Coupled SU(3)-structures and Supersymmetry
aa r X i v : . [ m a t h . DG ] M a y COUPLED
SU(3) -STRUCTURES AND SUPERSYMMETRY
ANNA FINO AND ALBERTO RAFFERO
Abstract.
We review coupled SU(3)-structures, also known in the literature as restrictedhalf-flat structures, in relation to supersymmetry. In particular, we study special classesof examples admitting such structures and the behaviour of flows of SU(3)-structures withrespect to the coupled condition. Introduction
In the physical literature, manifolds endowed with SU(3)-structures have been frequentlyconsidered to construct string vacua [13, 19, 20, 21, 25, 27, 28, 29, 39, 40].In this paper, we are mainly interested in the class of SU(3)-structures that are relevantfor N = 1 compactifications of type IIA string theory on spaces of the form AdS × N , whereAdS is the four-dimensional anti de Sitter space and N is a six-dimensional compact smoothmanifold. The requirement of N = 1 supersymmetry implies the existence of a globallydefined complex spinor on the internal 6-manifold N . As a consequence, the structure groupof N reduces to SU(3), which is equivalent to the existence on N of an almost Hermitianstructure ( h, J, ω ) and a complex (3,0)-form Ψ of nonzero constant length satisfying somecompatibility conditions. As shown in [29], in the case where the two SU(3)-structuresare proportional, imposing the Killing spinor equations for four-dimensional N = 1 stringvacua of type IIA on AdS constrains the intrinsic torsion of the SU(3)-structure to lie in W − ⊕ W − . Further constraints on the torsion forms are implied by the Bianchi identitiesfor the background fluxes in the absence of sources. Moreover, all these constraints are notonly necessary but also sufficient to guarantee the existence of solutions. Examples of thiskind of solutions were considered for instance in [10, 26, 29, 40].SU(3)-structures whose torsion class is W − ⊕ W − are known as coupled SU(3)-structures[38] in the mathematical literature and are characterized by the fact that they are half-flat
SU(3)-structures, i.e., both ψ + := ℜ (Ψ) and ω ∧ ω are closed forms, having dω proportionalto ψ + . Coupled structures were recently considered in [16, 30, 36]. They are of interest forinstance because their underlying almost Hermitian structure is quasi-K¨ahler and becausethey generalize the class of nearly K¨ahler SU(3)-structures, namely the half-flat structureshaving dω proportional to ψ + and dψ − proportional to ω ∧ ω , where ψ − := ℑ (Ψ).Up to now, very few examples of manifolds admitting complete nearly K¨ahler structuresare known. In the homogeneous case there are only finitely many of them by [9], while new Mathematics Subject Classification. complete inhomogeneous examples were recently found on S and S × S in [18]. Amongthe remarkable properties of nearly K¨ahler structures in dimension 6, it is worth recallinghere that the Riemannian metric h they induce is Einstein, that is, its Ricci tensor Ric( h )is a scalar multiple of h . It is then quite natural to ask whether coupled structures inducingEinstein metrics can exist or if requiring that a coupled structure induces an Einstein metricimplies that it is actually nearly K¨ahler. An attempt to find coupled Einstein structureson explicit examples was done in [36], where the existence of invariant coupled Einsteinstructures was excluded on the compact manifold S × S for Ad( S )-invariant Einsteinmetrics and on all the six-dimensional solvmanifolds. While writing this paper, we foundout that the work [40], which provides a family of AdS vacua in IIA string theory, containsan example of a coupled Einstein structure. This answers to the question and can be usedto construct examples of G -structures with non-vanishing torsion inducing Einstein andRicci-flat metrics.One of the main motivations to study half-flat structures is due to the role they playin the construction of seven-dimensional manifolds with holonomy contained in G . Morein detail, by a result of Hitchin [23], on a 6-manifold N it is possible to define a flow forSU(3)-structures, the so-called Hitchin flow , which can be solved for any given analytichalf-flat structure as initial condition. A solution to the flow equations consists of a familyof half-flat structures depending on a parameter t ∈ I ⊆ R and allows to define a torsionless G -structure on the product manifold I × N . One question that naturally arises is thenwhether coupled structures, which are in particular half-flat, are preserved by this flow.A generalization of the Hitchin flow can be introduced considering an SU(3)-structure,not necessarily half-flat, and using it to define a G -structure with torsion on the productmanifold I × N . The evolution equations for the differential forms defining the SU(3)-structure can then be obtained by requiring that the intrinsic torsion of the G -structurebelongs to a certain torsion class. Of course, the Hitchin flow equations can be recoveredas a special case of this generalized flow. This idea was considered for example in [14],where the generalized Hitchin flow was used as a tool to study the moduli space of SU(3)-structure manifolds constituting the internal compact space for four-dimensional N = domain wall solutions of heterotic string theory. In that case, the authors considered thenon-compact seven-dimensional manifold defined by combining the direction perpendicularto the domain wall and the internal 6-manifold and observed that it is possible to defineon it a G -structure whose non-vanishing intrinsic torsion forms can be recovered using theresults of [19, 28].Furthermore, homogeneous spaces admitting coupled structures were used to provideexamples of heterotic N = domain wall solutions with vanishing fluxes in [25] and anattempt to generalize this result in a more general case was done in [19].The present paper is organized as follows. In Section 2 we review some definitions andproperties regarding SU(3)- and G -structures. In Section 3 we study coupled structures inrelation to supersymmetry. In Section 4 we describe some explicit examples and in Section5 we study the behaviour of flows of SU(3)-structures with respect to the coupled condition. Acknowledgements . The authors would like to thank Thomas Madsen for useful conver-sations.
OUPLED SU(3)-STRUCTURES AND SUPERSYMMETRY 3 Review of
