IPhT-t14/180 Prepared for submission to JHEP
M-theory moduli spaces and torsion-free structures
Mariana Graña and C. S. Shahbazi Institut de Physique Théorique, CEA Saclay. E-mail: [email protected], [email protected]
Abstract: Motivated by the description of N = 1 M-theory compactifications to four-dimensions given by Exceptional Generalized Geometry, we propose a way to geometrize the M-theory fluxes by appropriately relating the compactification space to a higher-dimensional mani- fold equipped with a torsion-free structure. As a non-trivial example of this proposal, we construct a bijection from the set of Spin(7)-structures on an eight-dimensional S1-bundle
to the set of G2-structures on the base space, fully characterizing the G2-torsion clases when the total space is equipped with a torsion-free Spin(7)-structure. Finally, we elaborate on how the higher-dimensional manifold and its moduli space of torsion-free structures can be used to obtain information about the moduli space of M-theory compactifications.
ArXiv ePrint: 1410.8617 arXiv:1410.8617v2 [hep-th] 5 Apr 2015 Contents
1 Introduction1
2 Intermediate manifolds from Exceptional Generalized Geometry3
2.1 Seven-dimensional supersymmetric G2 structures3 2.2 Eight-dimensional intermediate manifolds5
2.3 Spin(7) holonomy on M8 9
3 Examples 10 3.1 Spin(7)-manifolds from trivial products of G2-structure manifolds and R 10 3.2 Spin(7)-manifolds as S1-bundles 13
4 Relating moduli spaces 18
5 Summary 21
1 Introduction
In the context of String/M-theory compactifications, supersymmetry implies the existence of a topological G-structure satisfying a particular set of differential conditions on the appropriate principal bundle over the internal manifold. In particular, this set of differential conditions can be written as first-order partial differential equations on the tensors that define the G-structure. The moduli space of a given supersymmetric compactification is an important space from the physics point of view; it is expected to be a finite-dimensional manifold equipped with a Riemannian metric closely related to the non-linear sigma-model appearing in the effective supergravity action of the compactification. In the simplest cases, namely in the absence of fluxes, the topological reduction is on the frame bundle of the internal space and the differential conditions implied by supersymmetry are equivalent to imposing the G-structure to be torsion-free [1,2], meaning that the reduction is not only topological but also geometric. Usually the moduli space of this torsion-free structures is well under control when the internal space is an oriented, compact manifold [3,4]. In particular, the Riemannian metric is known and thus it can be used to explicitly write the effective action of the compactification [5,6]. From the phenomenological point of view, realistic scenarios require the moduli spaces to be just isolated points, and for that the presence of non-vanishing fluxes is crucial1. This implies that the topological G-structure induced by supersymmetry will have non- vanishing torsion [9, 10], always subject to the corresponding differential equations. It is
1It is also possible to obtain phenomenologically viable models through flux-less compactifications on manifolds with the appropriate kind of singularities, see references [7,8].
– 1 – thus desirable to have a better understanding of the moduli space in the presence of fluxes, in order to find as explicitly as possible the four-dimensionl models coming from String/M- theory compactifications, that can ultimately be used to make contact with particle physics or cosmology. A huge effort has been devoted so far in understanding the supersymmetric compactifi- cations, see [11] and references therein. One approach, namely generalized geometry either in its complex [12, 13] or exceptional flavour [14, 15], has proven to be particularly interest- ing and fruitful, giving a unified geometrical description of the different flux backgrounds as well as their moduli spaces [15–20]. In this note we are going to argue that through the exceptional generalized geomet- ric description of the moduli space of N = 1 supersymmetric M-theory compactifications to four-dimensions, we can associate to every internal seven-dimensional manifold M7 an eight-dimensional manifold M8 equipped with a particular set of tensors S (given in Def- inition 2.6) that contain all the information about the topological reduction implied by supersymmetry. This eight-dimensional manifold (M8, S) was dubbed intermediate man- ifold in [21], where the study of its topological properties was initiated. The geometry of an intermediate manifold (M8, S) is completely specified once the tensors S are required to satisfy the differential conditions obtained from supersymmetry. In order to translate the supersymmetry conditions to the intermediate structure S, it is necessary first to write them in the exceptional generalized geometry language, which has not been done yet. Once they are known we will be able to fully uncover the role of intermediate manifolds in rela- tion to M-theory compactifications and their moduli spaces. However, it is clear that the moduli space of intermediate structures on M8 is closely related to the moduli space of the corresponding compactification, and therefore we can resort to this guiding principle in order to construct particular examples of M8. More precisely, here we will propose that it is natural to construct M8 from M7 in such a way that M8 is equipped with a torsion-free Spin(7)-structure induced by a one- to-one map from the G2-structure in seven dimensions, obtaining therefore a candidate for a map between G2-structures with non-zero torsion in seven dimensions and torsion-free structures in eight-dimensions. This is interesting because the moduli space of the latter is much more under control than the moduli space of the former, and thus it is reasonable to expect that using the map some new information can be obtained about the moduli space of G2-structures with non-vanishing torsion. Once we realize that one can embed structures with torsion in a given dimension into torsion-free structures in a higher dimension, there is no need to stop at eight-dimensions.
