Coupled SU(3)-structures and Supersymmetry

We review coupled ${\rm SU}(3)$-structures, also known in the literature as restricted half-flat structures, in relation to supersymmetry. In particular, we study special classes of examples admitting such structures and the behaviour of flows of ${\rm SU}(3)$-structures with respect to the coupled condition.

In this paper, we are mainly interested in the class of SU(3)-structures that are relevant for N = 1 compactifications of type IIA string theory on spaces of the form AdS 4 ×N , where AdS 4 is the four-dimensional anti de Sitter space and N is a six-dimensional compact smooth manifold. The requirement of N = 1 supersymmetry implies the existence of a globally defined complex spinor on the internal 6-manifold N . As a consequence, the structure group of N reduces to SU (3), which is equivalent to the existence on N of an almost Hermitian structure (h, J, ω) and a complex (3,0)-form Ψ of nonzero constant length satisfying some compatibility conditions. As shown in [29], in the case where the two SU(3)-structures are proportional, imposing the Killing spinor equations for four-dimensional N = 1 string vacua of type IIA on AdS 4 constrains the intrinsic torsion of the SU(3)-structure to lie in W − 1 ⊕ W − 2 . Further constraints on the torsion forms are implied by the Bianchi identities for the background fluxes in the absence of sources. Moreover, all these constraints are not only necessary but also sufficient to guarantee the existence of solutions. Examples of this kind of solutions were considered for instance in [10,26,29,40]. SU(3)-structures whose torsion class is W − 1 ⊕ W − 2 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 proportional to ψ + . Coupled structures were recently considered in [16,30,36]. They are of interest for instance because their underlying almost Hermitian structure is quasi-Kähler and because they generalize the class of nearly Kähler SU(3)-structures, namely the half-flat structures having 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 Kähler structures are known. In the homogeneous case there are only finitely many of them by [9], while new complete inhomogeneous examples were recently found on S 6 and S 3 × S 3 in [18]. Among the remarkable properties of nearly Kähler structures in dimension 6, it is worth recalling here 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 inducing Einstein metrics can exist or if requiring that a coupled structure induces an Einstein metric implies that it is actually nearly Kähler. An attempt to find coupled Einstein structures on explicit examples was done in [36], where the existence of invariant coupled Einstein structures was excluded on the compact manifold S 3 × S 3 for Ad(S 1 )-invariant Einstein metrics and on all the six-dimensional solvmanifolds. While writing this paper, we found out that the work [40], which provides a family of AdS 4 vacua in IIA string theory, contains an example of a coupled Einstein structure. This answers to the question and can be used to construct examples of G 2 -structures with non-vanishing torsion inducing Einstein and Ricci-flat metrics.
One of the main motivations to study half-flat structures is due to the role they play in the construction of seven-dimensional manifolds with holonomy contained in G 2 . More in detail, by a result of Hitchin [23], on a 6-manifold N it is possible to define a flow for SU(3)-structures, the so-called Hitchin flow, which can be solved for any given analytic half-flat structure as initial condition. A solution to the flow equations consists of a family of half-flat structures depending on a parameter t ∈ I ⊆ R and allows to define a torsionless G 2 -structure on the product manifold I × N . One question that naturally arises is then whether 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 2 -structure with torsion on the product manifold 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 2 -structure belongs to a certain torsion class. Of course, the Hitchin flow equations can be recovered as 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 = 1 2 domain wall solutions of heterotic string theory. In that case, the authors considered the non-compact seven-dimensional manifold defined by combining the direction perpendicular to the domain wall and the internal 6-manifold and observed that it is possible to define on it a G 2 -structure whose non-vanishing intrinsic torsion forms can be recovered using the results of [19,28].
Furthermore, homogeneous spaces admitting coupled structures were used to provide examples of heterotic N = 1 2 domain wall solutions with vanishing fluxes in [25] and an attempt 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 and properties regarding SU(3)-and G 2 -structures. In Section 3 we study coupled structures in relation to supersymmetry. In Section 4 we describe some explicit examples and in Section 5 we study the behaviour of flows of SU(3)-structures with respect to the coupled condition.