SU(3) -structures and G -structures An SU(3)-structure on a six-dimensional smooth manifold N is the data of a Riemannianmetric h , an orthogonal almost complex structure J , a 2-form ω related to h and J via theidentity ω ( · , · ) = h ( J · , · ) and a (3 , ψ + + iψ − whichis compatible with ω , i.e., ω ∧ Ψ = 0 , and satisfies the normalization condition i (cid:0) Ψ ∧ Ψ (cid:1) = 23 ω = 4 dV h , where dV h is the Riemannian volume form of h . At each point p ∈ N there exists an h -orthonormal frame ( e , . . . , e ) of T ∗ p N , called adapted frame for the SU(3)-structure, suchthat ω = e + e + e , Ψ = ( e + ie ) ∧ ( e + ie ) ∧ ( e + ie ) , and whose dual frame ( e , . . . , e ) is adapted for J , i.e., J e i = e i +1 , i = 1 , , . Remark . Here and hereafter, the notation e ijk ··· is a shortening for the wedge product e i ∧ e j ∧ e k ∧ · · · . Moreover, we will also use the notation θ n as a shortening for the wedgeproduct of a differential form θ by itself for n -times.Using the results of [22, 37], one can show that an SU(3)-structure actually depends onlyon the pair ( ω, ψ + ), let us recall briefly how. For each p ∈ N , let V := T p N , denote by A : Λ ( V ∗ ) → V ⊗ Λ ( V ∗ ) the canonical isomorphism given by A ( ξ ) = v ⊗ Ω, where i v Ω = ξ ,and define for a fixed ρ ∈ Λ ( V ∗ ) K ρ : V → V ⊗ Λ ( V ∗ ) , K ρ ( v ) = A (( i v ρ ) ∧ ρ )and λ : Λ ( V ∗ ) → (Λ ( V ∗ )) ⊗ , λ ( ρ ) = 16 tr K ρ . If λ ( ρ ) = 0, p | λ ( ρ ) | ∈ Λ ( V ∗ ) defines a volume form by choosing the orientation of V forwhich ω is positively oriented. Moreover, whenever λ ( ρ ) < J ρ := − p − λ ( ρ ) K ρ . An SU(3)-structure on N can then be defined as a pair ( ω, ψ + ) such that the 2-form ω is nondegenerate, i.e., ω = 0, the 3-form ψ + is compatible with ω and satisfies λ ( ψ + ( p )) < p ∈ N , the almost complex structure is J = J ψ + , the imaginary part of Ψ is given by ψ − := J ψ + , the normalization condition holds and h ( · , · ) := ω ( · , J · ) defines a Riemannianmetric. ANNA FINO AND ALBERTO RAFFERO
The intrinsic torsion τ of an SU(3)-structure is completely determined by the exteriorderivatives of ω, ψ + , ψ − , as shown in [12]. More in detail, we have(1) dω = − w − ψ + + w +1 ψ − + w + w ∧ ω,dψ + = w +1 ω − w +2 ∧ ω + w ∧ ψ + ,dψ − = w − ω − w − ∧ ω + J w ∧ ψ + , where w ± ∈ C ∞ ( N ), w ± ∈ Λ , ( N ), w ∈ Λ , ( N ), w , w ∈ Λ ( N ) are the intrinsic torsionforms of the SU(3)-structure. It is then possible to divide the SU(3)-structures in classesby seeing which torsion forms vanish. For example, if ω, ψ + and ψ − are all closed, thenall the torsion forms vanish and the manifold N is Calabi-Yau . If all the torsion forms but w − vanish, the SU(3)-structure is said to be nearly K¨ahler and we write τ ∈ W − . If both ψ + and ω are closed, then the torsion forms w +1 , w +2 , w , w vanish, the SU(3)-structureis said to be half-flat and we write τ ∈ W − ⊕ W − ⊕ W . As recently shown in [2], theSU(3)-structures can also be described in terms of a characterizing spinor and the spinorialfield equations it satisfies.In [4], it was shown that the Ricci and the scalar curvature of the metric h induced byan SU(3)-structure can be expressed in terms of the intrinsic torsion forms. In particular,if we consider the projections E : Λ ( N ) → Λ , ( N ) and E : Λ ( N ) → Λ , ( N ) given by E ( β ) = ( β + J β ) − ∗ (( ∗ ( β + J β ) + ( β + J β ) ∧ ω ) ∧ ω ) ω,E ( ρ ) = ρ − ∗ ( J ρ ∧ ω ) ∧ ω − ∗ ( ρ ∧ ψ − ) ψ + − ∗ ( ψ + ∧ ρ ) ψ − , where ∗ is the Hodge operator defined using h and the volume form dV h , then the tracelesspart of the Ricci tensor has the following expressionRic ( h ) = ι − ( E ( φ )) + γ − ( E ( φ )) , where the 2-form φ and the 3-form φ depend on the intrinsic torsion forms and theirderivatives and the maps ι : S ( N ) → Λ , ( N ) and γ : S − ( N ) → Λ , ( N ) are (pointwise) su (3)-modules isomorphisms (see [4] for the details). The Ricci tensor of h can then berecovered from the identity Ric( h ) = 16 Scal( h ) h + Ric ( h ) . Starting from an SU(3)-structure ( ω, ψ + ) on a 6-manifold N , it is possible to constructa G -structure on the 7-manifold I × N , where I ⊆ R is a connected open interval. Beforedescribing how, we recall that a G -structure on a seven-dimensional manifold M is char-acterized by the existence of a globally defined 3-form ϕ inducing a Riemannian metric g ϕ and a volume form dV g ϕ given by(2) g ϕ ( X, Y ) dV g ϕ = 16 i X ϕ ∧ i Y ϕ ∧ ϕ, for any pair of vector fields X, Y ∈ X ( M ). The intrinsic torsion of a G -structure ϕ iscompletely determined by the exterior derivatives of ϕ and ∗ ϕ ϕ , where ∗ ϕ is the Hodge OUPLED SU(3)-STRUCTURES AND SUPERSYMMETRY 5 operator defined using the metric g ϕ and the volume form dV g ϕ . More in detail, it holds [7](3) dϕ = τ ∗ ϕ ϕ + 3 τ ∧ ϕ + ∗ ϕ τ ,d ∗ ϕ ϕ = 4 τ ∧ ∗ ϕ ϕ + τ ∧ ϕ, where τ ∈ C ∞ ( M ), τ ∈ Λ ( M ), τ ∈ Λ ( M ) = { β ∈ Λ ( M ) : ∗ ϕ ( ϕ ∧ β ) = − β } , τ ∈ Λ ( M ) = { ρ ∈ Λ ( M ) : ϕ ∧ ρ = 0 and ∗ ϕ ϕ ∧ ρ = 0 } are the intrinsic torsion forms of the G -structure. Also in this case it is possible to classify the G -structures in termsof the non-vanishing torsion forms. For example, if ϕ is both closed and co-closed, thenall the torsion forms vanish, Hol( g ϕ ) ⊆ G and the G -structure is called parallel . If ϕ isa closed form, then all the torsion forms but τ vanish and the G -structure is said to be calibrated . If the only non-vanishing torsion forms are τ and τ , then at least locally themetric g ϕ is conformally equivalent to the metric induced by a calibrated G -structure andthe G -structure is called locally conformal calibrated . If the only vanishing torsion form is τ , then the G -structure is said to be integrable . In this case there exists a unique affineconnection with totally skew-symmetric torsion preserving the G -structure by [17].Consider now ( ω, ψ + ) and two smooth functions F : I → C − { } and G : I → R + , thefollowing 3-form defines a G -structure on I × N ([24]) ϕ = ℜ ( F Ψ) + G | F | ω ∧ dt, where t is the coordinate on I . Moreover, we have g ϕ = G dt + | F | h,dV g ϕ = G | F | dt ∧ dV h , ∗ ϕ ϕ = G ℑ ( F Ψ) ∧ dt + 12 | F | ω . For some particular choices of the interval I and the functions F and G , we obtain thefollowing remarkable manifolds: • the cylinder Cyl ( N ) with metric dt + h , if I = R and G, F ≡ • the cone C ( N ) with the metric dt + t h , if I = R + , G ≡ F ( t ) = t , • the sin-cone SC ( N ) with the metric dt + sin ( t ) h , if I = (0 , π ), G ≡ F ( t ) =sin( t ) e i t .Observe that with the choice G ≡
1, the manifold I × N with metric dt + | F | h is the warpedproduct of I and N with warping function | F | . Using the expression of the Ricci tensor ofthe warped product metric [35], it is possible to show the following general properties (seealso [6]). Proposition 2.2.
Let ( M m , g ) be a Riemannian manifold of dimension m . Then the conemetric dt + t g is Ricci-flat if and only if the metric g is Einstein with Ric( g ) = ( m − g . Proposition 2.3.
