M-theory Compactifications to Three Dimensions with M2-brane Potentials
aa r X i v : . [ h e p - t h ] N ov M-theory Compactifications to Three Dimensionswith M2-brane Potentials
Cezar Condeescu , , Andrei Micu , Eran Palti INFN, Sezione di Roma “Tor Vergata”,Via della Ricerca Scientifica 1, 00133 Roma, Italy Department of Theoretical Physics, IFIN-HH,Reactorului 30, 077125, Magurele/Ilfov, Romania Institut f¨ur Theoretische Physik, Ruprecht-Karls-Universit¨at,Philosophenweg 19, 69120, Heidelberg, Germany
Abstract
We study a class of compactifications of M-theory to three dimensions that preserve N = 2 supersymmetry and which have the defining feature that a probe space-time fillingM2 brane feels a non-trivial potential on the internal manifold. Using M-theory/F-theoryduality such compactifications include the uplifts of 4-dimensional N = 1 type IIB compact-ifications with D3 potentials to strong coupling. We study the most general 8-dimensionalmanifolds supporting these properties, derive the most general flux that induces an M2potential, and show that it is parameterised in terms of two real vectors. We study thesupersymmetry equations when only this flux is present and show that over the locus wherethe M2 potential is non-vanishing the background takes the form of a Calabi-Yau three-foldfibered over a 2-dimensional base spanned by the flux vectors, while at the minima of thepotential the flux vanishes. Allowing also for non-vanishing four-form flux with one leg inthe internal directions we find that the Calabi-Yau three-fold in the fibration is replaced byan SU (3)-structure manifold with torsion classes satisfying 2 W = − W .E-mails: [email protected] ; [email protected] ; [email protected] . ontents N = 2 supersymmetry . . . . . . . . 62.2.1 9-dimensional uplifts to Y = X × S . . . . . . . . . . . . . . . . . . . . 82.2.2 Loci of G structure: α = − SU (4)-structure: α = +1 . . . . . . . . . . . . . . . . . . . . . . . 92.3 Parametrisation in terms of a SU (3)-structure . . . . . . . . . . . . . . . . . . . . 9 N = 2 supersymmetry in 3D . . . . . . . . . . . . . . . . . . . . . 133.2 Flux induced variations of α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 M2-brane potentials and Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . 15 f = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 The case ˜ f = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 A.1 Completely antisymmetric Fierz identities . . . . . . . . . . . . . . . . . . . . . . 26A.2 Fierz identities with symmetric part . . . . . . . . . . . . . . . . . . . . . . . . . 30
B Relations satisfied by the SU (3) structure forms 30C Supersymmetry equations 34D Killing properties of the Majorana-Weyl components 36 In this paper we study compactifications of M-theory to three dimensions that preserve N = 2supersymmetry and which induce a potential for space-filling M2-branes. Though interestingas three-dimensional vacua in themselves, our primary motivation for studying them originatesin F-theory [1]. Four-dimensional N = 1 F-theory vacua can be defined as dual to a particular1imit of N = 2 M-theory compactifications to three-dimensions. This definition relies on theassumption that the 8-dimensional manifold on which M-theory is compactified is ellipticallyfibered. Then the appropriate limit where four-dimensional physics is recovered is that of avanishing fibre. This duality is most often used to construct F-theory backgrounds as dual toM-theory compactifications on elliptically fibered Calabi-Yau manifolds. Such compactifica-tions can exhibit realistic particle physics models and have been under intensive study in recentyears [4, 5, 6]. A well-understood generalisation of such models is to include a particular (2 , N = 2 compactifications of M-theory to three dimensions. This class ofcompactifications have the property that space-filling probe M2-branes on the M-theory side,which are dual to space-filling probe D3-branes on the F-theory side, are BPS at all points ofthe (conformal) CY and therefore feel no potential [8].There are a number of interesting departures from such backgrounds which can be char-acterised by the fact that a potential is generated for (probe) D3-branes. The most familiarare backgrounds of type IIB string theory that support gaugino condensation on D7-branes,or certain types of D3-instantons that contribute to the superpotential, as used for modulistabilisation in the KKLT and Large volume scenarios [9, 10]. The fact that these induce apotential for D3-branes can be shown by performing a 1-loop string calculation [11] or throughgravitational back-reaction [12]. As well as being used for moduli stabilisation, there is a directcosmological use for type IIB backgrounds that induce D3-potentials for inflation [13, 14]. Suchbackgrounds also play an important role in particle physics model building. In F-theory modelswhere all the generations are realised on a single matter curve the Yukawa coupling at a pointof intersection is rank one [15, 16] unless an appropriate deformation of the geometry which in-duces a D3 potential by flux [15] or non-perturbative effects [17] is present. Since all the listedinteresting IIB/F-theory backgrounds have potentials for D3-branes they are not within thebetter understood class of warped CY compactifications. Although an approach of neglectingthe backreaction of the effects that induced the D3 potential on the geometry and continuing touse a CY background may be a valid approximation for some purposes, an understanding of thebackreaction is essential for applications where the effects are large or when a treatment of thebackground in a 10-dimensional sense, rather than a 4-dimensional effective theory, is important.Studying such backgrounds, particularly at strong coupling, using F-theory/M-theory dualityshould therefore involve some understanding of three-dimensional M-theory backgrounds whichpreserve N = 2 supersymmetry and have a potential for M2-branes.Having identified the presence of a D3/M2 brane potential as the defining feature we areinterested in, it is essential to understand what are the properties of the background geometryand flux which induce such a potential. There is a rather general and neat answer to thisquestion in type IIB supergravity. The key property of the background is the structure groupof the metric once any flux/branes are back-reacted. Recall that a nowhere vanishing spinor ona manifold reduces its structure group. Since the presence of a nowhere vanishing internal spinoris a direct requirement for the background to preserve some supersymmetry a reduced structure Though see recent work on manifolds with
Spin (7) holonomy [2, 3]. Interestingly, such backgrounds also induce non-commutative deformations on the world-volume theories ofthe 7-branes [13, 15, 17]. N = 1 supersymmetryhave an SU (3) × SU (3) structure group [19, 20]. Here each SU (3) is associated to a spinor onthe manifold, but the two spinors are not everywhere orthogonal (which would lead to SU (2)-structure) or parallel (leading to SU (3)-structure) but rather the angle between them varies overthe manifold. Now an interesting result of [21, 22] is that backgrounds which support potentialsfor D3-branes are those which have such an SU (3) × SU (3) structure group, and further, thatthe minimum for the potential, where the D3-branes are BPS, occurs exactly on the loci wherethe two spinors become parallel, yielding a ‘local’ SU (3)-structure. This general result waschecked for the particular case of D7 gaugino condensation in [23] which confirmed that theirbackreaction indeed changes the structure group from SU (3) to SU (3) × SU (3). It was alsoconfirmed to some extent in [24, 25] for the case of D3-potentials induced by non-Imaginary-Self-Dual background flux, specifically by showing that the backreaction of D7 gaugino-condensationcan locally be viewed as such a background flux.One aim of this paper is to develop analogous relations between structure groups and M2-brane potentials in M-theory. The relation between structure groups and non-vanishing spinorsin 8 dimensions is rather different from the more familiar 6 and 7 dimensions. Consider acompactification on an 8-dimensional manifold X . A requirement for preserving some super-symmetry in 3-dimensions is the existence of a nowhere vanishing Majorana spinor on X .This is so that the 11-dimensional M-theory supersymmetry spinor decomposes into a productof a 3-dimensional and 8-dimensional Majorana spinor. However, the existence of a nowherevanishing Majorana spinor in 8 dimensions does not imply a reduction of the structure group. This only occurs in the presence of nowhere vanishing Majorana-Weyl spinors: a single suchspinor implies
Spin (7)-structure, while two such spinors imply G or SU (4)-structure if theyhave the opposite or same relative chirality respectively. Now the number of supersymmetriespreserved by an M2-brane in a background is given by the number of independent covariantlyconstant Majorana-Weyl spinors of fixed chirality [26]. Since the manifolds with fixed structuregroup have a fixed number of Majorana-Weyl spinors of fixed chiralities M2-branes are BPS overthe whole space and preserve the supersymmetries of the background. Therefore a connectionbetween structure groups and M2 potentials for 8-dimensional manifolds is not obvious.There is a useful way to think about the M2-potential in terms of 8-dimensional localstructure groups, i.e. submanifolds of X over which the spinors satisfy certain properties suchas being non-vanishing. The two nowhere vanishing Majorana spinors can be decomposed into4 Majorana-Weyl spinors, but any of these four may vanish on certain loci. In the genericpoint on X all four are non-vanishing and we have a local SU (3)-structure on X , over certainloci one of the Majorana-Weyl components in each Majorana spinor may vanish so that weare left with two Majorana-Weyl spinors of same or opposite chirality leading to local SU (4)or G -structures. M2-branes are calibrated by a Majorana-Weyl spinor of fixed chirality andtherefore on the generic SU (3)-structure loci they preserve no supersymmetry, on G loci theypreserve N = 1 supersymmetry and on SU (4) loci they preserve N = 2 supersymmetry. Theytherefore feel a potential in such backgrounds with minima at SU (4) and G -structure loci. We will use the notion of a ‘local’ structure group often in the paper. We define it as the structure groupthat would result should the properties of the spinors on a local submanifold where the local structure group isdefined be extended to the full space. The stabiliser group of a Majorana spinor is G . However we use the notion of a global structure group asmaximal over the manifold, while the G stabiliser may enhance over certain loci. X and global G-structures by defining an auxiliary 9-dimensional manifold, Y , whichis just the product of the 8-dimensional one with a circle Y = X × S . Now the existenceof an 8-dimensional nowhere vanishing Majorana spinor on X induces a nowhere vanishingMajorana spinor on Y and this is known to imply a reduction of the structure group of Y to Spin (7). Therefore it is quite natural to work with this auxiliary 9-dimensional manifold whenstudying the supersymmetry properties of the background. In this paper we are interestedin N = 2 vacua and so require two covariantly constant, and therefore nowhere vanishing,Majorana spinors. The requirement of Majorana spinors rather than Majorana-Weyl spinorsleads to the most general 8-dimensional backgrounds that preserves N = 2 in three dimensionsfrom M-theory, in this sense they are the analogs of SU (3) × SU (3) structure 6-dimensionalbackgrounds in IIB. As with the case of one spinor, the nowhere vanishing Majorana spinorsdo not induce a reduction of the structure group in 8 dimensions. It is possible to consideragain an uplift to Y , but although the structure group is reduced to at least Spin (7), there isin general no further reduction on Y due to the second spinor. Nonetheless, the 9-dimensionalapproach is useful for treating the 8-dimensional local structure groups in a unified way and wewill utilise it in this work.So far we have discussed only the geometry part of the compactification and not the energy-momentum tensor that sources it. In this work we will study the background flux that cansource M2-brane potentials. In relation to the previous discussion of physics sources for a D3-potential in IIB, this flux can be thought of either as non-trivial background flux or, in thespirit of [24, 25], as flux that is accounting for the backreaction of localised sources. We willbe able to give the form of the most general flux that generates an M2-potential in terms of8-dimensional SU (3)-structure geometric objects and two real one-forms that parameterise theflux. For the simple case where only this type of flux and four-form flux with one leg in theinternal directions, which we henceforth refer to as 1-form flux, are present the supersymmetryequations simplify considerably and we are able to present them as differential relations onthe SU (3)-structure forms and extract some key properties. We find that for this limited fluxconfiguration the compactification must be to 3-dimensional Minkowski space. If we furtherimpose the vanishing of the 1-form flux we find that the M2-potential is only along the directionsparameterised by the two singlet vectors of the 8-dimensional SU (3)-structure. The torsionclasses are such that on the generic SU (3)-structure locus the manifold can be described asa 6-dimensional Calabi-Yau fibered over a 2-dimensional base which is spanned by the singletvectors, and over which the M2-branes have a non-trivial potential. While over the special SU (4) and G -structure loci the flux vanishes. The more involved background where we alsoallow for non-vanishing 1-form flux leads to a similar configuration but the 6-dimensional fibre isnot Calabi-Yau but has non-vanishing torsion classes (which satisfy the relation 2 W = − W ).We will also present the supersymmetry equations in differential form for the most generalflux configurations that have other, non M2-potential inducing, fluxes turned on. Though thissubstantially more complicated system is difficult to analyse in as much detail.The outline of the paper is as follows. In section 2 we study the geometric properties of thebackground using G-structures. In particular we describe 8-dimensional manifolds with varyingstructure groups and M2-potentials. In section 3 we study the supersymmetry equations and4ormulate them in a way compatible with the 9-dimensional geometry. In particular we identifythe flux which is responsible for the M2-potentials, and parameterise it in terms of SU (3)-structure objects. In section 4 we study the implication of the supersymmetry equations forbackgrounds supporting such a flux. We summarise our results in section 5. Note 1:
The results presented in this paper rely on quite lengthy calculations. Althoughmost of these calculations can be done by hand we made extensive use of symbolic calculationapplications which are able to manipulate tensors and/or gamma matrices, in order to checkand derive some of our results. We acknowledge the use of the following resources: Mathematica[28], MathTensor [29], Cadabra, [30, 31], Gamma [32] and xTensor [33].
