, g p ) and another element which we denote by p . The latter acts on S p as minus the identity and hence it cannot be contained in H p . It follows that the restriction of q p to H p is injective and hence it gives an isomorphism from H p to G p . Thus the stabilizer stratification coincides with the stratification of M by the isomorphism type of G p H p . Let T be a stratum of the connected refinement of this stratification and let G T denote the isomorphism type of G p H p for p ∈ T . Endow T with the topology induced from M . The restriction Fr + (M )| T of the oriented frame bundle Fr + (M ) of M is a principal SO (8) bundle (in the sense of general topology) defined over the connected topological space T . Picking specific G p -orbits inside the fibers Fr p (M ) for p ∈ T specifies a G T -reduction of structure group of Fr(M )| T and such reductions for all connected strata T fit together into a "stratified G-structure" defined on M .
Remark. In the Physics literature, what we call a stratified G-structure is sometimes called a "local G-structure". In Mathematics, the word "local" refers to a structure or property which is defined/which holds for all points of some open subset of a topological JHEP11(2015)007 space. Since most strata of the stabilizer stratification are not open subsets of M , it is clear that a stratified G-structure cannot be a local G-structure in the sense used in Mathematics.

The case of compactifications to AdS 3
As an example, consider compactifications down to an AdS 3 space of cosmological constant Λ = −8κ 2 , where κ is a positive parameter. In this case, the eleven-dimensional background M is diffeomorphic with N × M , where N is an oriented 3-manifold diffeomorphic with R 3 and carrying the AdS 3 metric g 3 . The metric on M is taken to be a warped product: The warp factor ∆ is a smooth real-valued function defined on M while ds 2 3 is the squared length element of the AdS 3 metric g 3 . The Ansatz for the field strength G of elevendimensional supergravity is: where f ∈ Ω 1 (M ), F ∈ Ω 4 (M ) and ν 3 is the volume form of (N, g 3 ). The Ansatz for the supersymmetry generator is: and: Here ∇ S is the connection induced on S by the Levi-Civita connection of (M, g), while ν is the volume form of (M, g). Neither Q nor the connection D preserve the chirality decomposition S = S + ⊕ S − of S when κ = 0: It is not hard to check [38] that D is B-compatible:
1. An equivalent formulation of the Ansatz (1.20) is that the supersymmetry generators of the background span the space K 3 ⊗ K, where K 3 is the two-dimensional space of real Killing spinors on AdS 3 with positive Killing constant. Then ξ i in the Ansatz can be taken to form an orthonormal basis of K, while ζ i are arbitrary elements of K 3 , so that the Ansatz describes the general element of K 3 ⊗ K. Notice that one does not gain anything by decomposing ξ i into their positive and negative chirality parts in the Ansatz since D and Q do not preserve the sub-bundles S ± and hence K need not equal the direct sum of the intersections K ∩ Γ(M, S + ) and K ∩ Γ(M, S − ).
2. The amount N of supersymmetry preserved by the background may be larger than s in the limit Λ = 0, when AdS 3 reduces to the three-dimensional Minkowski space. In that limit, the results of [19,24] imply that all fluxes must vanish, thus F = f = κ = 0 and that d∆ = 0, which imply D = ∇ S and Q = 0, hence both D and Q preserve the sub-bundles S + and S − of S. A discussion of this phenomenon for the case s = 1 (which gives N = 1 for Λ < 0 and N = 2 for Λ = 0) can be found in [24, appendix B.1].

A toy example: the case s = 1
Let us illustrate the discussion above with the case s = 1. Then K is a one-dimensional locally non-degenerate subspace of Γ(M, S) while (K, D) is a trivial flat line sub-bundle of S. Assume that K is B-compatible. Then a B-compatible frame of K is given by a single Majorana spinor ξ ∈ Γ(M, S) which is everywhere of norm one; the same spinor gives a global normalized frame of K. The chiral projections K ± are the generalized sub-bundles of S generated by the positive and negative chirality parts ξ ± def. = P ± ξ of ξ. The chiral rank functions are given by: The chiral slices are: Since ξ(p) is everywhere non-vanishing, we have r + + r − ≥ 1, thus the allowed values are (r − (p), r + (p)) ∈ {(0, 1), (1, 0), (1, 1)}. Hence the chirality stratification takes the form:  Thus U is the non-chiral locus while S 10 and S 01 are the negative and positive chirality loci of [24]. The union of the latter is the chiral locus W = W − W + = S 10 S 01 of loc. cit. In this case, the stabilizer stratification is a coarsening of the chirality stratification, namely we have H p Spin (7) for p ∈ W and H p G 2 for p ∈ U . The stabilizer stratification coincides with the rank stratification of the cosmooth generalized distribution D def.
denote the scalar bilinear constructed from ξ. It was shown in [19,38] that the Fierz identities for ξ imply 1 − b 2 = ||V || 2 and hence b 2 ≤ 1. Thus the image of the map b is contained within the interval [−1, 1]. This interval is a semi-algebraic set given by the single polynomial inequality b 2 ≤ 1 for a variable b ∈ R. Its canonical Whitney stratification has a 0-dimensional stratum given by the two-point set {−1, 1} and a one-dimensional stratum given by the open interval (−1, 1). The connected refinement of the Whitney stratification has two connected 0-dimensional strata given by the one-point sets {+1} and {−1} and a connected 1-dimensional stratum given by the open interval (−1, 1). The Hasse diagram of the incidence poset of this stratification is depicted in figure 1. It was shown in [24] that the rank/stabilizer stratification coincides with the b-preimage of the canonical Whitney stratification of [−1, 1]: On the other hand, the chirality stratification coincides with the b-preimage of the connected refinement of the Whitney stratification: It was also shown in [24] that, for compactifications down to AdS 3 , the supersymmetry conditions (1.1) imply that the singular distribution D integrates to a singular foliation in the sense of Haefliger [33].
Remark. The compactifications studied in [17] correspond to the case M = W + .