Let ( M m , g ) be a Riemannian manifold of dimension m with Einsteinmetric g such that Ric( g ) = ( m − g . Then the sin-cone metric dt + sin ( t ) h is Einsteinwith Einstein constant m . ANNA FINO AND ALBERTO RAFFERO Coupled structures and Supersymmetry
In [29], the authors considered the problem of finding necessary and sufficient conditionsfor N = 1 compactification of (massive) IIA supergravity to four-dimensional anti-de Sitterspace on manifolds endowed with an SU(3)-structure. As a result, they obtained a setof constraints the intrinsic torsion forms of the SU(3)-structure ( ω, ψ + ) on the internalmanifold have to satisfy, we recall them here briefly. Supersymmetry equations and theBianchi identities constrain the intrinsic torsion to lie in the space W − ⊕ W − , i.e., the onlynon-vanishing intrinsic torsion forms are w − and w − . Furthermore, in absence of sources,the Bianchi identities provide a further constraint on the exterior derivative of w − (4) dw − ∝ ψ + , and the norms of w − and w − have to satisfy the following inequality [26](5) 3( w − ) ≥ | w − | , where | · | denotes the norm with respect to the metric h induced by the SU(3)-structure. Inthe massless limit, the solutions reduce to AdS × N , N being a compact 6-manifold endowedwith an SU(3)-structure with torsion in W − ⊕W − and for which (4) holds. Moreover, it wasobserved in [26] that the conditions (4) and (5) can be relaxed in the presence of sources.It is then worth studying from the mathematical point of view the properties of this kindof SU(3)-structures. In what follows, we suppose that the manifold N is connected.First of all, we recall that SU(3)-structures having torsion class W − ⊕ W − are knownas coupled structures [38] or restricted half-flat structures [27] in literature. They can bedefined as the subclass of half-flat structures having w ≡
0. In this case, dω is proportionalto ψ + , the intrinsic torsion form w − is constant [36] and has to be nonzero if we want theintrinsic torsion τ to belong to the class W − ⊕ W − . Thus, if we let c := − w − , we have(6) dω = cψ + ,dψ + = 0 ,dψ − = − cω − w − ∧ ω. The 2-form w − lies in the space Λ , ( N ), therefore it satisfies the following properties: w − ∧ ω = 0 , (7) w − ∧ ψ ± = 0 , (8) w − ∧ ω = − ∗ w − . (9)Using (9) and the expression of dψ − , it is easy to show that the 2-form w − is co-closed,that is δw − = ∗ d ∗ w − = 0. Remark . Observe that if a manifold admits a coupled structure ( ω, ψ + ) with coupledconstant c ∈ R − { } such that dω = cψ + , then one can choose a nonzero real constant r , define ˜ ω := r ω , ˜ ψ + := r ψ + and obtain a new coupled structure (˜ ω, ˜ ψ + ) with coupledconstant ˜ c = cr . In particular, it is always possible to find a coupled structure havingpositive coupled constant. OUPLED SU(3)-STRUCTURES AND SUPERSYMMETRY 7
From the results of [4], we have that the scalar curvature of the metric h induced by acoupled structure is given by(10) Scal( h ) = 152 ( w − ) − | w − | . Moreover, the forms φ and φ appearing in the traceless part of the Ricci tensor are(11) φ = ∗ ( w − ∧ w − ) + δ ( w − ψ + ) ,φ = − ∗ J ( dw − ) . Let us now focus on the condition (4). It forces the proportionality constant between dw − and ψ + to satisfy the following result. Proposition 3.2.
Let ( ω, ψ + ) be a coupled SU(3) -structure and suppose that dw − is pro-portional to ψ + , then it holds dw − = − | w − | ψ + . Moreover, the norm of w − is constant.Proof. First of all, observe that if dw − = kψ + for some function k ∈ C ∞ ( N ), then k has tobe constant. Indeed, taking the exterior derivatives of both sides we get dk ∧ ψ + = 0 , which implies dk = 0. Now suppose that dw − = kψ + . Then starting from w − ∧ ψ − = 0,taking the exterior derivatives of both sides and using the previous identities we have0 = dw − ∧ ψ − + w − ∧ dψ − = kψ + ∧ ψ − − w − ∧ w − ∧ ω = kω + w − ∧ ∗ w − = kω + | w − | ∗ kω + | w − | ω . Thus k = − | w − | . From the observation made at the beginning of the proof we also get that | w − | is constant. (cid:3) From Proposition 3.2 and the fact that w − is constant, we obtain the following constraint. Proposition 3.3.
Let ( ω, ψ + ) be a coupled SU(3) -structure such that dw − is proportionalto ψ + . Then the scalar curvature of the metric induced by the coupled structure is constant.Proof. Consider the expression (10) of the scalar curvature of h and conclude using the factthat both w − and | w − | are constant. (cid:3) Consider now condition (5), this implies a further constraint on the scalar curvature.
Proposition 3.4.
Let ( ω, ψ + ) be a coupled SU(3) -structure whose non-vanishing intrinsictorsion forms satisfy w − ) ≥ | w − | . Then the scalar curvature of the metric induced bythe coupled structure is positive. Moreover, it is also constant if dw − is proportional to ψ + . ANNA FINO AND ALBERTO RAFFERO
Proof.
Using the expression of the scalar curvature of a coupled structure and the inequality3( w − ) ≥ | w − | we get Scal( h ) = 152 ( w − ) − | w − | ≥ | w − | > . Moreover, if dw − is proportional to ψ + , then the scalar curvature is constant by Proposition3.3. (cid:3) It is also easy to characterize the coupled structures having dw − proportional to ψ + andinducing an Einstein metric: Proposition 3.5.
Let ( ω, ψ + ) be a coupled SU(3) -structure such that dw − is proportionalto ψ + . Then the induced metric h is Einstein if and only if the following identity holds ∗ ( w − ∧ w − ) = w − w − − | w − | ω. Proof.
Recall that a Riemannian metric h is Einstein if and only if Ric ( h ) = 0. We knowthat Ric ( h ) = ι − ( E ( φ )) + γ − ( E ( φ )) , where φ and φ for a coupled structure are given in (11). Now, using the fact that dw − isproportional to ψ + , one gets that also φ is proportional to ψ + . Thus E ( φ ) = 0, since ψ + belongs to a subspace of Λ ( N ) which is disjoint from Λ , ( N ). Moreover, φ = 14 ∗ ( w − ∧ w − ) −
12 ( w − ) ω − w − w − and E ( φ ) = 14 ∗ ( w − ∧ w − ) − w − w − + 112 | w − | ω. Therefore, Ric ( h ) = ι − ( E ( φ )) is zero if and only if E ( φ ) is zero, and from this theassertion follows. (cid:3) Examples
In this section, we examine some examples of 6-manifolds admitting an SU(3)-structuresatisfying (all or in part) the properties discussed in Section 3.4.1.
Nilmanifolds.
We recall here the definition of a nilmanifold and some useful proper-ties.
Definition 4.1.
Let G be a connected, simply connected, nilpotent Lie group and Γ acocompact discrete subgroup. The compact quotient manifold G/ Γ is called nilmanifold .In the general case, every left invariant tensor on G passes to the quotient defining aninvariant tensor on the nilmanifold G/ Γ. Moreover, all the 34 six-dimensional nilpotent Liealgebras existing up to isomorphisms [31] satisfy the following result
Proposition 4.2 ([32]) . Let g be a nilpotent Lie algebra and suppose there exists a basisof it such that the structure constants determined with respect to this basis are rationalnumbers. Then, denoted by G the simply connected nilpotent Lie group whose Lie algebrais g , there exists a discrete subgroup Γ of G such that G/ Γ is a nilmanifold. OUPLED SU(3)-STRUCTURES AND SUPERSYMMETRY 9
It then follows that there is a 1 − ω, ψ + ) on a nilmanifold and pairs ( ω, ψ + ) defining an SU(3)-structure on its nilpotent Liealgebra. This allows to work only with SU(3)-structures defined on nilpotent Lie algebras.Since every nilpotent Lie group is solvable, the following result by Milnor holds in thecase we are considering. Theorem 4.3 ([33]) . Let G be a solvable Lie group. Then every left invariant metric on G is either flat or has strictly negative scalar curvature. In particular, if a nilpotent Lie algebra is endowed with an inner product h , then Scal( h )is non-positive. As a consequence, using Proposition 3.4 it is immediate to show the Proposition 4.4.
There are no six-dimensional nilmanifolds admitting an invariant cou-pled structure satisfying the condition w − ) ≥ | w − | . Thus, we can only look for nilpotent Lie algebras endowed with a coupled structure( ω, ψ + ) having dw − proportional to ψ + . In [16], we showed that among the 34 non-isomorphic six-dimensional nilpotent Lie algebras there are only two of them admittinga coupled structure, we recall the result here. Proposition 4.5.
