Internal circle uplifts, transversality and stratified G-structures

We study stratified G-structures in N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2 $$\end{document} compactifications of M-theory on eight-manifolds M using the uplift to the auxiliary nine-manifold M^=M×S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widehat{M}=M\times {S}^1 $$\end{document}. We show that the cosmooth 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}$$ \widehat{\mathcal{D}} $$\end{document} on M^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widehat{M} $$\end{document} 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 and M^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widehat{M} $$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{U} $$\end{document} of M to the defining forms of the SU(3) structure which exists on an open subset U^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widehat{\mathcal{U}} $$\end{document} of M^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \widehat{M} $$\end{document}, thus providing a dictionary between the eight- and nine-dimensional formalisms.


Introduction
General N = 2 flux compactifications of eleven-dimensional supergravity [1] on eightmanifolds M have two independent internal supersymmetry generators ξ 1 , ξ 2 which are global sections of the rank sixteen bundle S of Majorana spinors on M . The class of such compactifications is little explored, with the notable exception of compactifications down to Minkowski 3-space [2], which arise when imposing the Weyl condition on ξ 1 and ξ 2 and which, as a consequence of no-go theorems, can only support a flux at the quantum, JHEP11(2015)174 rather than classical, level. Relaxing this condition leads to backgrounds which can support classical fluxes and which have a surprisingly rich geometry. Some aspects of such backgrounds were discussed in [3] using a formalism which uses the auxiliary nine-manifold M def.
= M × S 1 and the canonical liftsξ 1 ,ξ 2 toM of the internal supersymmetry generators (see also [4]). In that approach, one finds thatM is endowed with a stratified G-structure whose strata are defined by the isomorphism type of the stabilizer group inside (Spin(9)) of the pair of lifted spinors at various points ofM . The strata ofM correspond [3] to stabilizers isomorphic with SU(3), G 2 or SU (4). On the other hand, it was shown in [5] that the stabilizer stratification induced by ξ 1 and ξ 2 on M has SU(2), SU(3), G 2 and SU(4) strata, whose description is considerably more complex. This stratification of M coincides with a certain coarsening of the preimage of the connected refinement of the canonical Whitney stratification [6,7] of a four-dimensional compact semi-algebraic [8,9] body P ⊂ R 4 through a certain map B : M → R 4 whose image is contained in P. As shown in [5], this complicated stratification generalizes what happens in the much simpler case of N = 1 M-theory flux compactifications on eight-manifolds [10][11][12][13] (which extend the classically fluxless case of [14][15][16]), where the relevant semi-algebraic body is the interval [−1, 1], endowed with its Whitney stratification.
The complexity of the picture found in [5] may come as a surprise given the relative simplicity of the stabilizer stratification ofM . The purpose of this note is to explain this difference. Embedding M intoM as a hypersurface j(M ) located at some fixed point of S 1 , we show that the cosmooth [17] generalized distribution [18][19][20][21] D of [5] (which is the polar distribution defined by three 1-forms V 1 , V 2 , V 3 ∈ Ω 1 (M )) coincides with the intersection of T M with the restriction j * (D) ≡D| j(M ) of the polar distributionD which is defined onM by three 1-formsV 1 ,V 2 ,V 3 ∈ Ω 1 (M ). The latter can be expressed as bilinears inξ 1 andξ 2 . The algebraic constraints satisfied by V 1 , V 2 and V 3 as a result of Fierz identities for ξ 1 and ξ 2 are equivalent with the algebraic constraints satisfied byV 1 ,V 2 andV 3 as a result of Fierz identities forξ 1 andξ 2 . The intersection D = j * (D) ∩ T M may be pointwise transverse or non-transverse, giving rise to a disjoint union decomposition M = T N , where T is the transverse locus and N is the non-transverse locus of M . While D and j * (D) coincide when restricted to N , the ranks of their restrictions to T differ by one. The fact that T may be nonempty turns out to be responsible for the difference between the stabilizer stratifications of M andM and explains the increased complexity of the former when compared to the latter. In the special case when the transverse locus is empty (which turns out to be the case considered in [3]), the equality D = j * (D) holds globally on M and the stabilizer stratification of M is obtained directly from that ofM by intersecting every stratum of the latter with j(M ). In the generic case when T = ∅, the relation between the stabilizer stratifications of M andM can be understood using a version of known facts [22][23][24][25][26][27] regarding G-structures induced on orientable hypersurfaces of a G-structured manifold. On the open stratum U ⊂ M which carries an SU(2) structure (the "generic locus" of [5]), this observation allows one to give an explicit formula for the defining forms of the SU(2) structure in terms of the defining forms of the SU(3) structure which exists [3] on an open subsetÛ ofM .