Review of SU(3)-structures and G 2 -structures
An SU(3)-structure on a six-dimensional smooth manifold N is the data of a Riemannian metric h, an orthogonal almost complex structure J, a 2-form ω related to h and J via the identity ω(·, ·) = h(J·, ·) and a (3, 0)-form of nonzero constant length Ψ = ψ + + iψ − which is compatible with ω, i.e., ω ∧ Ψ = 0, and satisfies the normalization condition where dV h is the Riemannian volume form of h. At each point p ∈ N there exists an horthonormal frame (e 1 , . . . , e 6 ) of T * p N , called adapted frame for the SU(3)-structure, such that ω = e 12 + e 34 + e 56 , Ψ = (e 1 + ie 2 ) ∧ (e 3 + ie 4 ) ∧ (e 5 + ie 6 ), and whose dual frame (e 1 , . . . , e 6 ) is adapted for J, i.e., Remark 2.1. 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 wedge product of a differential form θ by itself for n-times.
The intrinsic torsion τ of an SU(3)-structure is completely determined by the exterior derivatives of ω, ψ + , ψ − , as shown in [12]. More in detail, we have are the intrinsic torsion forms of the SU(3)-structure. It is then possible to divide the SU(3)-structures in classes by seeing which torsion forms vanish. For example, if ω, ψ + and ψ − are all closed, then all the torsion forms vanish and the manifold N is Calabi-Yau. If all the torsion forms but w − 1 vanish, the SU(3)-structure is said to be nearly Kähler and we write τ ∈ W − 1 . If both ψ + and ω 2 are closed, then the torsion forms w + 1 , w + 2 , w 4 , w 5 vanish, the SU(3)-structure is said to be half-flat and we write τ ∈ W − 1 ⊕ W − 2 ⊕ W 3 . As recently shown in [2], the SU(3)-structures can also be described in terms of a characterizing spinor and the spinorial field equations it satisfies.
In [4], it was shown that the Ricci and the scalar curvature of the metric h induced by an SU(3)-structure can be expressed in terms of the intrinsic torsion forms. In particular, if we consider the projections E 1 : Λ 2 (N ) → Λ 1,1 0 (N ) and E 2 : Λ 3 (N ) → Λ 2,1 0 (N ) given by where * is the Hodge operator defined using h and the volume form dV h , then the traceless part of the Ricci tensor has the following expression where the 2-form φ 1 and the 3-form φ 2 depend on the intrinsic torsion forms and their derivatives and the maps ι : S 2 + (N ) → Λ 1,1 0 (N ) and γ : S 2 − (N ) → Λ 2,1 0 (N ) are (pointwise) su(3)-modules isomorphisms (see [4] for the details). The Ricci tensor of h can then be recovered from the identity Starting from an SU(3)-structure (ω, ψ + ) on a 6-manifold N , it is possible to construct a G 2 -structure on the 7-manifold I × N , where I ⊆ R is a connected open interval. Before describing how, we recall that a G 2 -structure on a seven-dimensional manifold M is characterized by the existence of a globally defined 3-form ϕ inducing a Riemannian metric g ϕ and a volume form dV gϕ given by for any pair of vector fields X, Y ∈ X(M ). The intrinsic torsion of a G 2 -structure ϕ is completely determined by the exterior derivatives of ϕ and * ϕ ϕ, where * ϕ is the Hodge operator defined using the metric g ϕ and the volume form dV gϕ . More in detail, it holds [7] (3) : ϕ ∧ ρ = 0 and * ϕ ϕ ∧ ρ = 0} are the intrinsic torsion forms of the G 2 -structure. Also in this case it is possible to classify the G 2 -structures in terms of the non-vanishing torsion forms. For example, if ϕ is both closed and co-closed, then all the torsion forms vanish, Hol(g ϕ ) ⊆ G 2 and the G 2 -structure is called parallel. If ϕ is a closed form, then all the torsion forms but τ 2 vanish and the G 2 -structure is said to be calibrated. If the only non-vanishing torsion forms are τ 1 and τ 2 , then at least locally the metric g ϕ is conformally equivalent to the metric induced by a calibrated G 2 -structure and the G 2 -structure is called locally conformal calibrated. If the only vanishing torsion form is τ 2 , then the G 2 -structure is said to be integrable. In this case there exists a unique affine connection with totally skew-symmetric torsion preserving the G 2 -structure by [17].
Consider now (ω, ψ + ) and two smooth functions F : I → C − {0} and G : I → R + , the following 3-form defines a G 2 -structure on I × N ( [24]) where t is the coordinate on I. Moreover, we have For some particular choices of the interval I and the functions F and G, we obtain the following remarkable manifolds: • the cylinder Cyl(N ) with metric dt 2 + h, if I = R and G, F ≡ 1, • the cone C(N ) with the metric dt 2 + t 2 h, if I = R + , G ≡ 1 and F (t) = t, • the sin-cone SC(N ) with the metric dt 2 + sin 2 (t)h, if I = (0, π), G ≡ 1 and F (t) = sin(t)e i t 3 . Observe that with the choice G ≡ 1, the manifold I ×N with metric dt 2 +|F | 2 h is the warped product of I and N with warping function |F |. Using the expression of the Ricci tensor of the warped product metric [35], it is possible to show the following general properties (see also [6]). Proposition 2.2. Let (M m , g) be a Riemannian manifold of dimension m. Then the cone metric dt 2 + t 2 g is Ricci-flat if and only if the metric g is Einstein with Ric(g) = (m − 1)g. Proposition 2.3. Let (M m , g) be a Riemannian manifold of dimension m with Einstein metric g such that Ric(g) = (m − 1)g. Then the sin-cone metric dt 2 + sin 2 (t)h is Einstein with Einstein constant m.