Note 2:
This paper has some overlap with a project which was initiated together with M. Ba-balic, I. Coman and C. Lazaroiu to whom we thank for insights in the subject of M-theorycompactifications to three dimensions.
The effective action of M-theory is described by eleven dimensional supergravity consistingof the following fields: metric g MN , three-form potential C with corresponding field strength G = dC and gravitino Ψ M . The action can then be written in the following way [34] S = 12 Z d x √− g (cid:18) R − G ∧ ∗ G − C ∧ G ∧ G (cid:19) . (2.1)We shall be interested in supersymmetric flux backgrounds. They correspond to fluxes forwhich the background gravitino vanishes together with its supersymmetry variation with 11-dimensional (Majorana) spinor parameter ǫδ Ψ M = ∇ M ǫ − (cid:0) Γ M NP QR − δ NM Γ P QR (cid:1) G NP QR ǫ = 0 . (2.2)The matrices Γ M are taken to satisfy the eleven-dimensional Clifford algebra with metric sig-nature ( − , + , ..., +).The supersymmetry equations should be supplemented by the Bianchi identity and equa-tions of motion for G which read dG = 0 , (2.3) d ⋆ G = − G ∧ G + 2 πT X . (2.4)The last term in (2.4) corresponds to a higher derivative gravitation correction [38], where T is the M5-brane tension and X is a known combination of the first and second Pontrjagin It is usually the case that the equations of motion and the supersymmetry equations imply the Einsteinequations, though we will not prove it here for the particular class of backgrounds under consideration. The compactification Ansatz is chosen by imposing a 3-8 split of the 11 dimensional man-ifold. We choose a metric which is a warped product of 3-dimensional space-time and an8-dimensional Euclidean manifold. ds = e ( ds , + ds ) . (2.5)The 11-dimensional index M decomposes into an external 3-dimensional index µ = 1 , ,
3, andan internal 8-dimensional index α = 1 , ..,
8. For the 4-form field strength G we choose the mostgeneral ansatz compatible with Lorentz invariance G = e (cid:16) ˜ f ∧ Vol + F (cid:17) , (2.6)where ˜ f is a one-form and F is a 4-form on the internal manifold, while Vol is the volumeelement of the external space-time.The eleven-dimensional Clifford algebra is decomposed according to the following equationsΓ µ = e ∆ ( γ µ ⊗ γ ) , Γ α = e ∆ ( ⊗ γ α ) , (2.7)with the 2 × { γ µ ; µ = 1 , , } generating the three-dimensional Clifford algebra Cl (2 , × γ α are taken to be real and symmetric. They generate the eight-dimensional Cliffordalgebra Cl (8 , ǫ , which is an 11-dimensionalMajorana spinor, according to the 3-8 split as ǫ = e − ∆2 η ⊗ ξ , (2.8)where η is a 3-dimensional Majorana spinor, while ξ is an 8-dimensional Majorana spinor.Each non-vanishing spinor ξ defines by the relation above one spinor η in three dimensions.Therefore for N = 2 supersymmetry we need two spinors ξ i on the internal manifold. Note thatin 8 dimensions there exist Majorana-Weyl spinors. However we do not impose any chiralitycondition on the internal spinors as from the supersymmetry equation it is clear that only theMajorana condition is necessary. Imposing the Weyl property is an additional constraint. Inmost studies of M-theory compactifications to 3 dimensions so far the Majorana-Weyl conditionwas imposed for simplicity but, as emphesised in [35, 27], this is not the most general case. N = 2 supersymmetry In this section we present a detailed description of the manifolds on which we compactifyM-theory. As explained before, such manifolds admit two independent, nowhere vanishing Note that for manifolds with 8-dimensional G -structure the Euler number is forced to vanish [39]. ξ , . We shall see further that the supersymmetry conditions imply that thenorm of these spinors is constant [35] and therefore without loss of generality we shall assumethat the two spinors are orthonormal ξ Ti ξ j = δ ij , i, j = 1 , . (2.9)Since in 8 dimensions the Majorana and Weyl conditions are compatible, we can split the twospinors into spinors of definite chirality ξ i = ( ξ + ) i + ( ξ − ) i . (2.10)However, the Majorana-Weyl components ( ξ ± ) i are no longer required to have constant normand moreover they can even vanish at certain points. The only requirement is the unit normof the spinors ξ i || ( ξ + ) i || + || ( ξ − ) i || = 1 . (2.11)In the case that all Majorana-Weyl components are everywhere non-vanishing we are actuallydealing with an 8-dimensional manifold with SU (3) structure which preserves (a maximum of) N = 4 supersymmetry in 3 dimensions. If some of the Majorana-Weyl components vanishidentically over the entire internal manifold, while the others are non-vanishing, then we aredealing with manifolds with G structure or manifolds with SU (4) structure depending on therelative chirality of the non-vanishing spinors. We see that the fact that the Majorana-Weylcomponents are allowed to vanish at certain points implies that such manifolds do not admita global reduction of the structure group. At generic points they look like SU (3) structuremanifolds, while at special points they look like G or SU (4) structure manifolds. Note that we did not consider the case where only one of the four Majorana-Weyl compo-nents vanishes. This is because on an 8-dimensional manifold once we are given three linearlyindependent, non-vanishing, Majorana–Weyl spinors which are not of the same chirality, onecan define a fourth spinor such that we end up with two spinors of one chirality and two ofthe other. To prove this, suppose that we have an eight-dimensional manifold which has twospinors, ξ and ξ of positive chirality and one, χ of negative chirality. The two spinors ofpositive chirality define a SU (4) structure and therefore one finds an almost complex structurewhich is given by J αβ = ξ T γ αβ ξ (2.12)We can define a fourth spinor χ which is of negative chirality χ = J αβ γ αβ χ , (2.13)and which has non-vanishing norm. This spinor is obviously orthogonal to ξ , and because thematrix γ αβ is antisymmetric in its spinorial indices, it is also orthogonal to χ . Therefore, inthis way we have 4 non-vanishing Majorana-Weyl spinors, two of positive and two of negativechirality, which define a SU (3) structure in 8 dimensions. It would be interesting to explore connections between these backgrounds and SU (4) × SU (4) backgroundsas studied in [40, 41]. .2.1 9-dimensional uplifts to Y = X × S It was pointed out in [27] that we can associate the existence of a nowhere vanishing Majoranaspinor on X to a reduced structure group by uplifting it to an auxiliary 9-dimensional manifold, Y , defined as the direct product of the 8-dimensional manifold under consideration X and acircle Y = X × S . For the case of 2 Majorana spinors this procedure is not as effectivesince, as we will show, the structure group does not reduce further on Y . However, uplifting to9-dimensions is still a useful procedure because it will allow us to describe the local structuregroups of X in a unified way. In particular the 8-dimensional structure groups, and the relatedM2 potential, will be mapped to an angle between two 9-dimensional vectors.We therefore go on to study 9-dimensional manifolds supporting two nowhere vanishingMajorana spinors. It is most common to study such manifolds from the perspective of spinorbilinears which can be constructed out of the spinors. Since 9-dimensional Euclidean gammamatrices can be chosen real (and therefore symmetric) the only spinor bilinears which can bedefined are the following( V ) m = ξ T γ m ξ , ( V ) m = ξ T γ m ξ , ( V ) m = ξ T γ m ξ = ξ T γ m ξ ,K mn = ξ T γ mn ξ = − ξ T γ mn ξ , Ψ mnp = ξ T γ mnp ξ = − ξ T γ mnp ξ , (Φ ) mnpq = ξ T γ mnpq ξ , (Φ ) mnpq = ξ T γ mnpq ξ , (Φ ) mnpq = ξ T γ mnpq ξ . (2.14)Here γ m are 9-dimensional gamma matrices, or generators of Cl (9 , m = 1 , ...,
9. Theseforms are not independent as certain products of such bilinears can be expressed in terms oflinear combinations of these bilinears. The complete set of relations which the forms abovesatisfy is given in appendix A.For an efficient description of such manifolds it is important to know the number of inde-pendent vectors. The forms defined above include three vectors. The Fierz relations for thevectors imply || V || = || V || = 1 , || V || = 12 (1 − α ) , (2.15) V · V ≡ α , V · V = V · V = 0 . (2.16)Here we define the usual contraction V · U = V m U m . Therefore the number of independentvectors is governed by a real parameter α which is the scalar product of the vectors V and V .Since the vectors V and V are of unit norm, this parameter α is in fact the cosine of the anglebetween these vectors and therefore can take values in the interval [ − , SU (3) structure. We denote this a ‘local structure’ because α varies over X and so particular values of it define certain submanifolds. Of course the notion of structuregroup only has a global meaning. However the terminology of a local structure is well definedby the properties of the spinors over the submanifold and will be used extensively in this work.For α = − V and V are no longer independent (they are anti-parallel) while V has unit norm. Therefore we are left with two independent unit vectors. This is the case ofa G structure. Finally, if α = 1, the vectors V and V are parallel while V vanishes. Thisis the case of an SU (4) structure. These local 9-dimensional structure groups dictated by theparameter α are directly inherited by the 8-dimensional manifold. In particular the physics we8re interested in, which is the potential for a probe space-filling M2-brane, is therefore directlyrelated to the variation of α over the internal manifold. We will show this in more detail insection 3.3.Before going into more details about these