JHEP11(2015)174
The note is organized as follows. Section 1 briefly recalls some results of [5], to which we refer the reader for further information. Section 2 discusses the stabilizer stratification ofM and compares its intersection with j(M ) with the B-preimage of the connected refinement of the canonical Whitney stratification of P. Section 3 takes up the issue of transversality of the pointwise intersection of j * (D) with T M and shows how the transverse or nontransverse character of this intersection explains the increased complexity of the stabilizer stratification of M as compared to that ofM . The same section shows how the stratified G-structure of M can be obtained by reducing that ofM along this intersection. Section 4 expresses the defining form of the SU(2) structure which exists on the generic locus of M in terms of the defining forms of the SU(3) structure which exists on an open subset ofM , while section 5 concludes.
Notations and conventions. We use the same notations and conventions as reference [5], to which we refer the reader for details. An equality which holds for any point of a subset A of a manifold is written as = A .

Brief summary of the eight-dimensional formalism
Let S denote the rank 16 vector bundle of Majorana spinors on M (which is endowed with the admissible [28,29] scalar product B) and ν denote the volume form of (M, g). Let γ : ∧T * M → End(S) be the structure morphism of S. Given two Majorana spinors ξ 1 , ξ 2 ∈ Γ(M, S) which are B-orthonormal everywhere, we define the 0-and 1-forms: = U B(ξ 1 , γ a γ(ν)ξ 2 )e a with i = 1, 2 and the linear combinations: It is convenient to consider the smooth map: The Fierz identities for ξ 1 , ξ 2 imply [5] that (1.1) satisfy the constraints: In view of the first two relations, we define:

JHEP11(2015)174
Consider the cosmooth generalized distributions: As shown in [5], the rank stratifications of M induced by D and D 0 have the same open stratum, the so-called generic locus of M : while the complement W def.
= M \ U (the non-generic locus) decomposes as: where: and Z 3 = ∅. The rank stratifications of M induced by D and D 0 are the disjoint union decompositions: It was shown in [5] that these stratifications can be described as different coarsenings of the B-preimage of the connected refinement of the canonical Whitney stratification of a semi-algebraic body P ⊂ R 4 , where B is the map defined through: a map whose image is contained in P. In particular, we have U = B −1 (IntP) and W = B −1 (∂P), while: where W 0 1 and W 1 1 are defined in loc. cit. and satisfy W 0 We refer the reader to [5] for the description of P and of its Whitney stratification, which we will freely use below. The description of W k and Z k as B-preimages of disjoint unions of various Whitney strata of the frontier of P can be found in loc. cit. It was also shown in [5] that the rank stratification of D 0 coincides with the stabilizer stratification of M , whose strata are defined by the isomorphism type of the common stabilizer group H p def.
2 Circle uplifts to an auxiliary nine-manifold  Figure 1. The canonical projections π 1 , π 2 ofM and the section j of π 1 . and π 1 and π 2 denote the canonical projections ofM onto M and S 1 , respectively (see figure 1). Consider the embedding j : M →M of M inM as the hypersurface given by the equation s = 0: This gives a section of the map π 1 :M → M , thus π 1 • j = id M , which implies that the pull-back map j * : Ω(M ) → Ω(M ) satisfies j * • π * 1 = id Ω(M ) . The differential j * def.
= dj : T M → TM | j(M ) is injective and identifies T M with the corank one sub-bundle j * (T M ) of the restriction of TM to j(M ). To simplify notation, we identify M with j(M ) and T M with j * (T M ) ⊂ TM | j(M ) . The unit circle S 1 is endowed with the exact one-form ds, dual via the musical isomorphism to the Killing vector field ∂ ∂s which generates rotations of S 1 . Let θ def.
Consider the following one-forms defined onM , where k = 1, 2, 3: Relations (2.2) and (2.3) imply thatV k coincide with the natural 1-forms constructed from the canonical liftsξ i of the Majorana spinors ξ i : whereê m is any local coframe ofM defined above an open subsetÛ ⊂M andγ m def. =γ(ê m ). The one-forms (2.4) are invariant under S 1 -rotations ofM , so their Lie derivatives with respect to ∂ ∂s vanish. Since π * 1 (V k ) are orthogonal to θ, we have: where we used the normalization property ||θ|| 2 = 1. Relations (2.6) imply that the first two rows of (1.3) are equivalent with the following system: which can also be written as: Relation (1.4) implies: , eqs. (2.5), (2.16), where they were obtained through direct computation starting from (2.5) and using Fierz identities for two spinors in nine dimensions. The common kernel ofV k defines a cosmooth generalized distribution onM : This distribution is invariant with respect to rotations ofM . However, notice thatD need not be orthogonal to the rotation generator θ and hence it cannot be written as the π 1 -pullback of a distribution defined on M . 2 We mention that the vector fields denoted here byV1,2,3 are denoted by V1,2,3 in loc. cit., while the vector fields denoted here byV± correspond to half of the vector fields denoted by V± in loc. cit., i.e. V here Compare [3], eq. (2.26) with our relationV± = 1 2 (V1 ±V2).