Let g be a six-dimensional, non-abelian, nilpotent Lie algebra endowedwith a coupled SU(3) -structure ( ω, ψ + ) . Then g is isomorphic to one of the following nilpo-tent Lie algebras I = (0 , , , , e + e , e − e ) , (12) N = (cid:0) , , , e , e + e , e − e − e (cid:1) . (13) Remark . Recall that the notation I = (0 , , , , e + e , e − e ) means that thereexists a basis of 1-forms ( e , . . . , e ) for I ∗ such that de = 0 , de = 0 , de = 0 , de = 0 , de = e + e , de = e − e , where d is the Chevalley-Eilenberg differential.Observe that the two Lie algebras I and N are isomorphic respectively to the Lie alge-bras labelled by n and n in the work [16]. Here they are given with different structureequations since in both cases the frame ( e , . . . , e ) is an adapted frame for the coupledSU(3)-structure. We emphasize some properties of these coupled structures in the followingexamples. Example 4.7. I = (0 , , , , e + e , e − e ) is (isomorphic to) the well known IwasawaLie algebra, which is the Lie algebra of the six-dimensional nilmanifold known in literatureas Iwasawa manifold (see for instance [1] for the definition). Since the frame ( e , . . . , e ) isadapted, we have that the pair ω = e + e + e ,ψ + = e − e − e − e , defines an SU(3)-structure on I . In this case dω = − ψ + and the non-vanishing intrinsictorsion forms are: w − = ,w − = − e − e + e . It is easy to check that condition (4) is satisfied dw − = − ψ + and that − | w − | = − , as we expected from Proposition 3.2. Finally, the scalar curvatureof the metric h induced by the coupled structure is Scal( h ) = − . Example 4.8.
Consider the Lie algebra N = (cid:0) , , , e , e + e , e − e − e (cid:1) . Sincethe frame ( e , . . . , e ) is adapted, we have that the pair ω = e + e + e ,ψ + = e − e − e − e , defines an SU(3)-structure on N . Moreover, dω = − ψ + and the non-vanishing intrinsictorsion forms are: w − = ,w − = − e − e + e − e + e . In this case dw − is not proportional to ψ + and the scalar curvature of the metric h inducedby the coupled structure is Scal( h ) = − . The fact that the Iwasawa manifold admits an invariant coupled structure was also ob-served in [29], where the authors wrote it was the unique nilmanifold admitting a coupledstructure they knew. Proposition 4.5 states that, up to isomorphisms, there are only twonon-abelian nilpotent Lie algebras admitting a coupled structure, one of which is the Iwa-sawa. Moreover, as observed in Example 4.7, the coupled structure on the Iwasawa Liealgebra satisfies condition (4), i.e., dw − is proportional to ψ + . Thus, it is a natural ques-tion to ask whether N admits a coupled structure satisfying (4) or not. In [10], the authorslooked for the possible nilmanifolds admitting an invariant coupled structure satisfying (4)and concluded that a systematic scan of all the possible six-dimensional nilmanifolds yieldsto two possibilities: the six-torus and the Iwasawa manifold. The six-torus has abelian Liealgebra, so it is not considered in Proposition 4.5. Moreover, the intrinsic torsion formsof any invariant SU(3)-structure defined on it are all zero. Anyway, this result seems toanswer negatively our question and we can prove this is actually what happens. Proposition 4.9.
There are no coupled
SU(3) -structures on N for which the exterior de-rivative of the intrinsic torsion form w − is proportional to ψ + .Proof. The idea is to describe all the possible coupled structures on N and see if there existsone of these whose intrinsic torsion form w − satisfies the required condition. Let us startconsidering a generic 2-form ω on N , we can write it as ω = X ≤ i
0, we can compute the almost complexstructure J induced by the stable form ψ + . Now, we change the basis from ( e , . . . , e ) toa basis ( E , . . . , E ) which is adapted for J . To do this, it suffices to define E i = e i and E i +1 = J e i for i = 1 , ,
5. With respect to ( E , . . . , E ), the matrix associated to J isskew-symmetric with non-vanishing entries given by J = 1 = J = J . We can thencompute the new structure equations with respect to the dual basis ( E , . . . , E ), obtaining dE i = 0 , i = 1 , , dE = ω √ B E ,dE = − ω ω E + √ Bω (cid:0) E + E (cid:1) ,dE = − ω ω E − ω ω E − ω ω + ω ω − ω − ω ω √ B E − ω √ B E − √ Bω E . Moreover, we have ψ + = − c Bω (cid:0) E − E − E − E (cid:1) ,ψ − = − c Bω (cid:0) E + E + E − E (cid:1) . We can write ω with respect to the new basis and impose it is of type (1 ,
1) with respectto J , obtaining 3 equations in the variables ω ij which can be solved under the constraint λ ( ψ + ) <
0. We can then consider the symmetric matrix H associated to h ( · , · ) = ω ( · , J · )with respect to the basis ( E , . . . , E ) and denote by P ⊂ A R the set on which it is positivedefinite. One can check that P = 0 when ( ω ij ) ∈ P . Now, if we let ( ω ij ) vary in the(non-empty) set Q := P ∩ { ( ω ij ) : λ ( ψ + ) < } , we have all the possible non-normalizedcoupled SU(3)-structures on N . The intrinsic torsion form w − is always − c , while w − canbe computed from its defining properties and the expression of dψ − . We are interested inthe coupled structures having w − such that dw − is proportional to ψ + . Thus, we can startwith a generic 2-form w of type (1 ,
1) with respect to J and write it as w = w E + w E + w E + w ( E + E ) + w ( E − E ) + w ( E + E ) w ( E − E ) + w ( E + E ) + w ( E − E ) , where w ij are real numbers. Then, we have to impose that w is primitive ( w ∧ ω = 0) andfulfills dψ − = − c ω − w ∧ ω and that dw is proportional to ψ + . The last condition gives rise to a set of polynomialequations in the variables w ij with coefficients depending on ω ij which can be solved in Q .The condition on dψ − gives 13 equations of the same kind as before, we can solve 4 of them,namely those obtained comparing the coefficients of E , E , E , E , but then we get that some of the remaining equations can be solved only if c = 0 or λ ( ψ + ) = 0. Theassertion is then proved. (cid:3) The previous results can be summarized as follows
Proposition 4.10.
Let g be a six-dimensional, non-abelian, nilpotent Lie algebra endowedwith a coupled SU(3) -structure ( ω, ψ + ) having dw − proportional to ψ + . Then g is isomor-phic to the Iwasawa Lie algebra. Twistor spaces.