manifolds it will be useful to understand thespecial cases of α = ± G structure: α = − V = − V = V , || V || = || V || = 1 , (2.17) K = V ∧ V , (2.18) V y Ψ = V y Ψ = 0 , (2.19)Φ = − V ∧ Ψ − ∗ ( V ∧ V ∧ Ψ) , (2.20)Φ = V ∧ Ψ − ∗ ( V ∧ V ∧ Ψ) , (2.21)Φ = V ∧ Ψ . (2.22)This description looks much like a G structure in 7-dimensions (which is given in terms ofthe 3-form Ψ) and two additional vectors which lift this G structure to 9-dimensions. This isexpected from the decomposition of the fundamental of SO (9) under G → ⊕ ⊕ . (2.23)A linear combination of these vectors gives the direction along the auxiliary circle used to upliftto 9-dimensions. SU (4) -structure: α = +1The Fierz relations for the case α = 1 read V = V = V , || V || = 1 , || V || = 0 , (2.24)Ψ = K ∧ V , Φ + = − K ∧ K , K m [ n (Φ − ) mpq ] = 2 (Φ ) npq , (2.25)where we define Φ ± = Φ ± Φ . Moreover it can be shown that when restricted to the subspacewhich is orthogonal to V , K is an almost complex structure and it is clear that we can organisethis orthogonal space as a space of SU(4) structure with Φ − and 2Φ playing the role of the realand imaginary parts of the complex four-form. The additional vector field should be understoodas the direction which we added to go to the auxiliary nine-dimensional manifold. SU (3) -structure The most interesting case is of when the angle α , between the vectors V and V varies. Thisoccurs over the generic patch on the manifold which manifests a local SU (3)-structure. Overthis locus we have that α = ±
1, and we will assume this in our present analysis and return9o the limit points later. In dealing with the more complicated SU (3)-structure case we areguided by the idea that, analogous to the SU (4) and G -structure loci, we expect to be ableto describe it in terms of an (uplifted) 6-dimensional SU (3)-structure. In order to unveil thisstructure let us first define V ± = V ± V . (2.26)Clearly V + and V − are mutually orthogonal and, since both V and V are orthogonal to V ,they are also orthogonal to V . However, these vectors no longer have unit norm but we find || V ± || = 2(1 ± α ) . (2.27)The next step is to decompose all the forms in (2.14) into forms of lower or equal rank formswhich are orthogonal to these vectors. Let us use as an example the decomposition of K . Wewrite K = J + aV + ∧ V − + bV + ∧ V + cV − ∧ V , (2.28)where a , b and c are coefficients which should be derived from imposing that J satisfies V ± y J = V y J = 0. From the Fierz relations with one free index we see that V + is already orthogonal to K and therefore a = b = 0. In order to find the coefficient c we contract with V − and use theFierz relations with one free index and obtain K = J + 11 − α V − ∧ V . (2.29)The other cases work in a similar way. However, when considering forms of higher rank, thenumber of terms that can be written on the right hand side increases rendering the calculationrather tedious. We find, eventually, the following equationsΨ = ϕ + 11 + α J ∧ V + + 12(1 − α ) V + ∧ V − ∧ V , (2.30)Φ + = −
21 + α J ∧ J −
21 + α ρ ∧ V + − − α J ∧ V − ∧ V , (2.31)Φ − = 41 − α ϕ ∧ V + 21 − α ρ ∧ V − + 21 + α J ∧ V + ∧ V , (2.32)Φ = − − α ϕ ∧ V − + 21 − α ρ ∧ V − α ) J ∧ V + ∧ V − . (2.33)In the above the three-form ρ is not independent, but can be expressed as ρ mnp = J rm ϕ npr . (2.34)Using the Fierz relation which involves K and Ψ one can check that the RHS of the aboveequation is indeed antisymmetric in all three indices as should be for the components of a3-form (see eq. (B.2)). Note also that ρ and ϕ are orthogonal to the vectors V i .Using the Fierz relations in appendix A it is not very hard to check the parametrisationabove. The terms which contain at least one vector field can be immediately verified by project-ing the entire relation on the corresponding vector and using the fact that vectors V ± and V areorthogonal. The only remaining problem is to determine the top forms which are orthogonalto the vectors. There are two different cases above. In (2.30), we denote this top form by ϕ and we shall further discuss its properties. In the remaining relations, the top form is no longer10n independent form, but is given in terms of J , as in (2.31), or it simply vanishes as in (2.32)and (2.33). The way to decide whether or not one can write additional terms in (2.31)–(2.33)is by computing the norms of the RHS and LHS of these relations. It is not difficult to checkthat these norms precisely agree, and therefore if we were to add some arbitrary form to theserelations, such forms would automatically have zero norm and thus vanish on an Euclideanspace.Naively the equations (2.29)–(2.33) diverge at the special points α = ±
1. To show thatactually there are no divergences in the SU (3) parameterisation in the SU (4) or G loci limits,we can extract the leading behaviour with respect to (1 ± α ) of the relevant SU (3)-structureforms V ∼ (1 − α ) , V − ∼ (1 − α ) , V + ∼ (1 + α ) ,ρ ∼ [(1 − α ) (1 + α )] , ϕ ∼ (1 − α ) , J ∼ (1 + α ) . (2.35)These relations can be inferred from the norms of these objects which we compute in appendix.The objects that define the geometry are the 9-dimensional bilinears. These are perfectlysmooth over the full range of α , though they take particular, different, forms over the SU (3), SU (4) and G loci.Note that apart from J , ϕ and ρ , which satisfy (2.34), no other form orthogonal to thevectors appears. We can in principle use in the parametrisation only ϕ and J , but it is moreintuitive to consider also ρ as these forms are used in general to describe a SU(3) structure.Indeed, the results obtained agree precisely with what is expected from a SU (3)-structure insix dimensions plus three additional directions orthogonal to it. However, in order to establishthe SU (3)-structure behind we still have to check a few more relations which the forms J and ϕ satisfy. Using the symmetric relation for K given in eq. (A.67) of the appendix we can computethe similar identity for JJ mn J np = −
12 (1 + α ) δ mp + 14 ( V + ) m ( V + ) p + 1 + α − α ) ( V − ) m ( V − ) p + 1 + α − α ( V ) m ( V ) p = 12 (1 + α ) [ − δ mp + ( P + ) mp + ( P − ) mp + ( P ) mp ] , (2.36)where P ± , denote the projectors on the directions +, − and 3 respectively. It follows that byan appropriate normalisation J can be indeed viewed as the almost complex structure on the6-dimensional subspace orthogonal to V ± and V .Making use of the Fierz identities listed in the appendix and of the eqs. (2.29)–(2.33) onecan show that the following identities must hold J y ϕ = J y ρ = 0 , (2.37) J ∧ ϕ = J ∧ ρ = 0 , (2.38) ϕ ∧ ρ = ∗ ( V + ∧ V − ∧ V ) , (2.39) J ∧ J ∧ J = 3(1 + α )2(1 − α ) ∗ ( V + ∧ V − ∧ V ) . (2.40)Furthermore, it is possible to show that eqs. (2.29)–(2.40) together with eq. (B.28) imply allthe Fierz identities for the bilinear forms listed in the appendix. By construction, the converseis also true. 11t is useful to construct from ϕ and ρ the following holomorphic and anti-holomorphic (withrespect J ) three-forms Ω = ϕ + i q α ρ , ¯Ω = ϕ − i q α ρ . (2.41)Indeed, it is easy to see that these forms obey the relations J mn Ω npq = i q α Ω mpq , J mn ¯Ω npq = − i q α ¯Ω mpq , (2.42)and therefore, up to some normalisation, Ω can be seen as a (3 ,
0) form with respect to thealmost complex structure J . It should now be clear that following a suitable normalisation theforms J and Ω (or its real components ϕ and ρ ) can be seen as the forms defining an SU (3)-structure on the space orthogonal to the vectors V ± and V . Note that the normalisation wehave depends on the parameter α and care must be taken over the α = ± SU (4) or G structures is more suitable. The G-structure technology introduced in the previous section is ideal for studying the su-persymmetry equations (2.2). In this section we rewrite the supersymmetry constraints asdifferential constraints on the appropriate forms and use these to extract general properties ofany solutions, in particular with respect to supporting a background with varying α .The supersymmetry variation of the gravitino (2.2) splits into internal and external compo-nents which read [35, 36, 37] D α ξ = ∇ α ξ + A α ξ = 0 , Qξ = 0 , (3.1)where we defined A α = λγ α γ + 124 F αβγδ γ βγδ + 14 ˜ f β γ αβ γ , (3.2) Q = − λγ + 12 ∂ α ∆ γ α − F αβγδ γ αβγδ −
16 ˜ f α γ α γ , (3.3)and the covariant derivative ∇ α is taken now with respect to the 8-dimensional metric definedin (2.5). The parameter λ is the cosmological constant for the 3-dimensional external space.These equations are valid for spinors ξ which live on an 8-dimensional manifold. As discussedpreviously we are interested in studying the background on a 9-dimensional manifold Y = X × S . To do this we uplift on a circle by adding a ninth direction so that now we have anindex range m = 1 , ...,
9. We uplift the gamma matrices by considering γ together with theother gamma matrices γ α , as the generators of the Clifford algebra Cl (9 , SO (9) covariant way.We therefore introduce a constant vector field θ such that γ = θ m γ m , m = 1 , . . . , . (3.4)12ith this the supersymmetry equations (3.1) have the same form where now A m = λθ n γ mn + 124 F mnpq γ npq + 14 ˜ f n θ p γ mnp , (3.5) Q = − λθ m γ m + 12 ∂ n ∆ γ n − F mnpq γ mnpq −