The distributionD 0
One can also lift W ∈ Ω 1 (M ) to the following one-form defined onM , which is everywhere orthogonal to θ:Ŵ def.

The stabilizer groups for M andM
Since the natural action of Spin(TpM ,ĝ p ) Spin(9) onŜ p induces an adjoint action on End(Ŝp) with respect to whichγ m (p) transform as the components of a one-form, it follows that the common stabilizer: whereqp : Spin(TpM ,ĝp) → SO(TpM ,ĝp) is the covering map. Notice that SO(TpM ,ĝp) does not stabilize θ(p). On the other hand, the common stabilizer: of ξ 1 (p) and ξ 2 (p) inside Spin(T p M, g p ) satisfies [5]: is the covering map. The relation: implies that the following holds for any point p ∈ M ≡ j(M ):
As we shall see in what follows, the isomorphism type ofĤp defines a stratification ofM which can be characterized as the pull-back through a smooth and rotationally-invariant mapα ∈ C ∞ (M , R) of the connected refinement of the canonical Whitney stratification of a closed interval.

The stratifications ofM and M induced byD
The rank function ofD gives a decomposition: The locusŴ decomposes further according to the corank ofD inside TM : Notice that we always have rkD(p) < 9, since ||V 1 || = ||V 2 || = 1 by (2.8) and hence the space spanned byV 1 (p),V 2 (p) andV 3 (p) has dimension at least one. We thus have a disjoint union decomposition: Also notice thatÛ ,Ŵ 1 ,Ŵ 2 andŴ are invariant under rotations of the circle and hence they have the forms: where U , W 1 , W 2 and W = W 1 W 2 are subsets of M which give a decomposition (see figure 2): As we shall see below, this decompositions of M induced byD is generally quite different from the first decomposition in (1.8) (which is induced by D). Using (2.8), the Gram determinant formula gives: where we introduced the function (this is denoted by α in [3]): Notice thatα is invariant under rotations of the circle and hence: Relation (2.23) implies that the decomposition (2.18) ofM coincides with theα-preimage of the canonical Whitney stratification of the closed interval [−1, 1]: while the first decomposition of M given in (2.22) coincides with the α-preimage of the same stratification: The following result (cf. [3]) shows that the rank stratification ofM induced byD coincides with theα-preimage of the connected refinement of the Whitney stratification of the interval, while the stratification of M given by the second decomposition in (2.22) coincides with the α-preimage of the same.
The following statement given in [3] follows from known facts about stabilizers of actions of Lie groups on spheres 3 : Proposition. The isomorphism type ofĤp is given by (see table 1): In particular, the stabilizer stratification ofM coincides with the rank stratification ofD and hence with theα-preimage of the canonical Whitney stratification of the interval [−1, 1] (see figure 3). 