In fact, for a general enough G2-structure in seven-dimensions we should not expect to obtain a torsion-free Spin(7)-structure in eight dimensions, but that could be possible if we consider instead a manifold of dimension large enough. This construction can be as well motivated from the Exceptional Generalized Geometry formulation of the compactification. The outline of this note is as follows. In section2 we review N = 1 M-theory compacti- fications to four dimensions, motivate the definition of intermediate manifolds and show how they can be constructed from the physical manifold M7. In section3 we present a few ex- amples of this construction. In particular, and also as a non-trivial example of intermediate
– 2 – manifold, we obtain a bijection from the set of Spin(7)-structures on an eight-dimensional 1 S -bundle and the set of G2-structures on the base space, fully characterizing the G2-torsion clases when the total space is equipped with an invariant torsion-free Spin(7)-structure. In section4 we define the moduli space of supersymmetric M-theory compactifications and ex- plain how the map between G2-structures in seven-dimensions and torsion-free structures in eight-dimensions can be used to obtain information about the former.
2 Intermediate manifolds from Exceptional Generalized Geometry
Let us consider the bosonic sector (g, G) of eleven-dimensional Supergravity on a space-time manifold M that can be written as a topologically trivial direct product
M = M1,3 × M7 , (2.1) where M1,3 is a four-dimensional Lorentzian oriented, spin manifold and M7 is a seven- dimensional, Riemannian, oriented, spin manifold. Although we will speak about compact- ifications, we are not going to assume that M7 is compact since it might be consistent to compactify in non-compact space-times with finite volume and appropriate behavior of the laplacian operator [22]. According to the product structure (2.1) of the space-time manifold M, the we choose the Lorentzian metric g on M to be given by
2A g = e g1,3 × g7 , (2.2)
2A ∞ where g7 is a Riemannian metric on M7, e ∈ C (M7) is the warp factor and g1,3 is a Lorentzian metric on M1,3. In addition, and in accordance again with the product structure of M and the choice of metric (2.2), we will assume that the four-form G is given by
∗ ∗ 4 G = µ pr1Vol4 + pr2G , µ ∈ R ,G ∈ Ωcl (M7) , (2.3) where pr1 : M → M1,3 and pr2 : M → M7 are the corresponding canonical projections and Vol4 denotes the Lorentzian volume form on M1,3. With the choice of metric and four-form given in equations (2.2) and (2.3) we guarantee that, given a particular Lorentzian four- dimensional manifold (M1,3, g1,3), the equations of motion that the bosonic fields of the theory (the full metric g and four-form G) need to satisfy, can written as a set of differential equation exclusively on M7. This will be important in section4 in order to properly define the moduli space of this kind of solutions. Although the set-up presented above is more general, we will be usually interested in compactifications on maximally symmetric spaces (M1,3, g1,3), which thus will be taken to be the AdS or Minkowski space. The dS-metric is discarded since it cannot be obtained supersymmetrically.