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As we shall see in the next sections, the situation is much more complicated when s = 2. In that case (assuming that K is B-compatible): 1. The chirality and stabilizer stratifications do not agree, in the sense that neither of them is a refinement of the other.
2. There exists a cosmooth singular distribution D (determined by the intersection of the kernels of three one-form valued spinor bilinears V 1 , V 2 and V 3 ) which integrates to a Haefliger foliation in the AdS 3 case. The rank stratification of D does not agree with the chirality stratification or with the stabilizer stratification.
3. The stabilizer stratification coincides with the rank stratification of a cosmooth singular sub-distribution D 0 ⊂ D (given by the intersection of D with the kernel of a fourth one-form spinor bilinear W ), but D 0 need not be integrable in the case of compactifications down to AdS 3 . The group G p = q p (H p ) is a subgroup of SO(D 0 (p), g p ) (and hence also a subgroup of SO(D(p), g p )) for any p ∈ M .
4. The chirality stratification coincides with the b-preimage of the connected refinement of the Whitney stratification of a three-dimensional semi-algebraic set R, where b ∈ C ∞ (M, R) is a map constructed using scalar spinor bilinears defined by an orthonormal basis of K. We have imb ⊂ R.
5. The stabilizer stratification and the rank stratification of D are different coarsenings of the B-preimage of the connected refinement of the canonical Whitney stratification of a four-dimensional semi-algebraic set P, where B : M → R 4 is another map constructed using an orthonormal basis of K. We have imB ⊂ P.
2 The generalized distributions D and D 0 in the case s = 2 Throughout this section, K denotes a B-compatible locally non-degenerate subspace of Γ(M, S).

Functions and one-forms defined by a basis of K
An orthonormal basis (ξ 1 , ξ 2 ) of K induces three smooth functions b i ∈ C ∞ (M, R) (i = 1, 2, 3), namely: It will be convenient to work with the combinations: Also consider the one-forms V i , V 3 , W ∈ Ω 1 (M ) (with i = 1, 2) given by: where the relations hold in any local coframe (e a ) defined above an open subset U ⊂ M . It will be convenient to work with the linear combinations:

The distributions D and D 0
The 1-forms V 1 , V 2 , V 3 generate a smooth generalized sub-bundle V (in the sense of [37]) of the cotangent bundle of M , which is also generated by V + , V − , V 3 . Let: denote the polar of V, which is a cosmooth generalized distribution on M , i.e. a cosmooth generalized sub-bundle of T M in the sense of [37]. Its orthogonal complement D ⊥ inside T M is a smooth generalized sub-bundle of T M which is spanned by the three vector fields obtained from V + , V − , V 3 by applying the musical isomorphism. Notice that D contains the cosmooth generalized distribution: Remark. When considering compactifications to AdS 3 , one can show that the supersymmetry conditions imply that D is an integrable distribution (namely, it integrates to a singular foliation in the sense of Haefliger) while D 0 may fail to be integrable. This is one reason for considering the generalized distribution D.

Behavior under changes of orthonormal basis of K
An orthonormal basis (ξ 1 , ξ 2 ) of K having the same orientation as (ξ 1 , ξ 2 ) has the form: (where u ∈ R) and defines the following 0-forms and 1-forms, where i = 1, 2: (2.7) Substituting (2.6) into these expressions, we find that b + , V + and W are invariant while each of the pairs b − , b 3 and V − , V 3 transforms in the fundamental representation of SO(2): (2.8)

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The improper rotation: which permutes ξ 1 and ξ 2 induces permutations b 1 ↔ b 2 and V 1 ↔ V 2 while V 3 , b 3 remain unchanged and W changes sign (to arrive at these conclusions, one uses the relations γ(ν) t = γ(ν), γ t a = γ a and the fact that γ(ν) anticommutes with γ a ). Hence (2.9) induces the transformations: (2.10) It follows that b + and V + depend only on K while W depends on K and on a choice of orientation of K. On the other hand, b − and V − change sign while b 3 and V 3 are invariant under a change of orientation of K. It also follows from the above that D and D 0 depend only on the space K and do not depend on the choice of basis (ξ 1 , ξ 2 ) for K.

The rank stratification of D
The compact manifold M decomposes into a disjoint union according to the rank of D: where the open set: will be called the generic locus while its closed complement: are linearly dependent} will be called the degeneration locus. The latter admits a stratification according to the corank of D(p): whose locally closed strata are given by: Combining everything gives the rank stratification of D: Definition. K is called generic if U = ∅ and non-generic otherwise. Notice that K is non-generic iff rkD(p) ≥ 6 for all p ∈ M , i.e. iff V 1 (p), V 2 (p) and V 3 (p) are linearly dependent for all p ∈ M .

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Remark. For any p ∈ M , let A p ∈ Hom(R 3 , T * p M ) denote the linear map which takes the canonical basis i of R 3 into V i (p):

This defines a smooth section
admits a Whitney stratification (the so-called canonical stratification [39,40]) whose strata are the Stiefel manifolds This induces a stratification of the total space of the bundle Hom(R 3 , T * M ), whose preimage through the section A is the stratification (2.13). The preimage of the stratum defined by rkA = 3 is the set U while the preimages of the strata defined by rkA = k with k = 0, 1, 2 are the sets W k .

The rank stratification of D 0
The generalized distribution D 0 induces a decomposition: where: is an open subset of M while: is closed. The latter stratifies according to the corank of D 0 : with locally closed strata given by: We shall see later 4 that we always have: so in particular rkD 0 (p) can never equal five. We thus obtain the rank stratification of D 0 :

Constraints on the stabilizer stratification
Since the action of Spin(T p M, g p ) on S p commutes with γ p (ν p ), relations (2.3) imply: where Spin(T p M, g p ) acts on T * p M by the dual of the vector representation. The action of Spin(T p M, g p ) on T * p M is obtained from that of SO(T p M, g p ) by pre-composing with the covering morphism q p : Spin(T p M, g p ) → SO(T p M, g p ). Hence (2.17) implies: (2.18) In particular, we have: The chirality stratification for s = 2 Let K be a two-dimensional B-compatible locally-nondegenerate subspace of Γ(M, S) and (K, D) be the associated trivial flat sub-bundle of S. Relations (1.13) imply (see figure 2): figure 2). The special locus decomposes as: where S kl = {p ∈ M |r − (p) = k, r + (p) = l}, while the chirality stratification is given by: where G is the non-special locus.
(b) The body R is the solid of revolution obtained by rotating ∆ around its hypotenuse, which lies on the b + axis; it is the union of two compact right-angled cones whose bases coincide.