16 ˜ f m θ n γ mn , (3.6)and we must impose the independence of physics quantities on the ninth direction θ · ˜ f = 0 , θ y F = 0 , θ · d ∆ = 0 . (3.7)These equations now hold after an arbitrary SO (9) rotation which no longer identifies θ withthe index value m = 9.Note the following simple consequences of the above. Firstly, the restrictions on the fluxesimply that one component of A , i.e. θ m A m , identically vanishes. This means that also thespinor ξ does not depend on this direction. Secondly, since we chose the 8-dimensional gammamatrices to be real and symmetric (this also holds for γ ), A m is a totally antisymmetric matrix.Contracting (3.1) with ξ T from the left we find0 = ξ T ∇ m ξ = ∇ m ( ξ T ξ ) = ∂ m || ξ || , (3.8)and therefore the supersymmetry equations impose that the internal spinors must have constantnorms as we already anticipated in the previous section.The auxiliary uplift direction should be given by a linear combination of the 9-dimensionalvectors. This can be seen from the analysis of the various particular cases. In particular forthe SU (4) case, in 8-dimensions we expect no singlet vector field and so the vector field whichsurvives should be interpreted as the additional ninth direction. In the G case in 8 dimensions,we expect a single vector field which is a singlet under the structure group. In 9 dimensionswe found two such vector fields and therefore a certain linear combination of them should beinterpreted as the additional direction. Finally, in the case of a SU (3) structure we expect twosinglet vector fields in eight dimensions, while we found three of them in 9 dimensions. Again,one combination of them should precisely give the additional direction. Thus we can write θ = α ) ( θ · V + ) V + + − α ) ( θ · V − ) V − + − α ( θ · V ) V . (3.9)Finally it is worth noting the practical matter that uplifting to 9-dimensions does not addany further complexity to the equations. This is because we are working with Majorana, ratherthan Majorana-Weyl, spinors in 8 dimensions which implies that they do not have any niceproperties under γ . Therefore including it in a higher dimensional Clifford algebra is useful,in particular because the complete set of gamma matrices means the Hodge star acts simplyallowing us to write all the bilinears in terms of forms of degree 4 or less. N = 2 supersymmetry in 3D For N = 2 supersymmetry in 3 dimensions, the supersymmetry equations (3.1) have to besatisfied by two spinors on the internal manifold, ξ , . In order to see what sort of constraintsthese equations impose on the 9-dimensional manifolds discussed in the previous section we13hould first rewrite them in terms of the spinor bilinears (2.14). Using the definitions (2.14)and (3.1) we can compute the derivatives of all the spinor bilinears to find [37] ∇ m V i n = 2 λθ n V i m − λ ( θ · V i ) δ mn − F mpqr Φ i npqr + 12 Φ i mnpq ˜ f p θ q (3.10) ∇ m K np = − λK m [ n θ p ] + 4 δ m [ n K p ] q θ q + 12 F m [ nqr Ψ p ] qr + Ψ m [ nq θ p ] ˜ f q − Ψ m [ nq ˜ f p ] θ q + δ m [ n Ψ p ] qr ˜ f q θ r (3.11) ∇ m Ψ npr = 6 λ Ψ m [ np θ r ] − λδ m [ n Ψ pr ] q θ q + 112 F mstu ( ∗ Ψ) nprstu + 32 F m [ npq K r ] q −
12 ( ∗ Ψ) mnprst ˜ f s θ t − K m [ n ˜ f p θ r ] (3.12)+ 3 δ m [ n ˜ f p K r ] q θ q − δ m [ n θ p K r ] q ˜ f q , ∇ m Φ i npqr = − λ Φ i m [ npq θ r ] + 8 λδ m [ n Φ i pqr ] s θ s + F m [ nst ( ∗ Φ i ) pqr ] st − F m [ npq V i r ] + 2( ∗ Φ i ) m [ npqs θ r ] ˜ f s − ∗ Φ i ) m [ npqs ˜ f r ] θ s (3.13)+ 2 δ m [ n ( ∗ Φ i ) pqr ] st ˜ f s θ t − δ m [ n V i p ˜ f q θ r ] , where the subscript i stands for any of 1 , , , + or − . It is interesting to note that the equationsabove imply that the derivatives along θ vanish identically upon using the condition on the fluxes(3.7).We should now consider the constraints which come from the external gravitino variation.These equations should be projected on a basis of spinors in order to find an equivalent setof equations. In 8 dimensions it is easy to find a basis of spinors in terms of Majorana-Weylsinglet spinors, but since the Majorana-Weyl components of the spinors we consider may vanishat certain points, we will not be able to use a basis constructed in such a way globally. Instead weshall project the supersymmetry equations on a larger set of spinors – not linearly independent– which are constructed by multiplying the spinors ξ , by arbitrary elements of the Cliffordalgebra. Specifically we shall project the spinor equations (3.1) on spinors of the form ξ i A = γ A ξ i , i = 1 , , γ A ∈ Cl (9 ,
0) (3.14)In practice we shall see that taking γ A = , γ m is enough and the other constraints are justconsequences of these ones. Therefore we shall use the following equations − λ ( θ · V i ) + 12 ( V i · d ∆) − F y Φ i = 0 (3.15)˜ f m θ n K mn = 0 (3.16) − λθ + d ∆ − ∗ ( F ∧ Φ + ) + 16 V + y ( ˜ f ∧ θ ) = 0 (3.17) − ∗ ( F ∧ Φ − , ) + 16 V − , y ( ˜ f ∧ θ ) = 0 (3.18)2 λθ y K − d ∆ y K + 16 Ψ y F −
13 ( ˜ f ∧ θ ) y Ψ = 0 (3.19)while the ones corresponding to other Clifford elements can be found in appendix C.14 .2 Flux induced variations of α The key property of the backgrounds that we are interested in is the variation of the parameter α , since this is the property that signals an M2-brane potential. We will explain this relation-ship in more detail in section 3.3. In this section we are interested in determining which flux isresponsible for inducing such a variation. In general not any flux will induce this variation, forexample we know that (2 ,
2) primitive flux as studied in the case of SU (4)-structure compact-ifications [7] will not induce such a variation. We will determine the flux which is relevant byusing the supersymmetry equations to directly evaluate the variation of α .The form of the supersymmetry equations as given in section 3.1 carries redundancies be-cause the bilinears (2.14) are not independent, but satisfy various relations coming from theFierz relations. In order to find a more tractable set of equations we should use a parametrisationfor the forms (2.14) which already makes use of the Fierz relations. A particular parameterisa-tion is the one given in (2.29)–(2.33) which is valid for the SU (3)-structure loci. We claim thatthis parameterisation is sufficient to capture all the variation of α . This follows from the simplereasoning that over the patches where it breaks down, the SU (4) and G loci, α is constant bydefinition. Further it is also at a maximum or minimum value over these loci so that derivativesalong directions leading away from the loci also vanish. Therefore over such loci dα = 0 andall the variation is within the SU (3)-structure locus.Using the supersymmetry equations (3.10), the variation of α can be generally computed as ∇ m α = 12 ∇ ( V n + V + n ) = V n + ∇ m V + n = − V n + F mpqr Φ + npqr + 12 Φ + mnpq ˜ f p θ q V n + . (3.20)Making use of the SU (3) parametrisation of the form Φ + , (2.31), and of the fact that the forms J , ρ , V − and V are all orthogonal to V + , together with the fact that θ should be a combinationof the vector fields V ± and V , we find that the second term above does not contribute to thevariation of the parameter α and so ∇ m α = − ρ pqr F mpqr . (3.21)Given again all the orthogonality properties of the SU (3)-structure forms we find that the fluxwhich is responsible for the variation of α can be written as F = ˜ h ∧ ρ + ˜ g ∧ ϕ , (3.22)where at this stage ˜ h and ˜ g are arbitrary one-forms on the internal manifold that parameterisethe flux.Note that ρ and ϕ are (depending on the conventions) the real and imaginary part of theholomorphic 3-form which can be defined on a manifold with SU (3)-structure. Over the SU (4)locus the flux therefore lifts to either (4 , ,
4) or (3 , ,
3) flux. Note that both thesefluxes are forbidden by supersymmetry in the case of pure SU (4)-structure where only (2,2)primitive fluxes are allowed [7], which is consistent with the understanding that the variationof α vanishes over that locus. At this stage it is worth going into some more detail regarding the relation between a potentialfor probe space-time filling M2-branes and the background geometry. In type IIB compactified15n manifolds with SU (3) × SU (3) structure, we know that space-time-filling D3 branes becomesupersymmetric at points where the manifold locally looks like a manifold with SU(3) structure(i.e. the two spinors defining the SU (3) × SU (3) structure are parallel) [21, 22]. In this sectionwe will find similar results for M2-branes in terms of local structure groups.Let us briefly recall some well-known facts about supersymmetry and M2 branes. We willutilise the form of the M2-brane action given in [44] S M = − T M Z d ζ √− G − T M Z d ζǫ ijk A kji + i T M Z d ζ √− G ¯ y (1 − Γ M )Γ i ˜ D i y , (3.23)where G ij is the induced metric on the brane, T M is the tension of the brane, A ijk is thepull-back of the supergravity 3-form on the brane, ˜ D i is the pull-back of the supercovariantderivative and y is an 11-dimensional Majorana spinor. Γ M is the brane chirality operatorwhich is given by Γ M = 16 √− G ǫ ijk Γ i Γ j Γ k . (3.24)Note that Γ M = 1 so that its eigenvalues are ± P ± = 12 (1 ± Γ M ) . (3.25)The action (3.23) is invariant under local κ transformations [44] δ κ y = (1 + Γ M ) κ + O ( y ) , δ κ X M = i y Γ M (1 + Γ M ) κ + O ( y ) (3.26)where κ is a 32-component spinor which may depend on the coordinates on the worldvolumeof the M2 brane, ζ i .For a background with Killing spinor ǫ satisfying (2.2), supersymmetry transformations actas δ ǫ y = ǫ + O ( y ) , δ ǫ X M = − i y Γ M ǫ + O ( y ) . (3.27)A bosonic brane configuration ( y = 0) is supersymmetric only if δ ǫ y = 0 and we see thatthis can not be satisfied unless ǫ = 0. However δ ǫ y = ǫ is compatible with supersymmetryonly if this transformation can be compensated by a κ transformation [26]. Therefore the onlysupersymmetry generators which are not broken by the bosonic brane configuration are thosewhich can be written as ǫ = δ κ y = (1 + Γ M ) κ = 2 P + κ . (3.28)This is equivalent to requiring [26] P − ǫ = (1 − Γ M ) ǫ = 0 . (3.29)For the compactification to 3 dimensions with space-time-filling M2-branes, the M2 brane chi-rality operator decomposes according to the the split of gamma matrices (2.7), asΓ M = ⊗ γ (3.30)where γ is the chirality operator on the 8d manifold. Using the spinor decomposition for thecompactification to 3 dimensions (2.8) we find we find that the condition above is equivalent to γ ξ = ξ . (3.31)16e see that the condition that the M2 brane is supersymmetric requires Killing spinors ofdefinite chirality.With this result in mind consider probe M2 branes in our supergravity background atthe different local SU (3), SU (4) or G -structure loci. Generally we have that there are 2Majorana Killing spinors. Over G loci, α = −