In the work [40], it was observed that on the twistor space Z overa self-dual Einstein 4-manifold ( M , g ) there exists a coupled structure. Moreover, for asuitable value of the scalar curvature of g , the metric induced by this structure is Einstein.Recall that given a four-dimensional, oriented Riemannian manifold ( M , g ), the set ofpositive, orthogonal almost complex structures on M forms a smooth manifold Z called the twistor space of M , which can be viewed as the 2-sphere bundle π : Z → M consistingof the unit − T M [34].On Z , it is possible to define two almost complex structures (see for example [3]), one ofwhich is never integrable as shown in [15]. Let us denote it by J .When the metric g is self-dual and Einstein, Xu showed in [41] that on ( Z, J ) there existsa basis of (1 , ε , ε , ε such that the first structure equations are:(14) d ε ε ε = − (cid:18) α − tr( α ) (cid:19) ∧ ε ε ε + ε ∧ ε ε ∧ ε σε ∧ ε , where α is a 2 × σ := Scal( g )24 . Using these, it is easyto show that the following pair of forms defines a coupled SU(3)-structure on Z [40] ω = i (cid:16) ε ∧ ε + ε ∧ ε + ε ∧ ε (cid:17) , Ψ = i ( ε ∧ ε ∧ ε ) . Observe that J is the almost complex structure induced by ℜ (Ψ) and that the metricinduced by ω and J takes the following form h = ε ⊙ ε + ε ⊙ ε + ε ⊙ ε . Moreover, the non-vanishing intrinsic torsion forms have the following expressions w − = ( σ + 2) ,w − = − i ( σ − (cid:16) ε ∧ ε + ε ∧ ε − ε ∧ ε (cid:17) ,dw − is proportional to ψ + dw − = −
83 ( σ − ψ + , and 3( w − ) ≥ | w − | if and only if − √ ≤ σ ≤ √ .We can consider a local frame ( e , . . . , e ) for Λ ( Z ) such that ε = e + ie , ε = e + ie , ε = e + ie and compute the Ricci curvature of the metric induced by the coupled structure OUPLED SU(3)-STRUCTURES AND SUPERSYMMETRY 13 using the results of [4]. What we get is that the scalar curvature of h isScal( h ) = − σ + 24 σ + 8and the traceless part of the Ricci tensor of h with respect to the considered frame has thefollowing form Ric ( h ) = −
23 ( σ − σ − , , , , − , − . Thus, the metric h is Einstein if and only if σ = 1 or σ = 2, that is if and only if the scalarcurvature of g is 24 or 48 respectively. In the first case the coupled structure is actuallynearly K¨ahler since w − = 0, while in the second case we get an example of a coupledSU(3)-structure inducing an Einstein metric. More in detail, the latter has the followingnon-vanishing intrinsic torsion forms w − = ,w − = − (cid:0) e + e − e (cid:1) . In particular, the coupled constant is c = − h ) = 48.Moreover, the characterization given in Proposition 3.5 is satisfied by this example.Recall that when Scal( g ) >
0, a compact, self-dual, Einstein 4-manifold ( M , g ) is iso-metric either to S or to CP with their canonical metrics [5, Thm. 13.30], thus Z is either CP or the flag manifold SU(3) /T .4.3. G -structures with special metrics induced by coupled Einstein structures. We can now use the coupled Einstein structure on Z to construct a G -structure with fullintrinsic torsion inducing an Einstein metric and a locally conformal calibrated G -structureinducing a Ricci-flat metric.First of all, we rescale the coupled Einstein structure on Z it in the following way˜ ω = ω, ˜ ψ + = (cid:0) (cid:1) ψ + . Then, (˜ ω, ˜ ψ + ) is a coupled structure with coupled constant c = −√
10 and inducing themetric ˜ h = h . Moreover, Scal(˜ h ) = 30 and Ric(˜ h ) = 5˜ h .As we observed in Section 2, starting from the coupled Einstein structure (˜ ω, ˜ ψ + ), wecan construct a G -structure ϕ on the sin-cone S ( Z ) inducing the sin-cone metric g ϕ = dt + sin ( t )˜ h . By Proposition 2.3, we then have that g ϕ is Einstein with Einstein constant6. Moreover, it is not difficult to show that the intrinsic torsion forms of the G -structureinduced on the sin-cone by a coupled structure with coupled constant c are τ = c +47 ,τ = (cid:0) − c (cid:1) cot( t ) dt,τ = − sin(2 t )2 w − ,τ = c − (cid:0) sin ( t ) ψ − − sin ( t ) cos( t ) ψ + + sin ( t ) dt ∧ ω (cid:1) − sin ( t ) dt ∧ w − . Thus, the coupled Einstein structure (˜ ω, ˜ ψ + ) induces a G -structure with full intrinsic tor-sion and Einstein metric on the sin-cone S ( Z ). If we consider the G -structure ϕ induced on the cone C ( Z ) by (˜ ω, ˜ ψ + ), then the metric g ϕ = dt + t ˜ h is Ricci-flat by Proposition 2.2. Moreover, the non-vanishing intrinsic torsionforms of the G -structure constructed on the cone from a coupled structure with coupledconstant c are τ = (cid:0) t − t c (cid:1) dt,τ = − tw − . Therefore, the coupled Einstein structure (˜ ω, ˜ ψ + ) induces a locally conformal calibrated G -structure on the cone C ( Z ) whose associated metric is Ricci-flat. Remark . It is worth observing here that calibrated G -structures inducing a Ricci-flatmetric are actually parallel [7]. The previous example shows that a result of this kind isnot true anymore for locally conformal calibrated G -structures.5. Flows
In this section, we study the behaviour of coupled structures with respect to knownevolution equations (flows) of SU(3)-structures.The
Hitchin flow , introduced in [23] as the Hamiltonian flow of a certain functional,allows to construct (non-complete) metrics with holonomy in G starting from a suitableSU(3)-structure. The idea is to consider a 6-manifold N endowed with an SU(3)-structure( ω, ψ + ) and define a G -structure on M := I × N for some interval I ⊆ R by ϕ = dt ∧ ω + ψ + , where ω and ψ + depend on the coordinate t on I . If we require the G -structure to beparallel, we get that for each t fixed the SU(3)-structure has to be half-flat and that, when t is not fixed, the following evolution equations have to hold(15) ( ∂∂t ψ + = dω ∂∂t ω ∧ ω = − dψ − . These equations are the so called Hitchin flow equations. A solution of them with initialcondition a given SU(3)-structure ( ω (0) , ψ + (0)) exists when the latter is half-flat and ana-lytic, but may not exist when the analytic hypothesis is dropped [8]. Moreover, it is easyto show that an SU(3)-structure ( ω ( t ) , ψ + ( t )) which is half-flat for t = 0 and evolves asprescribed in (15) stays half-flat as long as it exists.In the work [14], a generalization of the Hitchin flow was used to study the modulispace of SU(3)-structure manifolds. The starting point to define this flow is to considerthe embedding of an SU(3)-structure into a non-compact manifold endowed with an inte-grable G -structure. This is motivated by the subject the authors are interested in, namelyfour-dimensional domain wall solutions of heterotic string theory that preserve N = su-persymmetry (see also [19]). In this case, the internal six-dimensional manifold is endowedwith an SU(3)-structure and one can combine it with the direction perpendicular to thedomain wall in the four-dimensional non-compact space time to get a seven-dimensionalnon-compact manifold endowed with a G -structure. The physical setting provides furtherconstraints on the intrinsic torsion forms of the G -structure, which we will recall later.One can then study under which conditions a certain class of SU(3)-structures is preservedby this generalized flow. OUPLED SU(3)-STRUCTURES AND SUPERSYMMETRY 15
It is then a natural question to ask whether the coupled condition is preserved by theHitchin flow and, more in general, which constraints arise requiring that a solution to thegeneralized Hitchin flow is coupled as long as it exists. We begin giving the followingdefinition.
Definition 5.1.
Let ( ω ( t ) , ψ + ( t )) be a solution for the Hitchin flow defined on an interval I ⊆ R containing 0 and starting from a coupled structure at t = 0. If ( ω ( t ) , ψ + ( t )) isa coupled structure for each t ∈ I , that is dω ( t ) = c ( t ) ψ + ( t ) for some smooth function c : I → R , we call it a coupled solution for the Hitchin flow.Coupled solutions for the Hitchin flow can be easily characterized and induce an almostcomplex structure not depending on t . Proposition 5.2.
Let N be a six-dimensional manifold and suppose there exists on it asolution ( ω ( t ) , ψ + ( t )) for the Hitchin flow starting from a coupled structure ( ω (0) , ψ + (0)) and defined on some interval I ⊆ R containing 0. If ( ω ( t ) , ψ + ( t )) is a coupled solution,then there exists a smooth function f : I → R such that ψ + ( t ) = f ( t ) ψ + (0) . Conversely, if the pair ( ω ( t ) , ψ + ( t )) is a solution for the Hitchin flow with ψ + ( t ) = f ( t ) ψ + (0) ,then it is a coupled solution.Proof. If ( ω ( t ) , ψ + ( t )) is a solution for the Hitchin flow with ψ + ( t ) = f ( t ) ψ + (0) then fromthe flow equation ∂∂t ψ + ( t ) = dω ( t ) we obtain dω ( t ) = ∂∂t ( f ( t ) ψ + (0)) = (cid:18) ddt f ( t ) (cid:19) ψ + (0) . Thus the solution is a coupled structure with c ( t ) = ddt f ( t ). Suppose now that the solutionis coupled, dω ( t ) = c ( t ) ψ + ( t ). Then from the flow equation we obtain ∂∂t ψ + ( t ) = c ( t ) ψ + ( t ) . Working in coordinates on N , it is easy to show that ψ + ( t ) = f ( t ) ψ + (0) , where f ( t ) = e R t c ( s ) ds . (cid:3) Corollary 5.3.
Let ( ω ( t ) , ψ + ( t )) be a coupled solution for the Hitchin flow on a six-dimensional manifold N . Then the associated almost complex structure is J ( t ) = J (0) ,that is, it does not depend on t .Proof. We know that ψ + ( t ) = f ( t ) ψ + (0), therefore J ( t ) = J ψ + ( t ) = J f ( t ) ψ + (0) = J ψ + (0) = J (0) , since the almost complex structure induced by ψ + does not change if we rescale ψ + by areal constant. (cid:3) Coupled solutions on six-dimensional nilpotent Lie algebras.