The semi-algebraic body R
Consider the compact convex body (see figure 3): which is contained in the three-dimensional compact unit ball. Setting: one finds that R is the solid of revolution obtained by rotating the following isosceles right triangle around its hypothenuse: The compact interval: will be called the axis of R while the compact disk: will be called the median disk of R. The boundary ∂D of the median disk will be called the median circle (see figure 4). Notice that R is a semi-algebraic set, since it can be described by polynomial inequalities: Hence both R and its frontier ∂R (which is again a semi-algebraic set) admit [28] canonical 5 stratifications by semi-algebraic sets. Namely, the frontier:

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where: The set ∂ 1 R coincides with the median circle and hence it is connected. The set ∂ 0 R is disconnected, being a disjoint union of two singleton sets: will be called the left and right tips of R. We have: where S 2 denotes the unit sphere in the space R 3 . The set ∂ 2 R is relatively open in ∂R, being a disjoint union of two connected components: Recall that the canonical Whitney stratification of a semi-algebraic set is the coarsest stratification which satisfies the frontier conditions as well as Whitney's regularity condition (b). The strata of this stratification need not be connected. The general algorithm through which such stratifications can be obtained is due to [28] and is discussed in detail in [41]. JHEP11(2015)007 Figure 5. The connected refinement of the canonical Whitney stratification of ∂R. We use green for the median circle ∂ 1 R = ∂D, purple for ∂ − 2 R, yellow for ∂ + 2 R, blue for ∂ − 0 R and red for ∂ + 0 R. Theorem 1 of subsection 3.6 shows that the b-preimage of ∂ 1 R equals S 11 , while the b-preimages of ∂ + 2 R and ∂ − 2 R equal S 12 and S 21 respectively. The b-preimages of ∂ + 0 R and ∂ − 0 R are the sets S 02 and S 20 . where: will be called the left and right components of ∂ 2 R. The canonical Whitney stratification of ∂R has strata given by ∂ 0 R, ∂ 1 R and ∂ 2 R and corresponds to the decomposition (3.6), while its connected refinement (see appendix C) has strata given by ∂ ± 0 R, ∂ 1 R and ∂ ± 2 R and corresponds to the decomposition: The connected strata appearing in (3.8) are depicted in figure 5, while the values of b + and ρ on those strata are summarized in table 1. Together with IntR, the strata ∂ k R give the canonical Whitney stratification of R, whose connected refinement has strata IntR, ∂ ± 0 R, ∂ 1 R and ∂ ± 2 R. For later reference, let: denote the two closed halves of R lying to the left and right of the median disk. Notice that R ± are three-dimensional compact full cones. We have a disjoint union decomposition:

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We also define: which give the decomposition: Proof. Let us separate ξ i into positive and negative chirality parts: = P ± ξ i and i = 1, 2 .
The condition B(ξ i , ξ j ) = δ ij and the definitions of b 1 , b 2 and b 3 give the equations: which can be solved to give: The Gram matrix Γ of the ordered system (ξ + 1 , ξ + 2 , ξ − 1 , ξ − 2 ) takes the block diagonal form: 6 where: are the Gram matrices of the pairs (ξ ± 1 , ξ ± 2 ). A simple computation gives: The conclusion now follows from (3.13) upon using the fact that Γ ± are semipositive, which by Sylvester's theorem amounts to the conditions (Γ ± ) 11 ≥ 0, (Γ ± ) 22 ≥ 0 and det Γ ± ≥ 0. 6 The Gram matrices considered here are defined at every point p ∈ M , hence they are matrix-valued functions defined on M .

The map b
Consider the determinant line bundle det K = ∧ 2 K. The scalar product B| K induces a norm on det K which we denote by || ||. Since (ξ 1 , ξ 2 ) is an orthonormal frame of (K, B), we have ||ξ 1 ∧ ξ 2 || = 1 and hence ξ 1 ∧ ξ 2 is an orthonormal frame of det K. The generalized bundles K ± ⊂ S ± inherit the Euclidean scalar product B from S and hence ∧ 2 K ± are normed generalized vector bundles of rank at most one. The generalized bundle morphisms Proposition. We have: where || || op denotes the fiberwise operator norm on the generalized bundle Hom(∧ 2 K, ∧ 2 K ± ). In particular, det Γ ± depend only on the subspace K ⊂ Γ(M, S) and are independent of the choice of orthonormal basis for K.
Remark. The proposition allows one to give a different proof of the fact that the functions b + , ρ 2 ∈ C ∞ (M, R) depend only on K. This follows by taking the sum and difference of equations (3.13), which allows one to express ρ and b + in terms of det Γ + and det Γ − .
Proposition. The image of b is a subset of ∆.

Relation to the rank stratifications of D and D 0
Lemma. Let p ∈ S be a K-special point. Then: 1. When p ∈ S 11 S 12 , we can rotate the orthonormal basis of K such that either of the following holds, at our choice: 2. When p ∈ S 11 S 21 , we can rotate the orthonormal basis of K such that either of the following holds, at our choice: 3. When p ∈ S 11 , we can rotate the orthonormal basis of K such that either of the following holds, at our choice: Proof.
where λ 1 and λ 2 are real numbers, one of which may be zero. Under a rotation (2.6) of the basis of K, we have: It is easy to see that we can choose u such that either of the combinations λ 1 cos u 2 + λ 2 sin u 2 or −λ 1 sin u 2 + λ 2 cos u 2 vanishes, at our choice. The statements about the 1-form spinor bilinears follow immediately from the forms of ξ i after such a rotation.
2. The case p ∈ S 11 S 21 proceeds similarly.

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Proposition. Let p ∈ S be a K-special point. Then D 0 (p) = D(p) and we can rotate the basis of K such that either V 3 (p) = W (p) or V 3 (p) = −W (p), at our choice. Moreover: • For p ∈ S 20 S 02 , we have D(p) = T p M , hence rkD(p) = 8 • For p ∈ S 11 , we have rkD(p) = 7 • For p ∈ S 12 S 21 we have rkD(p) = 6.
Proof. Follows from the Lemma and from the remark above upon using the fact that D and D 0 are invariant under rotations of the basis of K. The proposition also follows from Theorem 1 below and from the results of subsection 4.3 and of appendix E.
Remark. It is shown in appendix E that, for p ∈ G, we have rkD(p) ∈ {5, 6, 7} and rkD 0 (p) ∈ {4, 6}, hence D 0 (p) and D(p) may differ; in fact, their ranks cannot be determined only from the value of b(p). Together with the Proposition, this gives: A precise description of the relation between the chirality stratification and the rank stratifications of D and D 0 can be found in section 5.