1, these become Majorana-Weyl Killing spinorsof opposite chirality. Therefore M2 branes preserve N = 1 supersymmetry on these loci.Over SU (4) loci, α = +1, the Killing spinors are Majorana-Weyl of same chirality and soM2 branes either preserve N = 2 supersymmetry or no supersymmetry. However, for a givenfixed chirality of the two Majorana-Weyl spinors either M2 or anti-M2 branes preserve N = 2supersymmetry, and we define M2 branes as the objects which preserve the supersymmetriesover SU (4) loci. Over the SU (3) loci, − < α <
1, we have two Killling Majorana spinorswhich contain 4 non-vanishing Majorana-Weyl component spinors. We have three possibilities:either all of the Majorana-Weyl components are Killing individually, in which case the M2branes preserves N = 2 supersymmetry, or two Majorana-Weyl components of one Majoranaspinor are Killing while the components of the other are not, in which case the M2 preserves N = 1 supersymmetry, or none of the components are Killing in which case an M2 brane isnon-supersymmetric. In appendix D we show that the flux which induces a varying α (3.22)precisely implies that the last possibility is realised and M2 branes preserve no supersymmetry.Indeed it is rather simple to see that this should be the case for any background which does nothave a global SU (3) or G -structure since if any Majorana-Weyl components were covariantlyconstant by themselves their norm could be set to unity over the full manifold, thereby implyingthat they are nowhere vanishing and induce a global SU (3) or G -structure.On general grounds we expect that supersymmetric loci are minima of the potential. Thismeans that on the generic (non-supersymmetric) locus a probe M2 brane should feel a potentialwhich drives it to a supersymmetric locus.Let us be more explicit about the condition for N = 2 supersymmetry of the M2 brane. Thecondition (3.31) has to be satisfied for two spinors ξ and ξ , which means that the two spinorsare actually Majorana–Weyl of positive chirality and therefore define a SU(4) structure. In thelanguage used in section 2.2 this means that the vectors V and V are equal and therefore V − = 0 . (3.32)Conversely, the condition V − = 0 implies that we are dealing with a SU(4) point and thereforethe two spinors are Majorana–Weyl. Finally, it is interesting to note that, in the spirit of theanalysis in [21, 22] of D3 superpotentials, the condition (3.32) hints that the 1-form V − may beassociated to the derivative of a world-volume potential. In the previous section we identified the particular flux that sources the variation of α . In thissection we study backgrounds that can support this flux. As an initial investigation we restrict Note that there exists the interesting possibility of multiple SU (4) loci with different chiralities in which caseneither M2 or anti-M2 can preserve supersymmetry on all of loci. We will not consider such configurations indetail in this work. In principle the chirality of the spinors can not be determined. However, if this is negative, and therefore,the condition (3.31) is not satisfied, this would be a point where anti M2 branes are supersymmetric. F to be solely composed of the flux of interest so that it takes the form (3.22),while the warp-factor ∆ and 1-form flux f are unconstrained. We leave a complete investigationallowing also for other types of 4-form fluxes in the background for future work.It is useful to decompose the 4-form flux along the three vectors as F = h ∧ ρ + g ∧ ϕ + α ) h + V + ∧ ρ + − α ) h − V − ∧ ρ + − α ) h V ∧ ρ + α ) g + V + ∧ ϕ + − α ) g − V − ∧ ϕ + − α ) g V ∧ ϕ . (4.1)In the above we defined h i = V i · ˜ h ≡ V mi ˜ h m , g i = V i · ˜ g ≡ V mi ˜ g m , i = ± , , (4.2)while g and h are defined as the components of ˜ g and ˜ h orthogonal to the vectors V ± and V .We therefore decompose the fluxes h , g and ˜ f as˜ h = h + 12(1 + α ) h + V + + 12(1 − α ) h − V − + 2(1 − α ) h V , (4.3)˜ g = g + 12(1 + α ) g + V + + 12(1 − α ) g − V − + 2(1 − α ) g V , (4.4)˜ f = f + 12(1 + α ) f + V + + 12(1 − α ) f − V − + 2(1 − α ) f V , (4.5)where f , f ± and f are defined in analogy with g and h by their relation to the vectors. Notethat there are no divergences at α = ± θ is a linearcombination of the vectors, this condition imposes the orthogonality of the one-form fluxes ˜ h ,˜ g and ˜ f on θ θ y ˜ h = θ y ˜ g = θ y ˜ f = 0 . (4.6)With these definitions, and using the various relations in appendix B, we find that (3.21)yields dα = − (1 − α )(1 + α ) h − − α ) g y J − (1 − α ) h + V + − (1 + α ) h − V − − α ) h V (4.7)Note that, as expected, dα = 0 on the SU (4) and G loci.To analyse the supersymmetry conditions for this particular flux Ansatz we shall start withthe constraints (3.15)-(3.19). Inserting the SU(3) parametrisation (2.30)-(2.33) and the flux18nsatz above, by using the relations in appendix B we find0 = − λ ( θ · V + ) + 12 ( V + · d ∆) −
16 (1 − α ) h + , (4.8a)0 = − λ ( θ · V − ) + 12 ( V − · d ∆) + 16 (1 + α ) h − + 23 g , (4.8b)0 = − λ ( θ · V ) + 12 ( V · d ∆) + 16 (1 + α ) h − g − , (4.8c)0 =( θ · V ) f − − f ( θ · V − ) , (4.8d)0 = d ∆ − λθ −
13 ( g − V − g V − ) + 16 f + θ −
16 ( θ · V + ) ˜ f , (4.8e)0 = − h + V − + h − V + − g + V + 2 g V + + f − θ − ( θ · V − ) ˜ f , (4.8f)0 = − h + V + 2 h V + + g + V − − g − V + + 2 f θ − θ · V ) ˜ f , (4.8g)0 = 2 λ − α ( θ · V − ) V − λ − α ( θ · V ) V − − d ∆ y J − − α ( d ∆ · V − ) V + 11 − α ( d ∆ · V ) V − + 16 (1 − α )˜ h y J −
16 (1 − α )˜ g −
112 1 − α α g + V + − g V − g − V − + ( θ · V + )3(1 + α ) ˜ f y J − − α ) ( ˜ f ∧ θ ) y ( V + ∧ V − ∧ V ) . (4.8h)Let us look more carefully at equation (4.8f). Contracting it with V − we immediately find that h + = 0. By contracting with V and using (4.8d) we find that g + = 0. Furthermore, projectingequations (4.8f) and (4.8g) on V + and on the 6-dimensional space orthogonal to the vectors V ± and V one obtains the following equations g = − h − − α ) f − ( θ · V + ) + 14(1 + α ) ( θ · V − ) f + , (4.9) g − = 2 h + 11 + α f ( θ · V + ) −
11 + α ( θ · V ) f + , (4.10)and ( θ · V − ) f = 0 , ( θ · V ) f = 0 . (4.11)Now, using (3.9), the relations coming from the orthogonality of θ on the fluxes ˜ f and ˜ g become h − ( θ · V − ) + 4 h ( θ · V ) = 0 , (4.12) h ( θ · V − ) − h − ( θ · V ) = 0 , (4.13)where we used (4.9), (4.10) and (4.8d). These equations can be viewed as a linear system for h − and h which has a non-trivial solution only if the corresponding determinant vanishes.Therefore, there are two cases to consider. One of them must have θ · V − = θ · V = 0 and theother one has to satisfy h − = h = 0 and, following (4.11), f = 0. We shall focus on the firstsolutions as we want to study loci with dα = 0. Indeed, it is easy to show that in the lattercase one must have that dα = 0. In order to prove it one needs also the identity g y J + 12 (1 + α ) h = 0 , (4.14)19esulting from eq. (4.8h) after contraction with J .Let us continue with the case θ · V − = θ · V = 0. Making use of eq. (3.9) we obtain that θ has to be in the direction of V + . We can therefore write θ = ( θ · V + )2(1 + α ) V + , (4.15)and the fact that the physics should not depend on θ now transfers to V + . Notice that this isconsistent with the expectation that the flux should vanish over G structure loci since in thiscase we have V + = 0 but the auxiliary direction θ must be non-vanishing.Together with equation (4.8a), (4.15) immediately implies that λ = 0, and therefore all suchcompactifications are to 3-dimensional Minkowski space. Furthermore, equations (4.8b), (4.8c),(4.8f) and (4.8g) allow to solve for the projections of d ∆ on V − and V in terms of f − and f .Altogether the equations (4.8) become0 = λ = f + = g + = h + = d ∆ · V + , (4.16a) d ∆ · V − = 1 − α h − + θ · V + α ) f − , d ∆ · V = 1 − α h + θ · V + α ) f , (4.16b) g = − h − − θ · V + α ) f − , g − = 2 h + θ · V + α f , (4.16c)0 = d ∆ −
13 ( g − V − g V − ) −
16 ( θ · V + ) ˜ f , (4.16d)0 = − d ∆ y J + 16 (1 − α )( h y J − g ) + ( θ · V + )3(1 + α ) f y J . (4.16e)Contracting (4.16d) with J we find d ∆ y J = ( θ · V + )6 f y J , (4.17)and then (4.16e) becomes g − h y J = ( θ · V + )1 + α f y J . (4.18) ˜ f = 0 At this point we split the analysis to two cases, the simpler case ˜ f = 0 is studied in thissection, while the more general case is studied in section 4.2. For this section we therefore set f = f + = f − = f = 0.Let us analyse more closely equation (4.18). Since J , up to normalisation, acts like analmost complex structure on the 6-dimensional space orthogonal to the vectors V ± and V , thisequation tell us that a particular combination of the fluxes g and h is holomorphic with respectto the almost complex structure J . It is not hard to see that the complex flux defined as h = h + i q α g , ¯ h = h − i q α g (4.19)20s holomorphic in that it obeys J mn h n = i q α h m , J mn ¯ h n = − i q α ¯ h m . (4.20)It is useful to use this flux combination together with objects which again have well-definedbehavior when contracted with J . In particular, we can construct a (4 ,