2.1 Seven-dimensional supersymmetric G2 structures Supersymmetric solutions of eleven-dimensional Supergravity have to satisfy the Killing spinor equation of the theory, given by
– 3 – 1 1 D ≡ ∇ + (ι G) + v[ ∧ G = 0 , ∀ v ∈ X (M) , (2.4) v v 6 v 12 where S → M the 32-dimensional real, symplectic spinor bundle over M, ∈ Γ(S) is a Majorana spinor, v[ is the one-form associated to v through the metric and the terms in parenthesis act on the spinor via the Clifford multiplication. Given the decomposition (2.1) and the choice of metric (2.2) we have to decompose the Spin(1, 10) spinor in terms of Spin(1, 3) × Spin(7) representations. Let us denote + − by ∆R the 32-dimensional symplectic real representation of Spin(1, 10), by ∆1,3 and ∆1,3 the positive and negative chirality complex Weyl representation of Spin(1, 3) ' Sl(2, C), R of complex dimension two, and by ∆7 the real Majorana representation of Spin(7). The branching rule Spin(1, 10) → Spin(1, 3) × Spin(7) is given by2
h i R + − R ∆1,10 = ∆1,3 ⊕ ∆1,3 ⊗ ∆7 , (2.5) and therefore
h i + − R = ξ ⊗ η , ξ ∈ Γ S1,3 ⊕ S1,3 , η ∈ Γ S7 , (2.6)
± R where S1,3 is the positive (negative) chirality spin bundle over M1,3 and S7 is the real spin bundle over M7. For each linearly independent eleven-dimensional spinor of the form (2.6) satisfying the Killing spinor equation (2.4), there is one out of 32 supersymmetries preserved. Therefore 1 ν = 32 supersymmetry requires the existence of a single Majorana Spin(7) spinor in the 3 internal space M7 . R The existence of a global spinor implies that the structure group of the spin bundle S7 is reduced from Spin(7) to G2. This structure can alternatively be defined by a positve 4 3-form ϕ , which additionally induces a Riemannian metric g7 that can explicitly be written as follows
1 V g (v, w) = ι ϕ ∧ ι ϕ ∧ ϕ , v, w ∈ X (M ) , (2.7) 7 7 3! v w 7 where V7 is the Riemannian volume form of M7. The supersymmetry conditions coming from the Killing spinor equation (2.4) can be translated into a set of differential conditions on ϕ. In the absence of fluxes, and for a Minkowski four-dimensional vacuum these amount to dϕ = 0, d ∗ ϕ = 0, which implies that the structure is torsion-free, or equivalenetly that the manifold has G2 holonomy. In the
2Let V be a complex representation of a Lie group G. If c: V → V is a real structure on V , namely an 2 invariant, antilineal map such that c = 1, then V = [V ] ⊗R C, where [V ] is a real representation of G of real dimension equal to the complex dimension of V . 3We could have also considered, as it is standard in the literature, the branching rule of the complexi-
fication of the supersymmetry spinor. Then, we would have ended up with two Majorana spinors on M7. 1 Notice however that in that case the solutions are generically ν = 16 supersymmetric. 4A 3-form ϕ is said to be positive [4] if at every point it can be written as ϕ = dx123 + dx145 + dx167 + 246 257 347 356 T mnp dx − dx − dx − dx . Given a Majorana spinor η and a metric g7, the form ϕ = η γmnpη dx is positive.