Relation to the stabilizer group
Proposition. Let p be any point of M . Then the following statements hold: 1. When p ∈ S 02 S 20 we have H p SU (4) 2. When p ∈ S 11 , we have H p G 2 3. When p ∈ S 12 S 21 , we have H p SU (3) 4. When p ∈ G, we have either H p SU(2) or H p SU(3), according to whether dim D 0 (p) = 4 or dim D 0 (p) = 6.

Proof.
1. In this case, ξ 1 and ξ 2 are chiral and of the same chirality at p, so their stabilizer inside Spin(8) equals SU(4).
2. After a rotation as in the Lemma given in the previous subsection, we have two nonvanishing spinors ξ 1 and ξ 2 of opposite chirality at p, whose stabilizer inside Spin (8) is isomorphic with G 2 .
The results of appendix F show that the first case arises iff rkD 0 (p) = 6 while the second case arises iff rkD 0 (p) = 4.
Remark. Appendix F gives an explicit construction of a one-parameter deformation of the pair (ξ 1 , ξ 2 ) which breaks the stabilizer group from SU(3) to SU(2).
Corollary. The stabilizer stratification coincides with the rank stratification of D 0 .
3.6 Characterizing the chirality stratification Theorem 1. The K-special locus is given by: Furthermore, we have: Moreover, we have G = b −1 (IntR) and hence the chirality stratification of M coincides with the b-preimage of the connected refinement of the canonical Whitney stratification of R.
Proof. Relation (3.13) implies: Since det Γ ± (p) ≥ 0, we have ρ(p) ≤ 1 ± b + (p) and hence ρ(p) ≤ 1 − |b + (p)|. Thus det Γ + (p) = 0 can be realized only for b + (p) ≤ 0 and det Γ − (p) = 0 can be realized only for b + (p) ≥ 0 and in both cases we have ρ(p) = 1−|b + (p)| i.e. b(p) ∈ ∂R. The case det Γ + (p) = det Γ − (p) = 0 occurs for b + (p) = 0 and ρ(p) = 1, i.e. on the median circle ∂D. We also have: = dim K ± (p) = 2 − r ∓ (p) of the chiral slices of K p count the number of linearly independent spinors inside the space K p which have chirality ±1. In the case of compactifications down to AdS 3 , σ + (p) can be interpreted [26] as the number of supersymmetries of the background which are preserved by a space-time filling M2-brane placed at p, while σ − (p) counts the number of supersymmetries preserved by a space-time filling M2-antibrane placed at p; these numbers are indicated in the last column of the table.

JHEP11(2015)007 4 Algebraic constraints
The Fierz identities for ξ ± 1 , ξ ± 2 imply that the following relations hold (see appendix B): In particular, the first two rows of (4.1) form the following system for V r , b r : Relations (4.2) constrain the norms ||V r || 2 and the angles θ rs = θ sr between V r and V s (a total of six quantities) in terms of the three quantities b r . Fixing the latter generally fails to completely determine the former.
Remark. For a general choice of V r , one cannot find b r such that (4.2) is satisfied. The conditions on V r under which it is possible to solve for b r are given in appendix D.

Reduction to a semipositivity problem
Let us define: β def.
as well as: ρ def.
and consider the map B : M → R 4 defined through: The second line in (4.2) gives: which shows that β contains the same information as the norm of V + , provided that b + is known. When β is fixed, the constraints (4.2) amount to the condition that the Gram matrix of V + , V − , V 3 be given by: JHEP11(2015)007 The system given by (4.3) and the last two rows of (4.2) has solutions V r iff the symmetric matrix G(b, β) is positive semidefinite; in this case, V r are determined by β and b r up to a common action of the group Γ(M, O(T M, g)). Furthermore, V + , V − and V 3 are linearly independent at p ∈ M iff G(p) def.
Remark. Relation The semipositivity conditions for G(b, β) can be analyzed using Sylvester's criterion, leading to a nonlinear programming problem whose solution is given in appendix D. To state the results concisely, we introduce a compact four-dimensional semi-algebraic body P which can be viewed as a singular segment fibration over R.