3 The stabilizer of a single non-vanishing spinor in the Majorana representation ∆9 R 16 of Spin(9) is a subgroup isomorphic with Spin (7), belonging to a certain conjugacy class of subgroups of Spin(9) which is usually denoted by Spin ∆ (7) (see, for example, [30]). With respect to this subgroup, we have the decomposition ∆9 = Λ7 ⊕ ∆7 ⊕ R, where Λ7 R 7 and ∆7 R 8 are the vector and real spinor representations of Spin(7), respectively. Stabilizingξ1(p) first, we can takeξ1(p) ∈ R andξ2(p) ∈ Λ7 ⊕ ∆7. ThusĤp Stab Spin ∆ (7) (ξ2(p)) is isomorphic with SU(4) Spin(6), G2 or SU(3). The first case arises when ξ2(p) ∈ Λ7, the second whenξ2(p) ∈ ∆7 and the third whenξ2(p) has non-vanishing projection on both Λ7 and ∆7. In the second and third case, we used the fact that Spin(7) acts transitively on the unit sphere S 7 ⊂ ∆7 with stabilizer G2 and the fact that G2 acts transitively on S 6 ⊂ Λ7 with stabilizer SU(3). Proposition. We have: and hence β(M ) ⊂ [0, 1] and: Moreover, the following relations express W 1 , W 2 , W and U in terms of the strata introduced in [5], subsection 5.3:  Also notice that U and W coincide with the generic and degeneration loci of the restricted distributionD| M :

The transverse and non-transverse loci of M
Recall that two subspaces K 1 and K 2 of a vector space K satisfy dim( The subspaces are called transverse when codim(K 1 ∩ K 2 ) = codimK 1 + codimK 2 , which is equivalent with codim(K 1 + K 2 ) = 0 i.e. with K 1 + K 2 = K. This condition defines a symmetric binary relation (the transversality relation) on the set of all subspaces of K. For p ∈ M , let p denote the transversality relation between subspaces of T j(p)M , and p denote its negation (the non-transversality relation).

JHEP11(2015)174
Definition. The transverse locus is the following subset of M : while its complement in M is called the non-transverse locus: where we identify p ∈ M with j(p) ∈M and T p M with the subspace j * ,p (T p M ) of T j(p)M .

Characterizing the transverse and non-transverse loci
Proposition. Let p ∈ M ≡ j(M ). Then: Moreover, the following statements are equivalent: In particular, we have dim D(p) = dimD(p) − 1 iff p ∈ T .
Corollary. Let p ∈ M ≡ j(M ). Then the following statements are equivalent: (a) p ∈ N .
(b) There exist λ 1 , λ 2 , λ 3 ∈ R such that: In particular, the non-transverse locus is contained in the degeneration locus of D and hence the generic locus of D is contained in the transverse locus: (3.5)

JHEP11(2015)174
P-locusD-stratum B −1 -stratum D0-stratum rkD rkD rkD0 transversalityĤp Hp  Proof. Follows immediately from (2.4) and from the characterization of non-transversality given at point (e) of the previous proposition, using the fact that θ(p) is orthogonal to V k (p).

Expressing
and we have the relations: Proof. Follows immediately by comparing the ranks ofD| M and D on various loci and using relations (2.28), the characterization of non-transversality given in the previous subsection and the results summarized in tables 5 and 6 of [5]. The situation is summarized in table 2.
Remark. The proposition implies: In particular, the SU(2) stratum U of M is the intersection of the SU(3) stratumÛ ofM with the locus j(T ) ⊂ j(M ), while the degeneration points of D (the points of the locus W ⊂ M ) are of three kinds: • The points p ∈ W 1 = W 0 W 0 1 (where β = 0 i.e. α = +1), which form the intersection of the SU(4) stratumŴ 1 ofM with j(M ). At such points, we have H p SU(4) or SU(3) according to whether p ∈ N or p ∈ T .
• The points of W \ W = W 2− 2 W 3 2 , which form the intersection of the SU(3) stratum U ofM with the locus j(N ) ⊂ j(M ). At such points, we have H p SU(3).