Working on six-dimensional nilpotent Lie algebras, it is possible to show that a coupled solution for theHitchin flow may not exist. As we recalled in Proposition 4.5, the only six-dimensionalnilpotent Lie algebras admitting a coupled structure are, up to isomorphisms, I and N . Ineach case, with respect to the frame we considered, the pair ( ω, ψ + ), where(16) ω = e + e + e ,ψ + = e − e − e − e , is a coupled structure with dω = − ψ + . For completeness, we observe also that ψ − = J ψ + = e + e + e − e . The following result shows our claim.
Proposition 5.4.
Consider the Hitchin flow on the six-dimensional nilpotent Lie algebras N and I . Then on I there exists a coupled solution starting from (16) at t = 0 , while on N there are no coupled solutions for the Hitchin flow starting from (16) .Proof. Let us start with I , it admits a coupled solution for the Hitchin flow already describedin [11]. We recover it here starting from a suitable pair ( ω ( t ) , ψ + ( t )) and requiring it satisfiesthe Hitchin flow equations. From Proposition 5.2, we know that ( ω ( t ) , ψ + ( t )) is a coupledsolution if and only if ψ + ( t ) = f ( t ) ψ + (0) = f ( t )( e − e − e − e ) , with f (0) = 1. It is also clear that ψ − ( t ) = f ( t ) (cid:0) e + e + e − e (cid:1) . Moreover, sincewe already know the form of the solution, we consider three smooth functions a ( t ) , a ( t ) , a ( t )with a i (0) = 1 and such that ω ( t ) = a ( t ) e + a ( t ) e + a ( t ) e . From now on we omit the t -dependence of the considered functions. The forms ω ( t ) and ψ ± ( t ) are compatible for each t and from the normalization condition we get(17) f = a a a . From the first Hitchin flow equation in (15) we obtain(18) ddt f = − a , while from the second one we have ddt ( a a ) = 0 , (19) ddt ( a a ) = 0 , (20) ddt ( a a ) = − f. (21)From (19), (20) and the starting conditions at t = 0 we deduce that a = a = 1 a . OUPLED SU(3)-STRUCTURES AND SUPERSYMMETRY 17
Using this result and (17), it holds necessarily f = 1 √ a . Thus the ODE (18) becomes ddt a = 2 a √ a and solving this we get a = (1 − t ) − . It is then easy to check that also (21) is satisfied. Then the pair ω ( t ) = (1 − t ) e + (1 − t ) e + (1 − t ) − e ,ψ + ( t ) = (1 − t ) ( e − e − e − e )is a coupled solution for the Hitchin flow.We consider now N , we will show that there are no coupled solutions starting from (16).Also in this case we need ψ + ( t ) = f ( t ) ψ + (0) = f ( t )( e − e − e − e ) , with f (0) = 1, while we consider 15 smooth real valued functions b ij ( t ), 1 ≤ i < j ≤
6, suchthat ω ( t ) = X ≤ i 0, which can not be possible. (cid:3) Generalized Hitchin flow. Since coupled solutions for the Hitchin flow may notexist in general as Proposition 5.4 states, we can consider the generalized Hitchin flow andinvestigate which properties the intrinsic torsion forms have to satisfy in order to preservethe coupled condition.In this case, we start with an SU(3)-structure ( ω, ψ + ) depending on a parameter t ∈ I ⊆ R and we construct a G -structure on M := I × N by ϕ = ν t dt ∧ ω + ℜ ( F Ψ) , where ν t ∈ C ∞ ( M ) and F is a complex valued smooth function defined on M and havingconstant module 1. Observe that the Riemannian metric defined by ϕ is g ϕ = ν t dt + h. As we already recalled, in the case of N = domain wall solutions the non-vanishingintrinsic torsion forms of the G -structure are τ , τ , τ . On M = I × N , τ and τ can bedecomposed as τ = u t dt + τ N ,τ = dt ∧ η t + τ N , where u t is a smooth function on M , τ N is a 1-form on N , η t is a 2-form on N dependingon t and τ N is a 3-form on N . Moreover, the following constraints hold u t = 12 ∂∂t φ,τ N = 12 dφ,d τ = 0 , where φ is the ten-dimensional dilaton, d denotes the exterior derivative on M and d denotes it on N .A general argument allows to write down the equations of the SU(3)-structure flow associ-ated to the embedding and some relations between the torsion forms of the SU(3)-structureand the G -structure. In particular w = 2 τ N , therefore, if we have an SU(3)-structure with vanishing w , we get dφ = 2 τ N = 0 . Following [14], we work in the gauge F = 1, in this case ϕ = ν t dt ∧ ω + ψ + . If we suppose that the structure is coupled for each t , i.e.,(22) dω ( t ) = c ( t ) ψ + ( t ) ,dψ + ( t ) = 0 ,dψ − ( t ) = − c ( t )( ω ( t )) − w − ( t ) ∧ ω ( t ) , where c : I → R is a smooth function such that w − ( t ) = − c ( t ), then the 2-form ω ( t )evolves as(23) ∂∂t ω ( t ) = λ t ω ( t ) + h t , where λ t = 2 u t − ν t w − ( t ) , (24) h t = ν t w − ( t ) − ∗ ( dν t ∧ ∗ ψ + ( t )) . (25)Moreover, it follows from a general argument involving the flow equations that dλ t = 0and using one of the constraints recalled earlier we get du t = 12 d (cid:18) ∂∂t φ (cid:19) = 12 ∂∂t ( dφ ) = 0 . Taking the exterior derivative of both sides of (24) we then have dν t = 0 , thus, ν t is actually a function of t and (25) becomes h t = ν t w − . Remark . With our convention, w − here is − w − in the work [14]. OUPLED SU(3)-STRUCTURES AND SUPERSYMMETRY 19 The flow equations for ψ + ( t ) and ψ − ( t ) determined in [14] reduce to the following in thecoupled case: ∂∂t ψ + ( t ) = 32 λ t ψ + ( t ) − τ ν t ψ − ( t ) − ν t γ, (26) ∂∂t ψ − ( t ) = 74 τ ν t ψ + ( t ) + 32 λ t ψ − ( t ) + ν t J γ, (27)where γ is a primitive 3-form of type (2 , 1) + (1 , 2) appearing in the expression of the Hodgedual of τ N on N .We derive now all the conditions that arise requiring these flow equations preserve thecoupled condition. We may sometimes omit the t -dependence of the forms for brevity.First of all, suppose that for each t the coupled condition dω ( t ) = c ( t ) ψ + ( t ) holds.Differentiating both sides with respect to t , we have d (cid:18) ∂∂t ω (cid:19) = ˙ cψ + + c (cid:18) λ t ψ + − τ ν t ψ − − ν t γ (cid:19) . Moreover, taking the exterior derivative of both sides of (23), using dν t = 0 and the hy-pothesis on the coupled condition, we obtain d (cid:18) ∂∂t ω (cid:19) = λ t cψ + + ν t dw − . Comparing the two equations it follows ν t dw − = ˙ cψ + + 12 cλ t ψ + − cτ ν t ψ − − cν t γ. Wedging both sides by ψ − and using the fact that γ ∧ ψ − = 0 since γ ∈ Λ , ⊕ Λ , , we get(28) ν t dw − ∧ ψ − = 23 ˙ c ω + 13 cλ t ω . Since for each t it holds dw − ∧ ψ − = −| w − | ω , where the norm is induced by h ( t ), equation(28) becomes − ν t | w − | ω c ω + 13 cλ t ω and the following result is proved. Proposition 5.6. Suppose that the generalized Hitchin flow preserves the coupled condition dω ( t ) = c ( t ) ψ + ( t ) , then the function c ( t ) must evolve in the following way ∂∂t c ( t ) = − c ( t ) λ t − ν t | w − ( t ) | h ( t ) . Moreover, for each