0) form by taking theexterior product of (2.41) with (4.19). Since J only acts on a 6-dimensional subspace, a (4 , h ∧ Ω = h ∧ ϕ −
21 + α g ∧ ρ + i q α ( h ∧ ρ + g ∧ ϕ ) = 0 . (4.21)Note that the imaginary part is proportional to the projection of the flux F orthogonal to thevectors. The fact that this part of the flux vanishes implies that the variation of α only dependson the projection of the ˜ h and ˜ g fluxes along the vectors V − and V .The equations (4.16) now become d ∆ · V − = 1 − α h − , d ∆ · V = 1 − α h , g = − h − , g − = 2 h , h y J − g = 0 (4.22)with all the rest of the flux components being zero. All the above relations greatly simplify thedifferential equations which now become dα = − (1 + α ) h − V − − α ) h V (4.23) dV + = h − V + ∧ V − + 2 h V + ∧ V , dV − = 2 h V − ∧ V , dV = − h − V − ∧ V , (4.24) dK = − h − J ∧ V − − h J ∧ V , (4.25) d Ψ = 21 + α J ∧ V + ∧ (cid:0) h − V − + h V (cid:1) − − α ρ ∧ ( h − V − h V − )+ 1 − α − α ϕ ∧ (cid:0) h − V − + h V (cid:1) , (4.26)where we made extensive use of the relations (B.33) for k = h which, in the case ˜ f = 0, implies˜ k = g . Note that the exterior derivative of J is the same as the derivative of K above, whilefor the derivative of ϕ we find dϕ = 21 − α ρ ∧ ( h V − − h − V ) + 1 − α − α ϕ ∧ (cid:0) h − V − + h V (cid:1) (4.27)In the above formula the term which is in the direction of V + in d Ψ drops out. This is preciselyas it should be, as ϕ has no components along V + and moreover its derivative along θ (which isidentified with V + in this case) vanishes. With a bit of more work one can compute the exteriorderivative of ρ as well. We find dρ = 3 − α − α ρ ∧ (cid:0) h − V − + h V (cid:1) + 1 + α − α ϕ ∧ ( h − V − h V − ) (4.28)At a first glance it seems that dϕ and dρ do not combine nicely into d Ω, but one has to takeinto account that d Ω contains an additional term of the form dα ∧ ρ due to the α -dependentfactor in front of ρ in the definition of Ω. With this we find d Ω = 1 − α − α Ω ∧ (cid:0) h − V − + h V (cid:1) + i α − α r
21 + α Ω ∧ ( h − V − h V − ) . (4.29)21t is important to notice that the exterior derivatives of the SU (3)-structure forms do not havecomponents strictly orthogonal to the vectors. Therefore one can conclude that the intrinsictorsion classes for the 6-dimensional manifold orthogonal to the vectors vanish and thereforethis is a Calabi–Yau manifold. We conclude that in the case ˜ f = 0 the supersymmetry equationsrequire that, over the SU (3) locus, the 8-dimensional manifold is a fibration of a 6-dimensionalCY manifold over a 2-dimensional base spanned by the vectors V − and V . While over any G or SU (4) loci the flux vanishes.Finally let us note that using the normalisation of V + we can write V + = p α ) θ , (4.30)taking θ to be constant we derive dV + = 12(1 + α ) dα ∧ V + , (4.31)which precisely agree with dV + found in (4.23). This is consistent with the interpretation thatthe auxiliary 9-dimensional manifold is a direct product of the original 8-dimensional manifoldand a circle. ˜ f = 0 Let us now return to the study of the ˜ f = 0 case. Compared to the case ˜ f = 0 we see thatnow the complex fluxes (4.19) are no longer (anti)holomorphic. Rather one should replace h → h + ( θ · V + )1+ α f in order to obtain a holomorphic combination. This means that following thesame argument of constructing a (4 ,
0) form on the space orthogonal to the vectors introducedin the previous section, we now find h ∧ ρ + g ∧ ϕ = − ( θ · V + )1 + α f ∧ ρ , (4.32)and so the flux F has also a component which is orthogonal to the vectors.For the variation of α we find dα = (1 − α )( θ · V + ) f − (1 + α )( h − V − + 4 h V ) . (4.33)One can compute again the derivatives of the forms and we find dV + = 1 − α α ) ( θ · V + ) f ∧ V + + 2 h V + ∧ V + 12 h − V + ∧ V − , (4.34) dV − = 2( θ · V + )(1 − α )(1 + α ) f V − ∧ V + 2 h V − ∧ V −
12 ( θ · V + ) f ∧ V − , (4.35) dV = − ( θ · V + )2(1 − α )(1 + α ) f − V − ∧ V − h − V − ∧ V −
12 ( θ · V + ) f ∧ V , (4.36) dK = − h − J ∧ V − − h J ∧ V − α α ( θ · V + ) f ∧ J − θ · V + − α ) f ∧ V − ∧ V − θ · V + α )(1 − α ) f − J ∧ V − − θ · V + (1 + α )(1 − α ) f J ∧ V . (4.37)22 Ψ can again be computed directly from its covariant derivative. Again, when deriving dϕ theterms in the direction of V + cancel and we are left with dϕ = 32 ( θ · V + ) ϕ ∧ f + ( θ · V + )2(1 − α ) ( f y ρ ) ∧ V − ∧ V + 1 − α − α ϕ ∧ (cid:0) h − V − + h V (cid:1) + 3( θ · V + )(1 − α )(1 + α ) ϕ ∧ (cid:0) f − V − + f V (cid:1) − − α ρ ∧ ( h − V − h V − ) (4.38) − ( θ · V + )2(1 − α )(1 + α ) ρ ∧ ( f − V − f V − ) .dρ can be computed from various combinations of spinor bilinears (e.g. dρ = ( V + y Φ + )) andwe find dρ = 2 α + 11 + α ( θ · V + ) ρ ∧ f + ( θ · V + )2(1 − α ) ( f y ϕ ) ∧ V − ∧ V + 3 − α − α ρ ∧ (cid:0) h − V − + h V (cid:1) + 3( θ · V + )(1 − α )(1 + α ) ρ ∧ (cid:0) f − V − + f V (cid:1) + 1 + α − α ϕ ∧ ( h − V − h V − ) (4.39)+ ( θ · V + )4(1 − α ) ϕ ∧ ( f − V − f V − ) . Again, taking into account the factors which multiply ρ in the definition of Ω we find for thelatter d Ω = 32 ( θ · V + )Ω ∧ f + i r
21 + α ( θ · V + )2(1 − α ) ( f y Ω) ∧ V − ∧ V + 1 − α − α Ω ∧ (cid:0) h − V − + h V (cid:1) + 3( θ · V + )(1 − α )(1 + α ) Ω ∧ (cid:0) f − V − + f V (cid:1) + i r
21 + α α − α Ω ∧ ( h − V − h V − ) (4.40)+ i r
21 + α ( θ · V + )2(1 − α )(1 + α ) Ω ∧ ( f − V − f V − ) . As in the case of f = 0 it will be instructive to find the torsion classes of the manifold with SU (3)-structure which is orthogonal to the vectors. Projecting on the 6-dimensional space theabove derivatives become P ( dJ ) = − α α ( θ · V + ) J ∧ f , (4.41) P ( d Ω) = 32 ( θ · V + )Ω ∧ f . (4.42)We see that now these derivatives are non-zero and in order to read off the torsion classes ofthe 6-dimensional manifold with SU (3)-structure we need to first normalise the forms J and Ω.First we normalise J to be a proper almost complex structure (i.e. its square to be − J ′ = r
21 + α J , (4.43)Furthermore we define ϕ ′ = r − α ϕ , (4.44)23hus the rescaling of ρ is the following( ρ ′ ) mnp = J ′ rm ( ϕ ′ ) npr = 2 p (1 − α )(1 + α ) ρ . (4.45)We then find the following expressions for the projections of dJ ′ and dϕ ′ on the 6 dimensionalspace orthogonal to the vectors P ( dJ ′ ) = − ( θ · V + )2 J ′ ∧ f , (4.46) P ( dϕ ′ ) = ( θ · V + ) ϕ ′ ∧ f . (4.47)According to [42] this means that the only non-vanishing torsion classes are W and W whichare given by W = 12 J ′ y ( dJ ′ ) = −
12 ( θ · V + ) f , (4.48) W = 12 ϕ ′ y ( dϕ ′ ) = ( θ · V + ) f , (4.49)and therefore satisfy 2 W + W = 0. Note that this relation also featured in the conditionsfound in [43] for non-K¨ahler backgrounds in heterotic string compactifications. In this paper we studied N = 2 compactifications of M-theory to three dimensions whichhave the defining property that a potential is induced for probe space-time filling M2-branes.Such backgrounds are specifically relevant for many model building applications, ranging frommoduli stabilisation to flavour physics, all of which rely on backgrounds that involve potentialsfor space-time filling probe D3-branes on the F-theory side, which by the F-theory/M-theoryduality implies an M2-potential. We showed that it is possible to translate the requirement ofan M2-potential to a specific property of the background geometry. Specifically, that the 8-dimensional background should uplift over a trivial circle fibration to a 9-dimensional manifoldin which the M2-potential can be related to variations of the angle α between two vectors.In terms of 8-dimensional geometry the variation of α implies that one can define a varying‘local’ structure group, so that over a generic point the manifold exhibits an 8-dimensional SU (3)-structure, while over special loci this changes to SU (4)-structure or G -structure.We studied the supersymmetry equations over such backgrounds and wrote them as differ-ential constraints on the 9-dimensional forms. We identified a specific 4-form flux that sourcesvariations of the angle α over the space, and showed that it vanishes over the special SU (4)-structure and G -structure loci, while over the generic SU (3)-structure locus it is parameterisedby two real one-forms h and g . We went on to study backgrounds which support this particular4-form flux as well as a further possible four-form flux with one leg in the internal directions, butno other 4-form fluxes. We showed that in this restricted case, over the generic SU (3)-structurelocus the background takes the form of a 6-dimensional SU (3)-structure manifold with torsionclasses W and W , satisfying 2 W = − W , fibered over a 2-dimensional base. In the casewhere the additional 1-leg flux is turned off we showed that the geometry is a 6-dimensional24alabi-Yau fibered over a 2-dimensional base which supports the flux and over which α varies.Since a major motivation for our work is an application to F-theory, and this requires that thebackground supports an elliptic fibration, it is encouraging that the simplest solutions are basedon a Calabi-Yau fibration since it is well known how to construct elliptically fibered Calabi-Yauthreefolds.The analysis of the supersymmetry equations performed in this work can form a guidefor finding full explicit solutions. This would involve also solving the equations of motionfor the supergravity fields, the Bianchi identities, and possibly, if they are not automaticallyimplied, Einstein’s equations. It is likely that imposing the full set of requirements for a stablesolution, and possibly a realistic vacuum, would imply the need to also incorporate furtherfluxes, perhaps the analogs of the primitive (2 ,