– 4 – presence of fluxes the structure is not torsion-free anymore. It is useful to decompose the torsion classes into G2 representations, namely
dϕ = τ0 ∗ ϕ + 3 τ1 ∧ ϕ + ∗τ3 , d ∗ ϕ = 4 τ1 ∧ ∗ϕ + τ2 ∧ ϕ , (2.8) where τ0,1,2,3 are a 0,1,2 and 3-form, respectively in the 1, 7, 14 and 27 representations of G2. Supersymmetry relates these torsion classes to the different representations of the 4- form flux G. For a Minkowski vacuum, and particular decomposition of the supersymmetry spinor, these relations can be found in [23]. In this paper we are interested in the set of all supersymmetric solutions of the form given in (2.2) and (2.3) as a space by itself; namely, we are interested in the moduli space of supersymmetric compactifications to four-dimensional AdS or Minkowski space. Due to the presence of a non-trivial warp factor e2A, a supersymmetric solution cannot be fully characterized by a set of first-order differential equations. Indeed, to obtain a supersym- metric solution to the full set of equations of motion of eleven-dimensional Supergravity one has to impose, on top of the supersymmetry conditions, the equation of motion for the flux. This is a second-order differential equation for e2A. Hence, the moduli space of supersymmetric solutions to eleven-dimensional Supergravity cannot be characterized in terms of first-order differential equations. Therefore, we will consider instead the moduli space of supersymmetric configurations, which depends exclusively on the Killing spinor equation 2.4, which is a first-order partial differential equation. We define then the concept of supersymmetric G2-structure, which will be used in section4, in the following way
Definition 2.1. Let (M1,3, g1,3) be a particular Lorentzian, four-dimensional, oriented and spin manifold and let M7 be a fixed seven-dimensional, oriented, spin manifold. A 2A supersymmetric G2-structure relative to (M1,3, g1,3) is a quadruplet ϕ, G, e , µ , where 2A ∞ ϕ ∈ Ω+ (M7) is a positive three-form, G ∈ Ωcl (M7) is a closed four-form, e ∈ C (M7) is a function and µ ∈ R, such that combined with the four-dimensional data in the form given by (2.2), (2.3), (2.6), it obeys the Killing spinor equation (2.4)5.
Remark 2.2. Notice that depending on the choice of four-dimensional manifold (M1,3, g1,3) the set of corresponding supersymmetric G2-structures may be empty, for example if (M1,3, g1,3) is dS-space, which is known to be non-supersymmetric. When talking about the set of super- symmetric G2-structures we will always omit the space they are relative to, assuming that it is one such that the set is non-empty.
2.2 Eight-dimensional intermediate manifolds
Using the tools of generalized geometry, compactifications of M-theory down to four- dimensions with N = 1 supersymmetry were geometrically characterised in [15] in terms of the space of sections of a real, 912-rank, vector bundle over M7
5The Killing spinor equation (2.4) is written in terms of the spinor η instead of the 3-form ϕ, but from it one can obtain equations for ϕ in terms of the components of G and the warp factor (see (3.25) in reference [23]).
– 5 – E → M7 , (2.9)
+ with structure group E7(7) × R , where E7(7) stands for the maximally non-compact real 6 + form of E7 acting in the 912 representation and R represents conformal rescalings. Let us denote by N912 the vector space corresponding to the 912 representation of + 7 E7(7) × R . In terms of Sl(8, R) representations, it decomposes according to
2 ∗ 3 ∗ 2 3 ∗ N912 ' S V ⊕ Λ V ⊗ V 0 ⊕ S V ⊕ Λ V ⊗ V 0 , (2.10) where V is an eight-dimensional real vector space where Sl(8, R) acts in the vector repre- sentation, S stands for symmetric and the subscript 0 denotes traceless. We can apply the decomposition (2.10) to the bundle E (2.9), since at every point p ∈ M7, Ep is the 912 irreducible representation of E7(7). Hence we can write (2.10) for E where now V would denote a rank-eight real vector bundle over M7
V → M7 , (2.11)
+ with structure group Sl(8, R) × R acting in the 8 of Sl(8, R). Therefore, every section φ ∈ Γ(E) can be decomposed as in (2.10) R η Given a spinor η ∈ ∆7 , there exists a canonical element ψ ∈ N912 which is stabilized by SU(7) ⊂ E7(7) [15, 20] and can be written as follows in terms of SU(8) representations (see footnote7) ψη = (η η, 0, 0, 0) . (2.12)
η Definition 2.3. An element ψ ∈ N912 is said to be E-admissible (or in the orbit of ψ ), if + there is a transformation g ∈ E7(7) × R such that
g · ψ = ψη . (2.13)
Definition 2.4. A section ψ ∈ Γ(E) is said to be E-admissible if, for every p ∈ M,
ψp ∈ N912 is an E-admissible element of N912.
+ Corollary 2.5. Let E → M be a rank-912 vector bundle with structure group E7(7) × R . If E is equipped with a E-admissible section ψ ∈ Γ(E), then it admits a topological reduction + of the frame bundle from E7(7) × R to SU(7).