The case T = ∅
The previous proposition immediately implies the following: Corollary. The condition T = ∅ is equivalent with the conditions W 0 1 = W 2+ 2 = U = ∅. When this condition is satisfied, we have andĤ p H p for any p ∈ M , both groups being isomorphic with SU(4), G 2 or SU(3) according to whether p ∈ W 0 , p ∈ W 1 1 or p ∈ W 2− 2 W 3 2 . Notice that T = ∅ implies B −1 (IntP) = U = ∅ and hence requires that the image of B be contained in the frontier ∂P of P. More precisely, we have: Remark. Reference [3] uses the assumption (see equation (3.9) of loc. cit.) that θ(p) is a linear combination ofV 1 (p),V 2 (p) andV 3 (p) for every point p ∈ M . By the characterization given at point (e) of the Proposition of subsection 3.3, this assumption is equivalent with the requirement that the transverse locus T be empty and hence that we are in the setting of the Corollary above. By the Corollary of subsection 3.3, the condition T = ∅ requires, in particular, that the 1-forms V 1 (p), V 2 (p) and V 3 (p) be linearly dependent at every point p ∈ M (cf. [5], appendix G). In was shown in [5] that, generically, we have U = ∅ and hence the transverse locus is not empty in the generic case.

Relation between the stabilizer stratifications of M andM
It is known that an orientable hypersurface in an 8-manifold with SU(4) structure carries a naturally induced SU(3) structure (see, for example, [22], section 4). An orientable hypersurface in a 7-manifold with G 2 structure carries a naturally induced SU(3) structure (see, for example, [23][24][25]). Finally, an orientable hypersurface of a manifold with SU(3) structure carries a naturally induced SU(2) structure [26]. Since these statements are purely algebraic, they extend immediately to the case of Frobenius distributions. Using these facts and the results above, we can understand how the stratified G-structure ofM induces the stratified G-structure of M . Namely, we have (see table 2): • The restriction D| N coincides withD| N and hence D| N carries the same structure group (namely SU(4), G 2 or SU(3)) asD| N on the components W 0 , W 1 1 and W 2− 2 W 3 2 respectively of the non-transverse locus.
• The restriction D| T is an orientable and corank one generalized sub-distribution of D| T and hence D| T carries the structure group SU(3), SU(3) and SU(2) on the components W 0 1 , W 2+ 2 and U respectively of the transverse locus T on whichD| T has the structure group SU(4), G 2 and SU(3) respectively.
These observations give a different way to understand the results of [5], provided that one knows the codimension of D(p) insideD(p) on the various strata (which follows from loc. cit.).
4 Explicit relation between the SU(3) structure onÛ and the SU (2) structure on U Since j identifies U = U ∩ T withÛ ∩ j(T ), the restriction ofD to the locus U ≡ j(U ) ⊂Û is a regular Frobenius distribution of rank six. SinceM is oriented with volume form (2.1), we can orientD| U using the volume form:

The projection of θ alongD on the generic locus
The one-form θ| U decomposes uniquely as: figure 4). Since U is a subset of T , the characterization at point (e) of the Proposition of subsection 3.3 gives θ| U ∈ V + | U ,V − | U ,V 3 | U and hence θ ⊥ = 0 and we can define the unit norm one-form: We orient the rank five Frobenius distribution D| U such that its volume form is given by: Proposition. We have D| U = (ker θ ⊥ )∩D| U , i.e. the normalized vector field n ∈ Γ(U ,D) is everywhere orthogonal to D| U insideD| U , where denotes the musical isomorphism of (M ,ĝ). Moreover, we have: and: Furthermore, we have: JHEP11(2015)174 and: where ρ was defined in (2.13).