The four-dimensional body P
Recall that the image of b is contained in R. The determinant of the Gram matrix (4.7) takes the form: where: (4.10) Thus det G(b, β) vanishes for β = 0 or β = f ± (b), where the functions f ± : R → R (which give the roots of the second order polynomial x 2 − (1 + ρ 2 − b 2 + )x + ρ 2 ) are defined through: The discriminant: is non-negative on ∆ and vanishes only for ρ = 1 − |b + |, i.e. on the left and right sides of ∆. The functions f ± satisfy: where:
(b) Graphs of the functions f + (b + , ρ) (green) and f − (b + , ρ) (red) for (b + , ρ) ∈ ∆ (blue). • the equality • the equality f − (b) = 0 is attained iff b ∈ I; • the equality f Notice that f ± depend only on b + and ρ and hence they can be viewed as functions defined on ∆ (see figures 6 and 7). In fact, they are symmetric under b + → −b + , so they depend only on |b + | and ρ.
Various special values of f ± are summarized in table 3. For every b ∈ R, consider the closed interval: Table 3. Special values for f + and f − . The values allowed for rkG on each region follow from Theorem 2 of subsection 5.1.
This interval degenerates to a single point for b ∈ ∂R, namely J| ∂R = { √ ρ}. Finally, consider the following four-dimensional compact body: which is fibered over R via the projection (b, β) π → b. The fiber over b ∈ R is the segment J(b), which, as mentioned above, degenerates to a point over ∂R.
The frontier of P. Let: be the full compact cone in R 3 with apex at the origin and base given by the disk D 2 × {1} and let: = F \ {(0, 0, 0)}. Notice that C is homeomorphic with the compact 3-dimensional ball, F is homeomorphic with S 2 andḞ is homeomorphic with R 2 (and hence with the interior of the unit disk D 2 ). Consider the function g :Ċ → R given by (see figure 8): The quantity under the square root is non-negative for (b − , b 3 , β) ∈ C and we have 0 ≤ Notice that g vanishes onḞ and is strictly positive in the interior of C.
Consider the following three-dimensional subsets of R 4 , each of which is homeomorphic withĊ: The intersection of the sets C ± is given by: and C ± are disjoint from I (since β = 0 on C ± while β = 0 on I). Notice that IntC + and IntC − are homeomorphic with IntĊ = IntC and hence with the interior of the unit 3-ball while F is homeomorphic with the interior of the two-dimensional disk. Let: be the compact right and left halves of I, which satisfy I + ∩ I − = {0 R 4 }. Figure 9 shows the sections of ∂P with the hyperplane b 3 = 0.
Proposition. The frontier of P is given by: where the components can be identified as: Moreover, I is closed (thus frI = ∅), while: 9 fr(IntC ± ) = F I ± , frF = {0 R 4 } . Remark. Topologically, fr(IntC ± ) = fr(C ± ) is obtained from the compact disk upon picking two opposite points on the boundary circle and identifying the resulting halves of the boundary to a segment corresponding to I ± ; the result is of course homeomorphic to a sphere.
Proof. We have: The frontier of P is the semi-algebraic set obtained by intersecting R with the hypersurface P (b, β) = 0. This equation can be written as: and requires that the right hand side be non-negative, which for b ∈ R is equivalent with the condition β ∈ [ρ, 1] i.e. (b − , b 3 , β) ∈ C. To study the solutions of (4.23), assume that this condition is satisfied and consider the cases: • β = 0. Then (4.23) requires ρ = 0 while b + is undetermined within the interval [−1, 1], which means that (b, β) belongs to the interval I.
The above shows that ∂P has the decomposition (4.20) and that (4.21) holds. The remaining statements follow from (4.21).
The body P is a semi-algebraic set, hence it admits a canonical Whitney stratification by smooth semi-algebraic subsets. To describe this stratification, notice that the set defined in (4.19) decomposes as: where: The frontier ∂P has the following decomposition into borderless manifolds of dimensions k = 0, 1, 2, 3: where the k-dimensional pieces are the following unions of connected components: with: The ten connected components listed above give the connected refinement of the canonical Whitney stratification of ∂P, whose incidence poset is depicted in figure 10. Using relations (4.22) and (4.24), we find: fr(IntI ± ) = ∂ 0 0 P ∂ ± 0 P , fr(∂D) = ∅ , which imply: fr(∂ 3 P) = ∂ 2 P ∂ 1 P ∂ 0 P = F I fr(∂ 2 P) = ∂D ∂ 0 0 P (4.30) fr(∂ 1 P) = ∂ 0 P = ∂I ∂ 0 0 P .
Remark. The canonical Whitney stratification of ∂P has six strata given by ∂ 3 P, ∂ 2 P, ∂D, IntI + IntI − , ∂I = ∂ + 0 P ∂ − 0 P and ∂ 0 0 P. The canonical Whitney stratification of P is obtained from this by adding the stratum IntP and similarly for its connected refinement.
The values of b + , ρ and β on the connected strata of ∂P are summarized in table 4. The following statement follows from the results of appendix D: Proposition. The locus β = 0 on P coincides with the compact segment I, while the locus β = 1 on P coincides with the compact disk D = D × {1}. The locus β = ρ on P coincides withĀ = A ∂D ∂ 0 0 P. In particular, the only locus on R where the value β = 0 can be attained is the interval I while the only locus on R where β = 1 can be attained is the median disk D.
Remark. Since ρ ≤ 1, we have β| ∂R = √ ρ ≥ ρ, with equality iff b ∈ ∂D, which corre- belonging to the unit disk. The section of P with the hyperplane b + = 0 is the compact full cone K = {0} × C contained between these two graphs, whose basis is the disk D (green). This disk coincides with the locus on P where β = 1. The apex of the cone is the midpoint of the interval I.
The section of P with the hyperplane b + = 0.5 is the body of revolution contained between these two graphs. The boundary of this body is the union of a cone with a "cap" (a curved disk). Sections of P with the hyperplanes b + = const. The sections of P with such hyperplanes are depicted in figure 12; they allow one to present P as a fibration over the interval [−1, 1]. In particular, the section with the hyperplane b + = 0 is the compact full 3-dimensional cone K = {0} × C, whose frontier equals F.
5 Description of the rank stratifications of D and D 0

Description of the rank stratification of D
The following result shows that the map B has image contained in P and that the rank stratification of D is a certain coarsening of the B-preimage of the connected refinement of the Whitney stratification of P.
Theorem 2. The image of the map B defined in (4.5) is contained in P: imB ⊂ P Furthermore, the following hold for p ∈ M : In particular, the rank stratification of D is given by: and we have W = B −1 (∂P).
Remark. The map b of (3.10) is related to the map B of (4.5) by: Using relations (4.31) and (4.32), this implies: The behavior of the one-forms V r on the locus W is given by the following result, whose proof can be found in appendix D: with ψ ∈ [0, 2π). Then V r and b r behave as follows: 1. When p ∈ W 2 , we have: (b) When p ∈ B −1 (A), we have:

Description of the rank stratification of D 0 and of the stabilizer stratification
The following result shows that the rank stratification of D 0 (which coincides with the stabilizer stratification) is given by another coarsening of the B-preimage of the connected refinement of the canonical Whitney stratification of P.
Theorem 4. For p ∈ M , we have: Hence the rank stratification of D 0 is given by: and the stabilizer group H p is given by: Proof. Follows immediately from Theorem 1 of section 3 together with the Lemma of appendix E.
The situation is summarized in table 5.
The b-image of the G-structure stratification is depicted in figure 13.

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P-description D-stratum D0-stratum rkD rkD0 Hp Table 5. The ranks of D and D 0 on various loci and the isomorphism type of H p .

Comparing the rank stratifications of D and D 0
Using relations (4.27), Theorem 2 shows that W k decompose as follows: where W 2 2 and W 3 2 decompose further as: so that: Finally, W 0 1 decomposes as: The components listed above give the B-preimage of the connected refinement of the canonical Whitney stratification of ∂P, to which we can add B −1 (IntP) to obtain the V -preimage JHEP11(2015)007 of the connected refinement of the Whitney stratification of P (see table 6). Theorems 2 and 4 give: In view of the last equality, we define Z ± 0 def.
= S + S − . The situation is summarized in table 7, where we remind the reader that the restrictions of D and D 0 to the special locus S coincide (see section 3).

Relation to previous work
Some aspects of N = 2 compactifications of eleven-dimensional supergravity down to AdS 3 were approached in [26] using a nine-dimensional formalism based on the auxiliary 9-manifoldM def.
= M × S 1 , but without carefully exploring the consequences of that formalism for the geometry of M . Sections 3-5 of [26] also discuss some consequences of the JHEP11(2015)007 P-description S-stratum D-stratum D0-stratum rkD rkD0 Hp supersymmetry equations (which were also derived in [25]) using the nine-dimensional formalism. Reference [26] makes intensive use of an assumption (equation (3.9) of loc. cit.) which, as we show in appendix G, can only hold when the SU(2) locus U of M is empty.
Since most results of [26] (including the count of the number of supersymmetries preserved by membranes transverse to M as well as the discussion of sections 3-6 of that reference) rely on that assumption, those results can apply only to the highly non-generic case when U = ∅. As we explain in detail in forthcoming work, failure of [26, eq. (3.9)] is related to the transversal vs. non-transversal character of the intersection of a certain distributionD defined onM with the pullback toM of the tangent bundle of M .