t it must hold dw − = − | w − | ψ + − cτ ψ − − cγ. In order to preserve the closedness of ψ + ( t ), we need d (cid:18) ∂∂t ψ + (cid:19) = 0 . Moreover, taking the exterior derivative of both sides of the flow equation (26) of ψ + wehave d (cid:18) ∂∂t ψ + (cid:19) = − τ ν t dψ − − ν t dγ. Comparing the two equations it then follows(29) dγ = − τ ν t dψ − . Observe now that dγ ∧ ω = 0, since γ is a primitive form of type (2 , 1) + (1 , ω and recalling that dψ − ∧ ω = − cω , we get τ ν t c = 0 , and then τ = 0 , since both c and ν t cannot be zero. In particular dγ = 0 . We can summarize the results in the following Proposition 5.7. If the closedness of ψ + is preserved by the generalized Hitchin flow, thenthe intrinsic torsion form τ vanishes and the 3-form γ is closed. Let us now consider the expression of dψ − in (22) and differentiate it with respect to t having in mind the results already obtained: d (cid:18) ∂∂t ψ − (cid:19) = (cid:18) − 23 ˙ c − cλ t (cid:19) ω + (cid:18) − cν t − λ t (cid:19) w − ∧ ω − ∂∂t w − ∧ ω − ν t w − ∧ w − . Taking the exterior derivative of both sides of the flow equation (27) of ψ − ( t ) we get d (cid:18) ∂∂t ψ − (cid:19) = − λ t cω − λ t w − ∧ ω + ν t d ( J γ ) . Comparing the two equations we obtain that the flow of w − must obey the following equation ∂∂t w − ∧ ω = 16 ν t | w − | ω + (cid:18) − cν t + 12 λ t (cid:19) w − ∧ ω − ν t w − ∧ w − − ν t d ( J γ ) . We also know that the following necessary conditions deriving from the Bianchi identity d ˆ H = 0 must hold dS X = 0 , (30) dS t = ∂∂t S X , (31)where ˆ H = dt ∧ S t + S X is the component of the ten-dimensional flux along M . Remark . The other constraint obtained from the Bianchi identities was recalled earlier,it is d τ = 0.Using the previous results, it follows from [14] that for a coupled structure S X = ν − t u t ψ − + J γ, S t = 0 . From the first identity (30) we then get(32) d ( J γ ) = − ν − t u t dψ − . OUPLED SU(3)-STRUCTURES AND SUPERSYMMETRY 21 Observe that d ( J γ ) ∧ ω = 0. Thus, if we wedge both sides of (32) by ω we obtain ν − t u t c = 0 , from which follows u t = 0 and, as a consequence, d ( J γ ) = 0. The second identity (31) thenreads ∂∂t ( J γ ) = 0 . We can summarize here some of the results obtained:i) the only non-vanishing intrinsic torsion form of the G -structure after imposing allconditions is τ . Moreover, ∗ τ X = γ and η t = 0.ii) dν t = 0.iii) dγ and d ( J γ ) = 0, thus γ is harmonic.iv) λ t = ν t c ( t ) . In particular, the evolution equations of the differential forms defining the coupled structurebecome ∂∂t ω ( t ) = 23 ν t c ( t ) ω ( t ) + ν t w − ( t ) ,∂∂t ψ + ( t ) = ν t c ( t ) ψ + ( t ) − ν t γ,∂∂t ψ − ( t ) = ν t c ( t ) ψ − ( t ) + ν t J γ. Moreover, the intrinsic torsion forms of the coupled structure must evolve as ∂∂t c ( t ) = − ν t ( c ( t )) − ν t | w − ( t ) | h ( t ) ,∂∂t w − ( t ) ∧ ω ( t ) = 16 ν t | w − ( t ) | h ( t ) ( ω ( t )) − ν t c ( t ) w − ( t ) ∧ ω ( t ) − ν t ( w − ( t )) , and for each t it must hold dw − = − | w − | ψ + − cγ. The Hitchin flow as a particular case of the generalized Hitchin flow. If wesuppose that ν t = 1 and γ = 0, then ϕ = dt ∧ ω + ψ + is a parallel G -structure. In this case, the evolution equations of the differential forms ω ( t ) , ψ + ( t ) , ψ − ( t ) read(33) ∂∂t ω ( t ) = c ( t ) ω ( t ) + w − ( t ) , ∂∂t ψ + ( t ) = c ( t ) ψ + ( t ) , ∂∂t ψ − ( t ) = c ( t ) ψ − ( t ) , the evolution equations of the intrinsic torsion forms of the coupled structure must be(34) ∂∂t c ( t ) = − ( c ( t )) − | w − ( t ) | h ( t ) , ∂∂t w − ( t ) ∧ ω ( t ) = | w − ( t ) | h ( t ) ( ω ( t )) − c ( t ) w − ( t ) ∧ ω ( t ) − ( w − ( t )) , and for each t the 2-form w − has to satisfy the following property(35) dw − = − | w − | ψ + , which is one of the conditions widely discussed in Section 3.A solution of these equations which is coupled for each t is also a coupled solution for theHitchin flow equations and vice-versa. For example, the coupled solution for the Hitchinflow on the Iwasawa Lie algebra I obtained in the proof of Proposition 5.4 satisfies (33) andthe conditions (34), (35). In the general case, the presence of w − ( t ) in the flow equationsmakes rather complicated any attempt to solve them. However, we can show that a solutionof them starting from a coupled SU(3)-structure stays coupled as long as it exists. Proposition 5.9. Let ( ω ( t ) , ψ + ( t ) , c ( t ) , w − ( t )) be a solution of the equations (33) , (34) , (35) , with initial condition a coupled structure ( ω (0) , ψ + (0)) satisfying dω (0) = c (0) ψ + (0) .Then ( ω ( t ) , ψ + ( t )) is a coupled structure as long as it exists.Proof. Consider dω ( t ) − c ( t ) ψ + ( t ), differentiating with respect to t and using the hypothesiswe get (omitting the t -dependence for brevity) ∂∂t ( dω − cψ + ) = d (cid:18) ∂∂t ω (cid:19) − ˙ cψ + − c ∂∂t ψ + = 23 cdω + dw − + 13 c ψ + + 14 | w − | ψ + − c ψ + = 23 c ( dω − cψ + ) . Thus, if we denote by ρ ( t ) = dω ( t ) − c ( t ) ψ + ( t ), we have that ∂∂t ρ ( t ) = c ( t ) ρ ( t ). There-fore, ρ ( t ) = q ( t ) ρ (0) , where q ( t ) = e R t c ( s ) ds . But ρ (0) = dω (0) − c (0) ψ + (0) = 0 since( ω (0) , ψ + (0)) is coupled. Then 0 = ρ ( t ) = dω ( t ) − c ( t ) ψ + ( t ) and, as a consequence, dψ + ( t ) = 0. (cid:3) Conclusions In this paper, we considered from the mathematical point of view the properties of SU(3)-structures which are of interest in the case of N = 1 compactification of type IIA stringtheory to four-dimensional anti-de Sitter space on 6-manifolds endowed with an SU(3)-structure, namely coupled structures satisfying (all or in part) the constraints given in (4)and (5).First of all, we derived some properties of such structures and some constraints impliedby them. These need to be taken into account when one looks for explicit examples.We then turned our attention to examples of 6-manifolds endowed with this kind ofSU(3)-structures. In the case of nilmanifolds, we already knew that up to isomorphismsthere are two non-abelian nilpotent Lie algebras admitting a coupled structure from [16].Here, we showed that for only one of these the condition (4) is satisfied while the condition(5) cannot be ever satisfied. However, since in the physical setting conditions (4) and (5)can be relaxed in the presence of sources, the nilmanifolds generated by I and N may beused to construct examples of the considered type of compactification. This was done forthe Iwasawa manifold in [10], thus it would be interesting to see what happens for the OUPLED SU(3)-STRUCTURES AND SUPERSYMMETRY 23 nilmanifold corresponding to N . We also recalled an example firstly described