Relation between SU(2) and SU(3) structures
An SU(2) structure on the oriented rank five Frobenius distribution D| U which is compatible with the metric g| D and with the orientation of D can be described by a normalized oneform α ∈ Ω 1 U (D) and three mutually orthogonal 2-forms ω 1 , ω 2 , ω 3 ∈ Ω 2 U (D) satisfying the equations (see [26]): where k, l = 1, 2, 3 and v is a non-vanishing four-form which satisfies:

JHEP11(2015)174
Namely, we have D 0 | U = ker α and (ω 1 , ω 2 , ω 3 ) is an orthogonal basis of the free C ∞ (U , R)module Ω 2+ U (D 0 ) of D 0 | U -longitudinal self-dual 2-forms. As explained in [26], this basis can be chosen such that it forms a positively-oriented frame of the rank three bundle ∧ 2+ D * 0 , where the latter is endowed with the orientation naturally induced from that of D 0 (which is given by the volume form 1 2 v). On the other hand, an SU(3) structure on the oriented rank six Frobenius distribution D| U which is compatible with the metricĝ| D and with the orientation ofD is determined [23] by an almost complex structure I ∈ Γ(U , End(D)) which is compatible with the metric and orientation ofD, together with a complex-valued three-form Ω ∈ Ω 2 U (D) ⊗ C which is of unit norm and has type (3, 0) with respect to I. The almost complex structure defines a two-form J ∈ Ω 2 U (D) through the relation: =ĝ(X, IY ) , ∀X, Y ∈ Γ(U ,D) (4.13) and this form satisfies:ν 14) The phase of the normalized (3,0)-form Ω is fixed through the convention: Decomposing Ω into its real and imaginary parts: relation (4.15) amounts to: The following proposition gives the relation between SU(3) structures on the rank six Frobenius distributionD| U def.
Proposition. There is a bijective correspondence between SU(3) structures onD| U which are compatible with the metric and orientation ofD| U and SU(2) structures on D| U which are compatible with the metric and orientation of D| U . This correspondence is given as follows, where n was defined in (4.3): (a) Given a metric-and orientation-compatible SU(3) structure onD| U with 2-form J ∈ Ω 2 U (D) and complex 3-form Ω ∈ Ω 3 U (D) ⊗ C, the following formulas give the canonical forms defining the corresponding metric-and orientation-compatible SU(2) structure on D| U , where i is the imaginary unit:

JHEP11(2015)174
(b) Given a metric-and orientation-compatible SU(2) structure on D| U which is defined by the canonical forms α ∈ Ω 1 U (D) and ω k ∈ Ω 2 U (D) (where k = 1, 2, 3), the following forms define the corresponding metric-and orientation-compatible SU(3) structure onD| U : ) Proof. This is an obvious adaptation of [26], proposition 1.4 to the case of Frobenius distributions. Notice that the signs agree with our choices of orientation. Indeed, we have:

Recovering the SU(2) structure on the generic locus of M
Reference [3] constructs an SU(3) structure on the rank six Frobenius distribution which is obtained by restrictingD to the open subsetÛ ⊂M , a set which (by the results of section 3) contains the π −1 1 (U ) of the generic locus. This SU(3) structure is described in loc. cit through certain differential forms denoted there by: As shown in appendix A, the canonically-normalized forms of that SU(3) structure are given by:Ĵ where:φ def.
Restricting everything to the subset U ≡ j(U ) ⊂Û ⊂ M , we obtain an SU(3) structure on the restricted Frobenius distributionD| U , whose canonically-normalized forms are given by: By definition, the 1-form θ ⊥ ∈ Ω 1 (U ) is the component of θ| U which is orthogonal to the sub- Hence the 1-form n (which is defined through (4.3)) is also orthogonal to this sub-bundle and thus ι nVk = 0 for all k. On the other hand, we have D| U = ker θ ∩D| U = ker n ∩D| U ⊂ ker n

Conclusions and further directions
We analyzed the stabilizer stratifications of internal eight-manifolds M which can arise in N = 2 M-theory flux compactifications down to three dimensions using the formalism based on the auxiliary nine-manifoldM def.
= M × S 1 , which can be viewed as a trivial circle bundle over M with projection π 1 . We showed how the complicated stratified G-structure of M which was determined in [5] relates to the much simpler stratified G-structure ofM . The increased complexity of the former arises from the fact that the cosmooth generalized distributionD whose rank determines the stabilizer stratification ofM may have pointwise transverse or non-transverse intersection with the π 1 -pull-back of the tangent bundle of M . We also gave an explicit construction of the defining forms of the SU(2) structure which exists on the generic locus U ⊂ M in terms of the defining forms of the SU(3) structure which exists on the locusÛ ⊂M .