Conclusions
We studied the conditions for "off-shell" extended supersymmetry in compactifications of eleven-dimensional supergravity on eight-manifolds M . We gave an explicit description of the stabilizer stratification induced by two globally-defined Majorana spinors as a certain coarsening of the preimage of the connected refinement of the Whitney stratification of a four-dimensional compact semi-algebraic set P through a map B : M → R 4 whose image is contained in P. We also described the chirality stratification as a coarsening of the preimage of the connected refinement of the Whitney stratification of a 3-dimensional compact semialgebraic set R through a smooth map b : M → R 3 whose image is contained in R. Unlike the case of N = 1 compactifications, the stabilizer and chirality stratifications do not agree. We found a rich landscape of reductions of structure group along the various strata, which we classified explicitly. The open strata of the chirality and stabilizer stratifications coincide and correspond to an open subset U ⊂ M which carries an SU(2) structure. This locus is present in generic N = 2 flux compactifications of eleven-dimensional supergravity on eight manifolds, for example in generic N = 2 compactifications down to AdS 3 spaces. We also discussed two natural cosmooth generalized distributions D and D 0 which exist on M when considering such backgrounds. These are defined by the four one-form spinor bilinears V 1 , V 2 , V 3 and W which are induced by two independent globally-defined Majorana spinors given on M , namely D is the intersection of the kernel distributions of V 1 , V 2 and V 3 while D 0 is the intersection of D with the kernel distribution of W . We showed that the rank stratification of D 0 coincides with the stabilizer stratification, while the rank stratification of D is another coarsening of the B-preimage of the connected refinement of the Whitney JHEP11(2015)007 stratification of P. The restriction of D to the open stratum U is a rank five regular Frobenius distribution which carries an SU(2) structure in the sense of [42], while the restriction of D 0 to U is a rank four Frobenius distribution (the almost contact distribution of [43]). Since the image G p = q p (H p ) ⊂ SO(T p M, g p ) of the pointwise stabilizer group H p of two independent Majorana spinors fixes the forms V 1 (p), V 2 (p), V 3 (p) and W (p), the distribution D 0 | U carries the SU(2) structure of D| U in the sense that G p is contained in the group SO(D 0 (p), g p ) SO(4) for any point p ∈ U . In this paper, we focused on the classification of spinor positions and stabilizer groups, which we treated in detail given its complexity. We mention that considerably more can be said about the chirality and stabilizer stratifications provided that one makes appropriate Thom-Boardman type genericity assumptions which allow one to apply results from the singularity theory of differentiable maps [44][45][46][47][48].
Since the manifolds M considered in this paper are eight-dimensional, it is not entirely clear how a description of such backgrounds may be given within the framework of exceptional generalized geometry [4][5][6][7][8][9][10], similar to the one given in [7][8][9] for 7-dimensional backgrounds of eleven-dimensional supergravity and in [35,36,49,50] for six-dimensional type II backgrounds. This stems from difficulties 10 in building an appropriate generalized connection in eight dimensions, which in turn relates to the presence of Kaluza-Klein monopoles in the U-duality algebra and hence to the problem of including "dual gravitons" at the nonlinear level in E 8(8) -covariant formulations of eleven-dimensional supergravity [11][12][13][14] (which is obstructed by the no-go results of [15,16]). A solution to this problem was recently proposed in [51] within the framework of exceptional field theory but, as pointed out in [52], that solution may be incomplete. It would be interesting to understand what light may be shed on our results by exceptional generalized geometry.
The results of this paper show that the rich landscape of G-structures arising in N = 2 flux compactifications of eleven-dimensional supergravity on eight-manifolds admits a natural description using stratification theory and standard constructions of real semi-algebraic geometry [30][31][32], thus giving clues about the mathematical tools required for general treatments of flux backgrounds. We note that the approach via cosmooth generalized distributions, stratified G-structures and semi-algebraic sets appears to be quite general and thus could be applied to flux backgrounds of any supergravity theory. In general, the complexity of the stratifications involved grows rather fast with the number of spinors (as implied by the results of [41]), but such stratifications can be computed algorithmically. We mention that powerful algorithms exist [32] for the study of semi-algebraic sets.

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09370102/2009. The work of C.I.L was supported by the research grants IBS-R003-G1 and IBS-R003-S1. C.I.L. acknowledges hospitality of the University of Nis and of the SEENET-MTP Office, where part of this work was completed. His visit to Nis, Serbia, was supported by the ICTP -SEENET-MTP network project PRJ-09 titled "Cosmology and Strings". E.M.B. acknowledges the invitation, financial support and hospitality of IPhT-CEA, Paris-Saclay, for her visit at the institute during the preparation of this article.

A Notations and conventions
Throughout this paper, (M, g) denotes a connected and compact smooth Riemannian eightmanifold, which we assume to be oriented and spin. The unital commutative R-algebra of smooth real-valued functions on M is denoted by C ∞ (M, R). The fact that M is orientable and spin means that its first two Stiefel-Whitney classes vanish, i.e. w 1 (M ) = w 2 (M ) = 0. All fiber bundles we consider are smooth. 11 We use freely the results and notations of [25,38,54], with the same conventions as there.
Recall that the set of isomorphism classes of spin structures of M is a torsor for the finite group H 1 (M, Z 2 ). Let (T * M, ) denote the Kähler-Atiyah bundle of (M, g), which is a bundle of unital associative R-algebras. = S + ⊕ S − denote the corresponding bundle of real pinors (a.k.a. Majorana spinors). Then S is a bundle of modules over the Kähler-Atiyah bundle (T * M, ) whose structure morphism is an isomorphism of bundles of algebras γ : (T * M, ) ∼ → (End(S), •) and hence the pair (S, γ) is an element of A. This gives a map which associates an element of A to every spin structure of M . It is easy to see that two spin structures are equivalent iff the corresponding pairs (S, γ) and (S , γ ) are equivalent in the sense described above, hence we have a bijection between H 1 (M, Z 2 ) and the set A/ ∼ . Throughout the paper, we assume that a spin structure has been chosen for M and we work with the corresponding pair (S, γ) ∈ A.
Up to rescalings by smooth nowhere-vanishing real-valued functions defined on M , the bundle S of Majorana spinors has two admissible pairings B ± (see [54][55][56]), both of which are symmetric. These pairings are distinguished by their types B ± = ±1. Throughout the paper, we work with B def.
= B + , which we can take to be a scalar product on S, denoting the induced norm on S by || ||.
Our convention for the Clifford algebra Cl(h) of a bilinear form h is that common in Physics, i.e. the generators satisfy e k e l + e l e k = 2h kl ; the convention common in Mathe-