6 Here 912 denotes the real representation induced on E7(7) by the 912 of the exceptional complex Lie group E7 = E7(7) ⊗ C. The 912 representation of E7 can be characterized as a particular subspace of V56 ⊗ e7, see appendix B in reference [15]. 7 The real group E7(7) contains two important subgroups, namely Sl(8, R) and SU(8)/Z2. The 912 decomposes 912 = 36 + 420 + 36 + 420 ,SU(8)/Z2 ⊂ E7(7) , 0 0 912 = 36 + 420 + 36 + 420 , Sl(8, R) ⊂ E7(7) , where [·] denotes the corresponding induced real representation.
– 6 – Off-shell N = 1 supersymmetry implies the existence of an E-admissible section ψ ∈ Γ(E), + and thus a topological reduction on E from E7(7) × R to SU(7) and vice-versa, such a topological reduction implies the existence of a globally defined spinor on M7. It is then necessary to take a closer look at the space of E-admissible sections, since it corresponds to the space of topological reductions on E implied by supersymmetry. Notice that the space E + of E-admisible elements N912 ⊂ N912 is transitive under the action of E7(7) ×R , and since every E-admissible element is stabilized by SU(7) we obtain
E N E ' 7(7) × + . (2.14) 912 SU(7) R E Hence, the space of E-admissible elements N912 is an 85-dimensional real homogeneous + 8 manifold times R , and it was conjectured to be a Kähler-Hodge manifold in reference [15] in order to make contact with the geometry of the non-linear sigma-model of N = 1 Supergravity. Let us define then the fibre bundle
EE → M7 , (2.15) E7(7) + ˜ with fibre EE p ' SU(7) × R , p ∈ M7. The set X of all E-admissible structures on E can be thus written as the space of sections of EE
X˜ = {ψ | ψ ∈ Γ(EE)} . (2.16) In order to properly define the moduli space of supersymmetric compactifications as a subset of X˜ there are two missing pieces. These are the differential conditions implied on
ψ ∈ Γ(EE) by supersymmetry, which come from the Killing spinor equation (2.4), and the quotient by the appropriate equivalence relation. If we had them we could readily characterize the moduli space of N = 1 compactifications to four dimensions as follows
n ˜ o X = ψ | dψ = 0 , ψ ∈ Γ(EE) /D . (2.17) where d˜ is an appropriate coboundary operator and D are generalized diffeomorphisms, combining diffeomorphisms and gauge transformations. However, given on one hand that we do not know the differential conditions, and on the other that there are not yet avail- able any mathematical results on deformations on these structures, we will pursue here a different path, more straightforward in some sense, though restricted to particular set of supersymmetric backgrounds. We have seen that off-shell N = 1 supersymmetry is equivalent to the existence of an
E-admissible section, which at every point p ∈ M7 can be written as as follows in terms of SL(8, R) representations (cf. Eq. (2.10)) [20]
−1 ˜ ψ = g8 ⊕ φ ⊕ g8 ⊕ φ , (2.18) where g8 is a Riemannian metric on V → M7 and φ, φ˜ are appropriate elements in the second and fourth components of (2.10). Therefore (g8, φ, φ˜) carry all the information of
8To the best of our knowledge this has not been proven yet
– 7 – the E-admissible section of E. However, E is an extrinsic bundle over M7, while it would be desirable to have a description given in terms of intrinsic bundles, namely tensor bundles of a given manifold, since they are easier to handle. Given that the exceptional bundle E can be written as the Whitney sum of rank-eight real vector bundles over the seven dimensional base M7, we propose an eight-dimensional manifold M8 such that it contains in its tensor bundles all the information that is present in the set of E-admissible sections of E. This way, the role of the rank-eight vector bundle
V → M7 is played by the tangent bundle T M8 of M8, which then must be equipped with the corresponding sections to contain a decomposition of the 912 representation of + E7(7) × R in terms of tensors over M8. This motivates the following definition.
Definition 2.6. Let M8 be an eight-dimensional, oriented, spin differentiable manifold. We say that M8 is an intermediate manifold if it is equipped with the following data