2) flux. Note that the backgrounds that arisein the simplified flux cases discussed above are naturally similar to the backgrounds studiedin [45], and the approach presented in that work of a two-stage reduction may be useful forfinding full solutions.As discussed in the introduction, M2 potentials can be sourced by non-perturbative effectsand it would be interesting, perhaps along the lines of [24, 25], to develop a map betweensuch non-perturbative effects and the flux presented in this work. A related extension of ourwork would be a more detailed understanding of the full form of the potential that is inducedfor the M2 branes. Along the same lines, a more detailed study of the applications to thephenomenological aims presented: moduli stabilisation, inflation, flavour physics, would beinteresting.
Acknowledgments
We would like to thank James Sparks, Calin Lazaroiu, Mirela Babalic and Ioana Coman foruseful discussions. The work of EP is supported by the Heidelberg Graduate School for Fun-damental Physics. The work of AM was partly supported by the UEFISCDI grant TE 93,77/4.08.2010 and by the “Nucleu” project.
A Fierz identities
Fierz relations represent identities of products of gamma matrices with reshuffled indices. Theyemerge as a direct consequence of the the fact that the elements of the Clifford algebra in d -dimensions form a basis for the square matrices (2 h d i × h d i ). Thus any square matrix M can be expanded in a basis of gamma matrices in the following way M = 12 h d i X A Tr(
M γ A ) γ A (A.1)Fierz identities take a simpler form in the case of Majorana spinors. We restrict to this caseand assume that the gamma matrices are real and symmetric. In order to write the generalquadratic Fierz identity one can define the following matrix M cd = ( γ A ) ac ( γ B ) bd (A.2)25or arbitrary fixed spinorial indices a, b and use eq. (A.1) to obtain( γ A ) ac ( γ B ) bd = 12 h d i X C ( γ A γ C γ TB ) ab ( γ C ) cd (A.3)The equation above can be used to generate all the necessary Fierz identities. For instance,by taking γ A = γ B = one obtains the well-known completeness relation for gamma matrices.Above, we have chosen to reshuffle the indices b and c . Similarly one can obtain relations withother indices reshuffled.We shall use the tensor form of the Fierz identities in eq. (A.3) which is obtained bycontracting with the invariant spinors on the nine-dimensional manifold. The relations weobtain are exhaustive as γ A , γ B run over all the basis elements of the Clifford algebra.An equivalent approach is to start with the completeness relation for the gamma matrices( ) ac ( ) bd = 12 h d i X C ( γ C ) ab ( γ C ) cd (A.4)and contract with arbitrary spinors. In our case the spinors are chosen as the invariant spinorson the nine-dimensional manifold multiplied by arbitrary elements of the Clifford algebra. Thesespinors do not form a basis on the space of spinors as they are not all linearly independent, butit is clear that they form a generating set. Therefore, the relations we obtain are exhaustiveand they are equivalent to the original statement that the gamma matrices form a basis.In this appendix we summarize the results obtained by performing a linear analysis of thesystem of equations generated in the way described above. We list the Fierz identities accordingto the number of space-time free indices. We also split the results into identities which havecompletely antisymmetric free indices and the ones which have symmetries corresponding toother Young tableaux. For the antisymmetric identities, the maximum number of free indiceswhich produces new results is four, as other relations with more antisymmetric free indices canbe obtained by contracting with the nine-dimensional ǫ symbol. For identities which have othersymmetries of the free indices, we also stop at four indices as in our calculations we do not needfurther relations. A.1 Completely antisymmetric Fierz identities
No free indices || V || = 1 h V , V i = α (A.5) || V || = 1 h V , V i = 0 (A.6) || V || = 12 ( 1 − α ) h V , V i = 0 (A.7) || K || = 12 (5 + 3 α ) (A.8) || Ψ || = 12 (11 − α ) (A.9)26 | Φ || = 14 h Φ , Φ i = − − α (A.10) || Φ || = 14 h Φ , Φ i = 0 (A.11) || Φ || = 12 (15 + α ) h Φ , Φ i = 0 (A.12)Notice that the real parameter α can take values only in the interval α ∈ [ − ,
1] (A.13)
One free index V m K mr = V r
13! Ψ mnp (Φ ) mnpr = 7 V r (A.14) V m K mr = − V r
13! Ψ mnp (Φ ) mnpr = − V r (A.15) V m K mr = −
12 ( V r − V r ) 13! Ψ mnp (Φ ) mnpr = −
72 ( V r − V r ) (A.16)12! K mn Ψ mnr = 2( V r + V r ) (A.17)14! (Φ ) mnpq ( ∗ Φ ) mnpqr = 14 V r
14! (Φ ) mnpq ( ∗ Φ ) mnpqr = − ( V r + V r ) (A.18)14! (Φ ) mnpq ( ∗ Φ ) mnpqr = 14 V r
14! (Φ ) mnpq ( ∗ Φ ) mnpqr = 7 V r (A.19)14! (Φ ) mnpq ( ∗ Φ ) mnpqr = 4( V r + V r ) 14! (Φ ) mnpq ( ∗ Φ ) mnpqr = 7 V r (A.20) Two free indices V m Ψ mrs = K rs − V [ r V s ]3 K mn (Φ ) mnrs = − K rs + 6 V [ r V s ]3 (A.21) V m Ψ mrs = K rs + 2 V [ r V s ]3 K mn (Φ ) mnrs = − K rs − V [ r V s ]3 (A.22) V m Ψ mrs = − V [ r V s ]2 K mn (Φ ) mnrs = 3 V [ r V s ]2 (A.23)13! Ψ mnp ( ∗ Φ ) mnprs = − K rs − V [ r V s ]3 (A.24)13! Ψ mnp ( ∗ Φ ) mnprs = − K rs + 8 V [ r V s ]3 (A.25)13! Ψ mnp ( ∗ Φ ) mnprs = − V [ r V s ]2 (A.26)273! (Φ ) mnp [ r (Φ ) mnps ] = 0 13! (Φ ) mnp [ r (Φ ) mnps ] = − V [ r V s ]2 (A.27)13! (Φ ) mnp [ r (Φ ) mnps ] = 0 13! (Φ ) mnp [ r (Φ ) mnps ] = 4 K rs − V [ r V s ]3 (A.28)13! (Φ ) mnp [ r (Φ ) mnps ] = 0 13! (Φ ) mnp [ r (Φ ) mnps ] = − K rs − V [ r V s ]3 (A.29) K m [ r K ms ] = 0 (A.30)Ψ mn [ r Ψ mns ] = 0 (A.31) Three free indices V m (Φ ) mrst = 0 V m (Φ ) mrst = 6 V [ r K st ] + 2 K m [ r Ψ mst ] (A.32) V m (Φ ) mrst = − V [ r K st ] + 2 K m [ r Ψ mst ] V m (Φ ) mrst = 0 (A.33) V m (Φ ) mrst = Ψ rst − V [ r K st ] V m (Φ ) mrst = − Ψ rst + 3 V [ r K st ] (A.34) V m (Φ ) mrst = − Ψ rst + 3 V [ r K st ] K mn ( ∗ Φ ) mnrst = − Ψ rst − V [ r K st ] (A.35) V m (Φ ) mrst = Ψ rst − V [ r K st ] K mn ( ∗ Φ ) mnrst = − Ψ rst − V [ r K st ] (A.36) V m (Φ ) mrst = − K m [ r Ψ mst ] K mn ( ∗ Φ ) mnrst = − V [ r K st ] (A.37)12 Ψ mn [ r (Φ ) mnst ] = − rst + 3 V [ r K st ] (A.38)12 Ψ mn [ r (Φ ) mnst ] = − rst + 3 V [ r K st ] (A.39)12 Ψ mn [ r (Φ ) mnst ] = 3 V [ r K st ] (A.40)13! (Φ ) mnp [ r ( ∗ Φ ) mnpst ] = 0 13! (Φ ) mnp [ r ( ∗ Φ ) mnpst ] = 4 V [ r K st ] (A.41)13! (Φ ) mnp [ r ( ∗ Φ ) mnpst ] = 0 13! (Φ ) mnp [ r ( ∗ Φ ) mnpst ] = 2Ψ rst + 2 V [ r K st ] (A.42)13! (Φ ) mnp [ r ( ∗ Φ ) mnpst ] = 0 13! (Φ ) mnp [ r ( ∗ Φ ) mnpst ] = − rst − V [ r K st ] (A.43)13! (Φ ) mnp [ r ( ∗ Φ ) mnpst ] = − V [ r K st ] (A.44)13! (Φ ) mnp [ r ( ∗ Φ ) mnpst ] = − rst − V [ r K st ] (A.45)13! (Φ ) mnp [ r ( ∗ Φ ) mnpst ] = 2Ψ rst + 2 V [ r K st ] (A.46)28 our free indices V m ( ∗ Φ ) mrsuv = (Φ ) rsuv (A.47) V m ( ∗ Φ ) mrsuv = − (Φ ) rsuv − V [ r Ψ suv ] − K [ rs K uv ] (A.48) V m ( ∗ Φ ) mrsuv = (Φ ) rsuv − V [ r Ψ suv ] (A.49) V m ( ∗ Φ ) mrsuv = − (Φ ) rsuv + 8 V [ r Ψ suv ] − K [ rs K uv ] (A.50) V m ( ∗ Φ ) mrsuv = (Φ ) rsuv (A.51) V m ( ∗ Φ ) mrsuv = (Φ ) rsuv + 4 V [ r Ψ suv ] (A.52) V m ( ∗ Φ ) mrsuv = 4 V [ r Ψ suv ] (A.53) V m ( ∗ Φ ) mrsuv = − V [ r Ψ suv ] (A.54) V m ( ∗ Φ ) mrsuv = 12 (Φ + Φ ) rsuv + 3 K [ rs K uv ] (A.55) K m [ r (Φ ) msuv ] = (Φ ) rsuv − V [ r Ψ suv ] (A.56) K m [ r (Φ ) msuv ] = − (Φ ) rsuv − V [ r Ψ suv ] (A.57) K m [ r (Φ ) msuv ] = −
12 (Φ − Φ ) rsuv − V [ r Ψ suv ] (A.58)12 Ψ mn [ r ( ∗ Φ ) mnsuv ] = (Φ ) rsuv + 2 V [ r Ψ suv ] (A.59)12 Ψ mn [ r ( ∗ Φ ) mnsuv ] = − (Φ ) rsuv + 2 V [ r Ψ suv ] (A.60)12 Ψ mn [ r ( ∗ Φ ) mnsuv ] = −
12 (Φ − Φ ) rsuv + 2 V [ r Ψ suv ] (A.61)12 K mn ( ∗ Ψ) mnrsuv = −
12 (Φ + Φ ) rsuv + 3 K [ rs K uv ] (A.62)Ψ m [ rs Ψ muv ] = −
12 (Φ + Φ ) rsuv − K [ rs K uv ] (A.63)12 (Φ ) mn [ rs (Φ ) mnuv ] = − ) rsuv
12 (Φ ) mn [ rs (Φ ) mnuv ] = − K [ rs K uv ] (A.64)12 (Φ ) mn [ rs (Φ ) mnuv ] = − ) rsuv
12 (Φ ) mn [ rs (Φ ) mnuv ] = − (Φ ) rsuv (A.65)12 (Φ ) mn [ rs (Φ ) mnuv ] = −
12 (Φ + Φ ) rsuv + K [ rs K uv ]
12 (Φ ) mn [ rs (Φ ) mnuv ] = − (Φ ) rsuv (A.66)29 .2 Fierz identities with symmetric part We list here the Fierz identities which can have a symmetric part, that is the ones which liein the tensorial algebra. We restrict only to necessary identities, that is the ones involving theforms V , K, Ψ and Φ . The rest can be obtained from these ones making also use of theantisymmetric identities already given earlier in the appendix. Two indices K mr K ms = 12 (1 + α ) δ rs − V ( r V s )2 + V r V s (A.67)Ψ mnr Ψ mns = (4 − α ) δ rs + 6 V ( r V s )2 − V r V s (A.68)(Φ ) mnpr (Φ ) mnps = 3(7 + α ) δ rs − V ( r V s )2 − V r V s (A.69) Three indices K mr Ψ mst = δ r [ s ( V t ]1 + V t ]2 ) + K m [ r Ψ mst ] (A.70)Ψ mnr (Φ ) mnst = − δ r [ s ( V s ]1 − V s ]2 ) + 12 V [ r K st ] − V r K st (A.71)13! (Φ ) mnpr ( ∗ Φ ) mnpst = 4 δ r [ s ( V t ]1 + V t ]2 ) (A.72) Four indices Ψ mrs Ψ muv = (1 − α ) δ rsuv + 2 δ [ r [ u (cid:16) V s ]1 V v ] + V v ] V s ]2 − V s ]3 V v ] (cid:17) (A.73) −
12 (Φ + Φ ) rsuv − K [ rs K uv ] + K rs K uv (A.74)12 (Φ ) mnrs (Φ ) mnuv = (3 + α ) δ rsuv − δ [ r [ u (cid:16) V s ]1 V v ] + V v ] V s ]2 + V s ]3 V v ] (cid:17) (A.75) −
12 (Φ + Φ ) rsuv + 3 K [ rs K uv ] − K rs K uv (A.76)(A.77) B Relations satisfied by the SU (3) structure forms In the main text we derived the parametrisation of the spinor bilinears (2.14) in terms offorms defining a SU(3) structure in six dimensions and three additional vectors V ± and V .Here we shall give more details about the relations which these forms satisfy, including alsobrief indications on how to derive such relations. The crucial relation which we shall use almost30verywhere is the symmetric relation obtained by contracting J with itself over one index whichcan be derived from (A.67) J mn J np = −