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matics has a minus on the right hand side. One recovers the Mathematics convention by multiplying all e k with the imaginary unit i; accordingly, the Killing constant of a Killing spinor is multiplied by i. Unlike in some of the literature on flux compactifications, we reserve the name "Killing spinor" for the mathematically consecrated notion, i.e. for a spinor ξ which satisfies ∇ k ξ = λe k ξ, where λ is the Killing constant and the right hand side involves Clifford multiplication; spinors which satisfy generalizations of this equation in which the right hand side contains a polynomial in e i are called generalized Killing spinors, as usual in the Mathematics literature.
The generalized distributions [37,53] D and D 0 considered in this paper are cosmooth in the sense of [37] rather than smooth. 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. a singular foliation in the sense of Haefliger) while a smooth generalized distribution integrates to a singular foliation in the sense of [62,63].
We use the "mostly plus" convention for pseudo-Riemannian metrics of Minkowski signature. Given a subset A of M , we letĀ denote the closure of A in M (taken with respect to the manifold topology of M ). The frontier (also called topological boundary) of A is defined as ∂A =Ā \ A. When considering the canonical Whitney stratification of a semi-algebraic set, we always work with its connected refinement (see appendix B). In some references (such as [41]) it is this connected refinement which is called the canonical Whitney stratification of that semi-algebraic set.

B Algebraic constraints for V r , W and b
Relations (4.1) can be obtained through direct computation using Fierz identities. Here, we give a proof which relies on reducing (4.2) to a Fierz identity satisfied by a single spinor. Consider the Majorana spinor: and the corresponding one-form: where U ⊂ M and x i± are arbitrary real numbers. This satisfies the relation [38]: The relations γ t a = γ a and γ(ν) t = γ(ν) give:

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for all i, j = 1, 2 and all α, β ∈ {−, +}. Using these as well as B(ξ ± i , ξ ∓ j ) = 0, we find: Using (3.11), these relations become: Substituting these expressions into (B.1) gives an algebraic equation which must hold for all x iα , i.e. a certain polynomial in the variables x iα must vanish identically. This means that the coefficients of all monomials in x iα in that polynomial must vanish, giving the relations: Using , we can write (B.2) in the form (4.1). The system (4.1) can also be written as: Using (3.11), we find: This allows us to write the norms of V i , V ± 3 given in (4.1) in the form: Proposition. Assume that ξ ± j does not vanish anywhere on the open subset U ⊂ M which supports the local orthonormal coframe (e a ) a=1...8 of (M, g). Then (γ a ξ ± j ) a=1...8 is an orthogonal frame of S ∓ defined above U which satisfies ||γ a ξ ± j || 2 = ||ξ ± j || 2 and we have:
Remark. Under the assumption of the second part of the proposition, relations (B.5) show that V 1 , V 2 and V ± 3 are nowhere-vanishing on U and that the following rescaled 1-forms have unit norm everywhere on U , where i = 1, 2: Using these normalized 1-forms, relations (B.6) can be written as: NoticeV 1 ,V 2 ,V ± 3 square to one in the Kähler-Atiyah algebra of (U, g) and hence the endomorphisms γ(V i ), γ(V ± 3 ) square to the identity automorphism of the bundle S| U . Also note that the second part of the proposition applies to any open subset of the non-special locus G ⊂ M which supports a local orthonormal coframe of (M, g).
Remark. The sub-system (4.2) can be obtained more directly as follows. An arbitrary norm one element ξ of K has the form: where u ∈ R is constant on M . This induces a function b(u) ∈ C ∞ (M, R) and a one-form V (u) ∈ Ω 1 (M ) given by: Since ||ξ(u)|| = 1, relation (B.1) implies that the following equality must hold for all u (cf. [19,23,24]): Substituting (B.10) into (B.11), we can separate the Fourier components in u, using the fact that {1, cos(u), sin(u)|n ∈ N * } form an orthogonal basis of the Hilbert space L 2 (S 1 ) of (complex-valued) square integrable functions on the circle. This leads to a system of algebraic constraints for b r and V r which is equivalent with (B.11). Expanding in Fourier components, one finds after some computation that (B.11) is equivalent with (4.2).

JHEP11(2015)007 C Stratified spaces
We recall some basic notions from stratification theory in order to fix terminology. In this paper, a finite stratification of a topological space X is understood in the most general sense, i.e. as a finite partition of X into non-empty subsets called strata. We say that the stratification is connected if all strata are connected. We let Σ ⊂ P(X) (where P(X) is the power set of X) denote the set of all strata, thus:

C.1 Incidence poset of a stratification
Consider the partial order relation ≤ defined on Σ through: Then (Σ, ≤) is a finite poset called the incidence poset of the stratification. We let < denote the transitive binary relation defined on Σ through: i.e.: where fr(S) denotes the small frontier of S (see appendix A). For any S ∈ Σ, let C(S) denote the strict lower set of S: For all S ∈ Σ, we have the obvious inclusion: S ∈C(S) S ⊆ fr(S) . (C.1)

C.2 The adjointness relation
We say that a stratum S adjoins a stratum S (and write S S) if the intersection S ∩S is non-empty. This defines a reflexive (but generally non-transitive) binary relation on Σ. We say that S strictly adjoins S (and write S S) if S S and S = S i.e. if S intersects frS. We have:

The frontier condition
We say that the stratification satisfies the frontier condition if the small frontier of each stratum is a union of strata. This amounts to the requirement that equality is always realized in (C.1): fr(S) = S ∈C(S) S , ∀S ∈ Σ and with the condition that equality is realized in (C.2). This happens iff the binary relations < and coincide, in which case ≤ and also coincide i.e. iff S ∩S = ∅ implies S ⊆S.
When the frontier condition is satisfied, the small frontier of any stratum can be determined immediately by looking at the Hasse diagram of the incidence poset of the stratification.