JHEP11(2015)174
An important aspect of our analysis is that the simplification obtained by uplifting to the nine-dimensional cylinderM is only apparent. While the stabilizer stratification ofM is indeed simpler than that of M , a large part of the complexity of the stratified G-structure of M is 'hidden' in the character of the intersection ofD| M and T M . Therefore, passing from M toM does not simplify the problem in any fundamental way, but merely divides differently the complexity of the stabilizer stratification of M : part of that complexity is encoded by the simpler stabilizer stratification ofM while the remainder is captured by the character of the intersectionD| M ∩ T M . In particular, the expectation that the analysis of stratified G-structures of flux compactifications simplifies upon passing to some uplifted description fails to materialize in our case. Part of the reason for this phenomenon is the high dimensionality (namely eight) of the internal space M and one may speculate that this relates to the problem of "dual gravitons" which is relevant for such backgrounds of M-theory.
One could also consider higher dimensional uplifts, for example to a ten-dimensional manifold of the formM def.
= M × Σ, where Σ is a (compact) Riemann surface. Such higher uplifts may lead to even simpler stabilizer stratifications ofM . However, the character of the intersection of j(M ) with this stratification would be more complex, since the number of essentially distinct cases grows with the dimension of the uplift. Hence uplifting to higher dimensions simplifies one aspect of the problem (namely, the stabilizer stratification of the uplift) while complicating the other (namely, the analysis of the intersection of j(M ) with the uplift stratification). Since one is ultimately interested in describing the stabilizer stratification of M rather than that of the auxiliary spaceM , it is unclear that there is much gain in such repackaging of the complexity. In our opinion, the best approach is that of [5], which gives a direct and complete solution using the tools of stratification theory and semi-algebraic geometry, without introducing any auxiliary space constructions.
It is natural to wonder what insight could be gained by replacing the cylinderM with a topologically non-trivial S 1 -bundle over M . Since the character of the intersection D| M ∩ T M is determined pointwise, it is clear that modifying the global topology ofM will affect the topology of its stratification, without changing the stratified G-structure of M (which is the physically relevant object), hence without leading to any true simplification of the problem. In general, questions regarding the precise relation between the topology of possible uplifts and the topology of the relevant stratifications require detailed analysis which falls outside the scope of the present work.
A related problem is whether (as suggested by the results of [31,32]) one can find a compact manifoldM (of dimension higher than nine) which fibers over M , such that the stratified G-structure of M uplifts to a globally-defined reduction of structure group ofM . In full generality, the answer to this question is non-obvious, since the topology of sucĥ M would have to depend rather non-trivially on the topology of the stratified G-structure of M . Another interesting question is to find a physics interpretation of the interaction between uplifts, stratified G-structures and intersection theory which appears to govern many of the phenomena uncovered in this paper and in the closely related work of [31,32]. This requires further conceptual development on which we hope to report in future work.
relation (2.42) of [3] becomes the condition thatΩ isÎ-pseudoholomorphic: whereÎ (1) denotes the action ofÎ on the first "slot" ofΩ. On the other hand, the analogue of relation (4.17) shows thatΩ satisfies the analogue of (4.15) onÛ . Combining everything, we conclude thatĴ andΩ are the canonically-normalized forms of the SU(3) structure which was constructed in [3] on the rank six Frobenius distribution obtained by restrictinĝ D to the locusÛ ⊂M .
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 4 Notice that there is a typo in [3], eq. (2.40), in that the right hand side of that equation should equal