in [40], thisis of particular interest not only because it answers a question arising from [36], but alsobecause allows to construct examples of G -structures with torsion inducing remarkablemetrics.In the last section, we considered the behaviour of the coupled condition with respectto the Hitchin flow and one of its possible generalizations determined starting from four-dimensional domain wall solutions of heterotic string theory preserving N = supersym-metry. We observed that it is not always possible to find coupled solutions for the Hitchinflow by working on explicit examples and derived the conditions implied by requiring thatthe coupled condition is preserved by the generalized Hitchin flow. An interesting openquestion would be to see whether there exist any example of a 1-parameter family of SU(3)-structures which solves the Hitchin flow equations and is coupled for at least one but notfor all t . References [1] Abbena, E., Garbiero, S., Salamon, S.: Hermitian geometry on the Iwasawa manifold. Boll. Un. Mat.Ital. B (7) 11, no. 2, suppl., 231–249 (1997)[2] Agricola, I., Chiossi, S., Friedrich, T., H¨oll, J.: Spinorial description of SU(3)- and G -manifolds.arXiv:1411.5663 (preprint)[3] Apostolov, V., Grantcharov, G., Ivanov, S.: Hermitian structures on twistor spaces. Ann. Glob. Anal.Geom. 16, no. 3, 291–308 (1998)[4] Bedulli, L., Vezzoni, L.: The Ricci tensor of SU(3)-manifolds. J. Geom. Phys. 57(4), 1125–1146 (2007)[5] Besse, A.: Einstein manifolds. Springer, Ergeb. Math. vol. 10 (1987)[6] Boyer, C.P., Galicki, K.: Sasakian geometry, holonomy, and supersymmetry. Handbook of pseudo-Riemannian geometry and supersymmetry. 39–83, IRMA Lect. Math. Theor. Phys., 16, Eur. Math.Soc., Z¨urich (2010)[7] Bryant, R.L.: Some remarks on G -structures. Proceedings of G¨okova Geometry-Topology Conference2005, 75–109, G¨okova Geometry/Topology Conference (GGT), G¨okova (2006)[8] Bryant, R.L.: Non-embedding and non-extension results in special holonomy. The many facets of ge-ometry, Oxford Univ. Press, Oxford, pp. 346–367 (2010)[9] Butruille, J.-B.: Classification des vari´et´es approximativement k¨ahleriennes homog`enes. Ann. GlobalAnal. Geom. 27 (3), 201–225 (2005)[10] Caviezel, C., Koerber, P., K¨ors, S., L¨ust, D., Tsimpis, D., Zagermann, M.: The effective theory of typeIIA AdS compactifications on nilmanifolds and cosets. Classical Quantum Gravity 26, no. 2, 025014,66 pp. (2009)[11] Chiossi, S., Fino, A.: Conformally parallel G structures on a class of solvmanifolds. Math. Z., 252,825–848 (2006)[12] Chiossi, S., Salamon, S.: The intrinsic torsion of SU(3) and G structures. In: Valencia (ed.) Geometry,Differential 2001, pp.115–133. World Scientific Publishing, River Edge (2002)[13] de Carlos, B., Micu, A., Gurrieri, S., Lukas, A.: Moduli stabilisation in heterotic string compactifica-tions. J. High Energy Phys., no. 3, 005, 36 pp. (2006)[14] de la Ossa, X., Larfors, M., Svanes, E.E.: Exploring SU(3)-Structure Moduli Spaces with Integrable G Structures. arXiv:1409.7539 (preprint)[15] Eells, J., Salamon, S.: Twistorial construction of harmonic maps of surfaces into four-manifolds. Ann.Scuola Norm. Sup. Pisa Cl. Sci. (4) 12, no. 4, 589–640 (1985)[16] Fino, A., Raffero, A.: Einstein Locally Conformal Calibrated G -structures. arXiv:1303.6137 (preprint)[17] Friedrich, T., Ivanov, S.: Parallel spinors and connections with skew-symmetric torsion in string theory.Asian J. Math. 6, no. 2, 303–335 (2002) [18] Foscolo, L., Haskins, M.: New G holonomy cones and exotic nearly K¨ahler structures on the 6-sphereand the product of a pair of 3-spheres. arXiv:1501.07838 (preprint)[19] Gray, J., Larfors, M., L¨ust, D.: Heterotic domain wall solutions and SU(3) structure manifolds. J. HighEnergy Phys. 8, 099, front matter + 36 pp. (2012)[20] Gurrieri, S.: Compactifications on half-flat manifolds. Fortschr. Phys. 53, no. 3, 278–336 (2005)[21] Gurrieri, S., Lukas, A., Micu, A.: Heterotic string compactified on half-flat manifolds. Phys. Rev. D (3)70, no. 12, 126009, 18 pp. (2004)[22] Hitchin, N.: The geometry of three-forms in six dimensions. J. Differential Geom. 55, no. 3, 547–576(2000)[23] Hitchin, N.: Stable forms and special metrics. In: Global Differential Geometry: The MathematicalLegacy of Alfred Gray. Contemp. Math. Amer. Math. Soc., vol. 288, pp. 70–89 (2001)[24] Karigiannis, S., McKay, B., Tsui, M.-P., Soliton solutions for the Laplacian co-flow of some G -structureswith symmetry. Differential Geom. Appl. 30, 318?333 (2012)[25] Klaput, M., Lukas, A., Matti, C.: Bundles over Nearly-K¨ahler Homogeneous Spaces in Heterotic StringTheory. J. High Energy Phys., no. 9, 100, 34 pp. (2011)[26] Koerber, P., L¨ust, D., Tsimpis, D.: Type IIA AdS compactifications on cosets, interpolations anddomain walls. J. High Energy Phys., no. 7, 017, 38 pp. (2008)[27] Larfors, M.: Revisiting toric SU(3) structures. Fortschr. Phys. 61, no. 12, 1031–1055 (2013)[28] Lukas, A., Matti, C.: G-structures and Domain Walls in Heterotic Theories. J. High Energy Phys., no.1, 151, 31 pp. (2011)[29] L¨ust, D., Tsimpis, D.: Supersymmetric AdS compactifications of IIA supergravity. J. High EnergyPhys., no. 2, 027, 25 pp. (2005)[30] Madsen, T.B., Salamon, S.: Half-flat structures on S × S . Ann. Global Anal. Geom. 44, 369–390(2013)[31] Magnin, L.: Sur les alg`ebres de Lie nilpotentes de dimension ≤ 7. J. Geom. Phys. 3, no.1, 119–144(1986)[32] Malcev, A.I. On a class of homogeneous spaces. Amer. Math. Soc. Translation, no. 39, 33 pp. (1951)[33] Milnor, J.: Curvatures of Left Invariant Metrics on Lie Groups. Advances in Math. 21, no. 3, 293–329(1976)[34] Muˇskarov, O.: Almost Hermitian structures on twistor spaces and their types. Atti Sem. Mat. Fis.Univ. Modena 37, no. 2, 285–297 (1989)[35] O’Neill, B.: Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics,103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York. xiii+468 pp. (1983)[36] Raffero, A.: Half-flat structures inducing Einstein metrics on homogeneous spaces. Ann. Glob. Anal.Geom. (2015). doi:10.1007/s10455-015-9457-1[37] Reichel, W.: ¨Uber die Trilinearen Alternierenden Formen in 6 und 7 Ver¨anderlichen. Dissertation,Greifswald (1907)[38] Salamon, S.: A tour of exceptional geometry. Milan J. Math. 71, 59–94 (2003)[39] Strominger, A.: Superstrings with Torsion. Nucl. Phys. B, 274–253 (1986).[40] Tomasiello, A.: New string vacua from twistor spaces. Phys. Rev. D 78, no. 4, 046007, 9 pp. (2008)[41] Xu, F.: Geometry of SU(3) manifolds. Ph. D. Thesis, Duke University (2008) Dipartimento di Matematica “G. Peano”, Universit`a di Torino, via Carlo Alberto 10, 10123Torino, Italy E-mail address : [email protected] E-mail address ::