C.4 Refinements and coarsenings
We say that a stratification Σ is a refinement of Σ if any stratum of Σ is a union of strata of Σ . In this case, we also say that Σ is a coarsening of Σ . The connected refinement of Σ is the refinement whose strata are the connected components of the strata of Σ; it is the coarsest connected stratification which is a refinement of Σ. We say that two stratifications Σ and Σ agree if one of them is a refinement of the other.

D The semipositivity conditions for G
Consider the Gram matrix (4.7). We use the notation G [ij|ij] for the 2 by 2 submatrix of G obtained by keeping only the i-th and j-th rows and columns of G, where 1 ≤ i < j ≤ 3. By Sylvester's criterion: • G is positive semidefinite, iff each of its principal (unsigned) minors: is non-negative.
• G is positive definite iff each of its leading principal minors det G, det G [12|12] and G 11 is positive; in this case, the non-leading principal minors are automatically positive.
Remark. When G is positive semidefinite, Kosteljanski's inequality [64] gives: where G[I] denotes the unsigned principal minor defined by keeping only those rows and columns of G indexed by elements of the subset I of the set {1, 2, 3}. For I ∩ J = ∅, this reduces to Fisher's inequality: which gives:

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To study Sylvester's conditions, we start by computing the determinants of the various submatrices of G. Consider the polynomial (4.10), which we reproduce here for convenience: Notice that: Direct computation gives: When viewing P as a quadratic polynomial in β 2 , its discriminant equals the function h(b + , ρ) defined in (4.12).
It follows that for any b ∈ R we can factorize P (b, β) as: where f ± (b) are given in (4.11). This allows us to write:

D.1 Proof of Theorem 2
Theorem 2 is an immediate consequence of Lemmas A, B and C proved below.
Proposition. The following inequality holds for b ∈ R: with equality iff b + ρ = 0.

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Proposition. For b ∈ R, we have: The first and third inequalities in (D.8) are both strict unless b + ρ = 0, in which case both of them become equalities. In particular, we have J(b) ⊂ [ρ, 1 − b 2 + ], where the interval J(b) was defined in (4.13).
Proof. The middle inequality is obvious, while the first and third inequalities are both equivalent with (D.7). The other statements follow immediately.
Proposition. For any b ∈ R, we have: In particular, the condition Proof. Inequalities (D.10) follow immediately from the Lemma. When β 2 ∈ [f − (b + , ρ), f + (b + , ρ)], these inequalities imply that β lies between the two roots of P (b + , b − , 0; β)) and P (b + , 0, b 3 ; β) (viewed as polynomials in β 2 ), which shows that det G Proof. Follows immediately from the previous proposition upon recalling that the body P is a fibration over R with fiber given by the interval J(b) defined in (4.13).
Proof of Theorem 2. The following result follows by combining Lemmas A, B and C: It is strictly positive iff B ∈ IntP. In particular, we have rkG(B) < 3 at a point p ∈ M iff B(p) ∈ ∂P. When B ∈ ∂P, we have: We know from subsection 3.2 that imb ⊂ R. Combining this with Theorem 2', we find that the image of B is contained in P. Theorem 2 now follows immediately.

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E The rank ofĜ Direct computation using (4.8) gives: The determinants of the 3 by 3 principal minors ofĜ are given by: where P (b, β) was defined in (4.10): while the determinants of the 2 by 2 principal minors are: Lemma. The rank ofĜ(B) is given as follows:  The Lemma follows by combining these results.

F.1 A family of special deformations
Consider a locally non-degenerate and B-compatible two-dimensional subspace K ⊂ Γ(M, S) and let (ξ 1 , ξ 2 ) be an orthonormal basis of K. Thus ξ 1 (p) and ξ 2 (p) form an orthonormal system of Majorana spinors for any p ∈ M . Let G denote the non-special locus of K, i.e. the set consisting of those points p ∈ M such that the positive chirality components ξ + 1 (p) and ξ + 2 (p) are linearly independent and such that the same holds for the negative chirality components ξ − 1 (p) and ξ − 2 (p). Consider the special class of deformations of the pair (ξ 1 , ξ 2 ) to another pair of Majorana spinors (ξ 1 ,ξ 2 ) such that only ξ − 1 changes: Recall that ξ ± 1 and ξ ± 2 generate the chiral projections K ± of the spinor sub-bundle K associated to S. Under a special deformation obeying (F.1), the positive chirality projection is invariant while the negative chirality projection may change: As a result, the bundle K changes toK and the space K changes to the spaceK = Rξ 1 + Rξ 2 ⊂ Γ(M, S). We require that the system (ξ 1 ,ξ 2 ) is everywhere orthonormal, so thatK is again a two-dimensional and B-compatible locally-nondegenerate subspace of Γ(M, S).

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andb ± ,Ṽ ± 3 denote the associated quantities defined as in section 2. Letβ ∈ C ∞ (M, R + ) denote the function defined according to (4.3). Notice thatξ − 1 has the form: where α 1 , α 2 ∈ C ∞ (M, R) and ζ ∈ Γ(M, S − ) is the projection ofξ 1 onto the Borthocomplement of K − inside S − . Hence ζ is a section of S − which is everywhere orthogonal to K − and whose norm we shall denote by: Recall that b(p) ∈ IntR for any p ∈ G.
Proposition. The one-forms U 1 and U 2 satisfy the following relations on the locus G: whileβ is given by the following expression on the same locus: (F.14)
Substituting (F.13) intoĜ, we find thatĜ equals the matrixĜ(b,β) of (F.15), whereβ is given by (F.14). It follows that there are no further constrains on U 1 and U 2 and hence that equality ofĜ(α 1 ) and (F.15) is equivalent with relations (F.13) and (F.14) on the locus G 0 . These relation also hold on G \ G 0 since U 1 , U 2 and λ vanish on that locus.
Since U 1 depends continuously on α 1 , relation (F.14) shows that: 14 At this step we used Mathematica R , which we acknowledge here.
Proof. Follows immediately from the results of the previous subsection.
Remark. Together with the results of subsection 3.5, the proposition implies that, for every value B 0 ∈ P and every point p ∈ M , there exists a pair of everywhere-orthonormal Majorana spinors (ξ 1 , ξ 2 ) on M whose function B satisfies B(p) = B 0 . In particular, all points of P can be realized by some two-dimensional and B-compatible locally-nondegenerate subspace K ⊂ Γ(M, S).