12 (1 + α ) δ mp + 14 ( V + ) m ( V + ) p + 1 + α − α ) ( V − ) m ( V − ) p + 1 + α − α ( V ) m ( V ) p = 12 (1 + α ) ( − δ mp + ( P + ) mp + ( P − ) mp + ( P ) mp ) . (B.1)In the main text we claimed that ρ defined as in (2.34) is totally antisymmetric. Using theFierz relations (with 3 free indices) which give the contractions of the vectors V , with theforms Φ , we can construct the object V + y Φ + . Using (2.31) we find J mn ϕ mpq = K m [ n Ψ mpq ] , (B.2)Let us continue by computing the norms of the SU(3) forms. Note that since we have done anorthogonal decomposition in terms of the vector fields, the terms on the RHS of (2.29)–(2.33)are independent, in the sense that total contractions of different terms vanish by definition.Using the norms of the vector fields which were listed at the beginning of this appendix, wecan immediately derive the norm of J as J y J = J mn J mn = (1 + α ) (B.3)Taking the square of (2.30) we find in a similar way the norm of ϕϕ y ϕ = ϕ mnp ϕ mnp = 2(1 − α ) . (B.4)From (2.34) and using (B.1) we find ρ y ρ = ρ mnp ρ mnp = (1 − α )(1 + α ) (B.5)Using the Fierz relation involving the contraction of K and Ψ over two indices, we immediatelyfind J y ϕ = 0 = J y ρ (B.6)where the second equality holds due to the fact that ρ in (2.34) is totally antisymmetric. Usingthe above relation and the Fierz identity involving the contraction of Ψ and Φ , over threeindices, we obtain ϕ y ρ = 0 . (B.7)Most of the other relations we shall need involve a Hodge ∗ operation and are somehowmore complicated. Let us look at the Fierz relation which gives the contraction of Φ with ∗ Φ over four indices. This can be rewritten in form notation as ∗ (Φ ∧ Φ ) = 4 V + . (B.8)Using (2.33) we find4(1 − α ) ϕ ∧ ρ ∧ V − ∧ V − − α )(1 + α ) ρ ∧ J ∧ V + ∧ V − ∧ V = 4 ∗ V + . (B.9)31learly, the second term on the LHS must vanish, as it contains V + while on the RHS we onlyfind ∗ V + . J and ρ are orthogonal to the vectors, and the only way this term can vanish is if ρ ∧ J = 0 . (B.10)The remaining relation can be rewritten, by contracting V − and V , as ϕ ∧ ρ = ∗ ( V + ∧ V − ∧ V ) . (B.11)From Fierz identities involving the contraction of Φ , with Φ we find similar relations, andagain, by contracting the appropriate vectors we find J ∧ J ∧ J = 32 1 + α − α ∗ ( V + ∧ V − ∧ V ) . (B.12)Further relations with Hodge star can be derived form these ones by contracting with appro-priate forms and making use of the orthogonality conditions and of the norms of the variousquantities. We find ∗ J = 12(1 + α )(1 − α ) J ∧ J ∧ V + ∧ V − ∧ V (B.13) ∗ ϕ = 1(1 + α )(1 − α ) ρ ∧ V + ∧ V − ∧ V (B.14) ∗ ρ = − α ) ϕ ∧ V + ∧ V − ∧ V (B.15)Other useful relations which can be derived easily from the ones already written so far are ∗ ( J ∧ V + ∧ V − ∧ V ) = (1 − α ) J ∧ J (B.16) ∗ ( J ∧ V − ∧ V ) = (1 − α )2(1 + α ) J ∧ J ∧ V + (B.17) ∗ ( J ∧ V + ∧ V ) = − J ∧ J ∧ V − (B.18) ∗ ( J ∧ V + ∧ V − ) = 2 J ∧ J ∧ V (B.19)and ∗ ( ρ ∧ V + ∧ V ) = 12 (1 + α ) ϕ ∧ V − ∗ ( ρ ∧ V ) = 14 ϕ ∧ V + ∧ V − (B.20) ∗ ( ρ ∧ V − ∧ V ) = −
12 (1 − α ) ϕ ∧ V + ∗ ( ρ ∧ V + ) = 1 + α − α ϕ ∧ V − ∧ V (B.21) ∗ ( ρ ∧ V + ∧ V − ) = − α ) ϕ ∧ V ∗ ( ρ ∧ V − ) = − ϕ ∧ V + ∧ V (B.22)One can also compute the Hodge duals of the original spinor bilinear forms and express them32n terms of the SU (3) parametrisation ∗ K = 12(1 + α )(1 − α ) J ∧ J ∧ V + ∧ V − ∧ V + 12(1 + α )(1 − α ) ϕ ∧ ρ ∧ V + (B.23) ∗ Ψ = 1(1 + α )(1 − α ) ρ ∧ V + ∧ V − ∧ V + 1(1 + α )(1 − α ) J ∧ J ∧ V − ∧ V + 12(1 − α ) ϕ ∧ ρ (B.24) ∗ Φ + = − α )(1 − α ) J ∧ V + ∧ V − ∧ V − − α ϕ ∧ V − ∧ V −
11 + α J ∧ J ∧ V + (B.25) ∗ Φ − = − α )(1 − α ) ρ ∧ V + ∧ V − − − α ϕ ∧ V + ∧ V −
11 + α J ∧ J ∧ V − (B.26) ∗ Φ = − α )(1 − α ) ρ ∧ V + ∧ V + 12(1 − α ) ϕ ∧ V + ∧ V − −
11 + α J ∧ J ∧ V (B.27)From the symmetric Fierz identity (A.68) involving the contraction of Ψ with itself over oneindex we can derive a similar relation for ϕϕ mrs ϕ mtu = (1 − α ) δ rstu + 2 1 − α α J r [ t J u ] s − − α α δ [ r [ t V + u ] V s ]+ − δ [ r [ t V − u ] V s ] − − δ [ r [ t V u ] V s ]3 + 12(1 + α ) V +[ t V − u ] V [ r + V s ] − + 21 + α V +[ t V u ] V [ r + V s ]3 + 21 − α V − [ t V u ] V [ r − V s ]3 . (B.28)By contracting this relation with J we can find similar relations for ρ or ρ and ϕρ mrs ϕ mtu = 2(1 − α ) δ [ r [ t J u ] s ] + 1 − α α J [ r [ t V + u ] V s ]+ + J [ r [ t V − u ] V s ] − + 4 J [ r [ t V u ] V s ]3 (B.29) ρ mrs ρ mtu = (1 − α )(1 + α ) δ rstu + (1 − α ) J r [ t J u ] s −
12 (1 − α ) δ [ r [ t V + u ] V s ]+ −
12 (1 + α ) δ [ r [ t V − u ] V s ] − − α ) δ [ r [ t V u ] V s ]3 + V +[ t V − u ] V [ r + V s ] − + V +[ t V u ] V [ r + V s ]3 + 1 + α − α V − [ t V u ] V [ r − V s ]3 . (B.30)Finally, by contracting a pair of indices in the above relations it is easy to find ϕ mnr ϕ mns = 2(1 − α ) δ sr − − α α V + r V s + − V − r V s − − V r V s ; ϕ mnr ρ mns = 2(1 − α ) J rs (B.31) ρ mnr ρ mns = (1 − α )(1 + α ) δ sr − − α V + r V s + − α V − r V s − − α ) V r V s . Before we end this section, let us note the following fact which is useful during the calcula-tions. Consider a 1-form k which is orthogonal to all the vector fields, i.e. k · V i = 0, and define˜ k = k y J . By contracting J we find the equivalent relation˜ k y J = − α k . (B.32)33urthermore, by contracting with ϕ and taking into account the definition of ρ from (2.34) wefind ˜ k y ϕ + k y ρ = 0 , α k y ϕ − ˜ k y ρ = 0 (B.33)Taking now the exterior product with ρ and ϕ we find˜ k ∧ ϕ + ( k y ϕ ) ∧ J = 0 , ˜ k ∧ ρ + ( k y ρ ) ∧ J = 0 − (˜ k y ρ ) ∧ J + 1 + α k ∧ ρ = 0 , − (˜ k y ϕ ) ∧ J + 1 + α k ∧ ϕ = 0where we used the identity ( k y ϕ ) ∧ J − ϕ ∧ ( k y J ) = k y ( ϕ ∧ J ) = 0 and similar ones for ρ and˜ k . Adding up the equations in the same column in such a way to obtain the combinations in(B.33) we find ˜ k ∧ ϕ + k ∧ ρ = 0 , ˜ k ∧ ρ − α k ∧ ϕ = 0 (B.34)These relations can be intuitively understood in a simple way. From the 1-forms k and ˜ k wecan construct a complex (1 ,
0) form k = k + i r
21 + α ˜ k (B.35)Then the expressions in (B.34) are nothing but the real and imaginary components of the (4,0)form k ∧ Ω. But this form should vanish identically since it only lives on the 6-dimensional spaceorthogonal to the vectors and this gives the relations in (B.34).
C Supersymmetry equations
We summarize in this appendix the supersymmetry algebraic constraints arising from the vari-ation of the external components of the gravitino. For the N = 2 flux background that weconsider one has to satisfy the equations Qξ j = 0 j = 1 , Q is given in eq. (3.6) for the auxiliary 9d manifold Y . We translate theequations above into constraints on the fluxes ˜ f and F involving the spinor bilinears defined ineq. (2.14). Specifically, we contract eq. (C.1) with the following generating set of the spinorialrepresentation γ A ξ i i = 1 , γ A ∈ { , γ m , γ mn , γ mnp , γ mnpq } (C.2)It is then convenient to represent the algebraic constraints in eq. (C.1) in the the followingequivalent form ξ Ti (cid:2) Qγ A ± γ TA Q T (cid:3) ξ j = 0 i, j = 1 , Q given in eq. (3.6) and expanding the products of gammamatrices one can express the resulting equations in terms of the spinor bilinears in eq. (2.14).34he result is the following (a number of these expressions first appeared in [37]) − λ ( θ · V i ) + 12 ( V i · d ∆) − F y Φ i = 0 ; ˜ f m θ n K mn = 0 (C.4) − λθ + d ∆ − ∗ ( F ∧ Φ + ) + 16 V + y ( ˜ f ∧ θ ) = 0 (C.5) − ∗ ( F ∧ Φ − , ) + 16 V − , y ( ˜ f ∧ θ ) = 0 (C.6)2 λθ y K − d ∆ y K + 16 Ψ y F −
13 ( ˜ f ∧ θ ) y Ψ = 0 (C.7)2 λθ [ m V + n ] − ∂ [ m ∆ V + n ] + 136 F [ mpqr Φ + n ] pqr −
16 Φ + mnpq ˜ f p θ q + 23 ˜ f [ m θ n ] = 0 (C.8)2 λθ [ m ( V − ) n ] − ∂ [ m ∆( V − ) n ] + 136 F [ mklp (Φ − ) n ] klp −
16 (Φ − ) mnkl ˜ f k θ l = 0 (C.9) − λθ y Ψ + d ∆ y Ψ + 16 ∗ ( F ∧ Ψ) + 16 K y F + 13 ( θ y K ) ∧ ˜ f −
13 ( ˜ f y K ) ∧ θ = 0 (C.10) − λθ k Φ i mnpk + 12 ∂ k ∆Φ i mnpk − F [ pklq ( ∗ Φ i ) mn ] klq + 112 F mnpk V ki − ∗ (Φ i ∧ ˜ f ∧ θ ) mnp + 16 ( V i ∧ ˜ f ∧ θ ) mnp = 0 (C.11) − λ ( K ∧ θ ) mnp + K ∧ d ∆ mnp + 16 ∗ ( F ∧ K ) mnp + 14 Ψ [ mkl F np ] kl −
13 [( θ y Ψ) ∧ ˜ f ] mnp + 13 [( ˜ f y Ψ) ∧ θ ] mnp = 0 (C.12) − λ ∗ (Φ + ∧ θ ) mnpq + 12 ∗ (Φ + ∧ d ∆) mnpq − ∗ ( F ∧ V + ) mnpq + 14 F [ mnrs (Φ + ) pq ] rs − F mnpq + 16 ( θ y Φ + ) ∧ ˜ f −
16 ( ˜ f y Φ + ) ∧ θ = 0 (C.13) − λ ∗ (Φ − ∧ θ ) mnpq + 12 ∗ (Φ − ∧ d ∆) mnpq − ∗ ( F ∧ V − ) mnpq + 14 F [ mnrs (Φ − ) pq ] rs + 16 ( θ y Φ − ) ∧ ˜ f −
16 ( ˜ f y Φ − ) ∧ θ = 0 (C.14) − λ (Ψ ∧ θ ) mnpq + (Ψ ∧ d ∆) mnpq + 19 ( ∗ Ψ) [ mnprst F q ] rst + 23 K [ mr F npq ] r + 13 ∗ (Ψ ∧ ˜ f ∧ θ ) mnpq + 13 ( K ∧ ˜ f ∧ θ ) mnpq = 0 (C.15)Notice that since we are using a set of generators { γ A ξ i } instead of a basis, the equations aboveare not independent. In fact, as it is done explicitly for the special flux analysed in the paper,one only needs the constraints arising from contraction with ξ i and γ m ξ i .35 Killing properties of the Majorana-Weyl components
Given that a background supports a covariantly constant Majorana spinor ξ , the requirementfor its Majorana-Weyl components to also solve the Killing spinor equation is[ D m , θ r γ r ] ξ = 0 , (D.1)[ Q, θ r γ r ] ξ = 0 . (D.2)Here D m and Q are defined in (3.1), and we recall that the eight-dimensional chirality matrix γ was given in terms of the 9-dimensional basis as γ = θ r γ r . One can easily show that thecommutators with γ are expressed as[ A m , θ r γ r ] = 2 λ ( γ m − θ m θ n γ n ) + 112 F mnpq θ r γ npqr , (D.3)[ Q, θ r γ r ] = ∂ n ∆ θ r γ nr + 136 F mnpq θ [ m γ npq ] + 23 ˜ f m θ n θ [ m γ n ] . (D.4)We now impose the orthogonality of the auxiliary direction θ on the fluxes and the fact that λ = 0 for our specific choice of flux[ A m , θ r γ r ] = 112 F mnpq θ r γ npqr , (D.5)[ Q, θ r γ r ] = ∂ n ∆ θ r γ nr −
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