Coupled structures and Supersymmetry
In [29], the authors considered the problem of finding necessary and sufficient conditions for N = 1 compactification of (massive) IIA supergravity to four-dimensional anti-de Sitter space on manifolds endowed with an SU(3)-structure. As a result, they obtained a set of constraints the intrinsic torsion forms of the SU(3)-structure (ω, ψ + ) on the internal manifold have to satisfy, we recall them here briefly. Supersymmetry equations and the Bianchi identities constrain the intrinsic torsion to lie in the space W − 1 ⊕ W − 2 , i.e., the only non-vanishing intrinsic torsion forms are w − 1 and w − 2 . Furthermore, in absence of sources, the Bianchi identities provide a further constraint on the exterior derivative of w − 2 (4) dw − 2 ∝ ψ + , and the norms of w − 1 and w − 2 have to satisfy the following inequality [26] where | · | denotes the norm with respect to the metric h induced by the SU(3)-structure. In the massless limit, the solutions reduce to AdS 4 ×N , N being a compact 6-manifold endowed with an SU(3)-structure with torsion in W − 1 ⊕W − 2 and for which (4) holds. Moreover, it was observed 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 kind of 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 − 1 ⊕ W − 2 are known as coupled structures [38] or restricted half-flat structures [27] in literature. They can be defined as the subclass of half-flat structures having w 3 ≡ 0. In this case, dω is proportional to ψ + , the intrinsic torsion form w − 1 is constant [36] and has to be nonzero if we want the intrinsic torsion τ to belong to the class The 2-form w − 2 lies in the space Λ 1,1 0 (N ), therefore it satisfies the following properties: Using (9) and the expression of dψ − , it is easy to show that the 2 Observe that if a manifold admits a coupled structure (ω, ψ + ) with coupled constant c ∈ R − {0} such that dω = cψ + , then one can choose a nonzero real constant r, defineω := r 2 ω,ψ + := r 3 ψ + and obtain a new coupled structure (ω,ψ + ) with coupled constantc = c r . In particular, it is always possible to find a coupled structure having positive coupled constant.
From the results of [4], we have that the scalar curvature of the metric h induced by a coupled structure is given by Moreover, the forms φ 1 and φ 2 appearing in the traceless part of the Ricci tensor are . Let us now focus on the condition (4). It forces the proportionality constant between dw − 2 and ψ + to satisfy the following result. Proposition 3.2. Let (ω, ψ + ) be a coupled SU(3)-structure and suppose that dw − 2 is proportional to ψ + , then it holds , then k has to be constant. Indeed, taking the exterior derivatives of both sides we get dk ∧ ψ + = 0, which implies dk = 0. Now suppose that dw − 2 = kψ + . Then starting from w − 2 ∧ ψ − = 0, taking the exterior derivatives of both sides and using the previous identities we have . From the observation made at the beginning of the proof we also get that |w − 2 | is constant.
From Proposition 3.2 and the fact that w − 1 is constant, we obtain the following constraint.
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 fact that both w − 1 and |w − 2 | are constant.
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 intrinsic torsion forms satisfy Then the scalar curvature of the metric induced by the coupled structure is positive. Moreover, it is also constant if dw − 2 is proportional to ψ + .
Proof. Using the expression of the scalar curvature of a coupled structure and the inequality Moreover, if dw − 2 is proportional to ψ + , then the scalar curvature is constant by Proposition 3.3.
It is also easy to characterize the coupled structures having dw − 2 proportional to ψ + and inducing an Einstein metric:

Proof. Recall that a Riemannian metric h is Einstein if and only if Ric
, where φ 1 and φ 2 for a coupled structure are given in (11). Now, using the fact that dw − 2 is proportional to ψ + , one gets that also φ 2 is proportional to ψ + . Thus E 2 (φ 2 ) = 0, since ψ + belongs to a subspace of Λ 3 (N ) which is disjoint from Λ 2,1 0 (N ). Moreover, is zero if and only if E 1 (φ 1 ) is zero, and from this the assertion follows.

Examples
In this section, we examine some examples of 6-manifolds admitting an SU(3)-structure satisfying (all or in part) the properties discussed in Section 3.

Nilmanifolds.
We recall here the definition of a nilmanifold and some useful properties.
Definition 4.1. Let G be a connected, simply connected, nilpotent Lie group and Γ a cocompact 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 an invariant tensor on the nilmanifold G/Γ. Moreover, all the 34 six-dimensional nilpotent Lie algebras existing up to isomorphisms [31] satisfy the following result ). Let g be a nilpotent Lie algebra and suppose there exists a basis of it such that the structure constants determined with respect to this basis are rational numbers. Then, denoted by G the simply connected nilpotent Lie group whose Lie algebra is g, there exists a discrete subgroup Γ of G such that G/Γ is a nilmanifold.
It then follows that there is a 1 − 1 correspondence between invariant SU(3)-structures (ω, ψ + ) on a nilmanifold and pairs (ω, ψ + ) defining an SU(3)-structure on its nilpotent Lie algebra. 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 the case we are considering. 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 coupled structure satisfying the condition 3(w − 1 ) 2 ≥ |w − 2 | 2 . Thus, we can only look for nilpotent Lie algebras endowed with a coupled structure (ω, ψ + ) having dw − 2 proportional to ψ + . In [16], we showed that among the 34 nonisomorphic six-dimensional nilpotent Lie algebras there are only two of them admitting a coupled structure, we recall the result here.
Observe that the two Lie algebras I and N are isomorphic respectively to the Lie algebras labelled by n 28 and n 9 in the work [16]. Here they are given with different structure equations since in both cases the frame (e 1 , . . . , e 6 ) is an adapted frame for the coupled SU(3)-structure. We emphasize some properties of these coupled structures in the following examples. It is easy to check that condition (4) is satisfied  The fact that the Iwasawa manifold admits an invariant coupled structure was also observed in [29], where the authors wrote it was the unique nilmanifold admitting a coupled structure they knew. Proposition 4.5 states that, up to isomorphisms, there are only two non-abelian nilpotent Lie algebras admitting a coupled structure, one of which is the Iwasawa. Moreover, as observed in Example 4.7, the coupled structure on the Iwasawa Lie algebra satisfies condition (4), i.e., dw − 2 is proportional to ψ + . Thus, it is a natural question to ask whether N admits a coupled structure satisfying (4) or not. In [10], the authors looked for the possible nilmanifolds admitting an invariant coupled structure satisfying (4) and concluded that a systematic scan of all the possible six-dimensional nilmanifolds yields to two possibilities: the six-torus and the Iwasawa manifold. The six-torus has abelian Lie algebra, so it is not considered in Proposition 4.5. Moreover, the intrinsic torsion forms of any invariant SU(3)-structure defined on it are all zero. Anyway, this result seems to answer negatively our question and we can prove this is actually what happens. where ω ij are real numbers. We may think the 15-tuple (ω 12 , . . . , ω 56 ) =: (ω ij ) as a point in the affine space A 15 R − {0}. The homogeneous polynomial P of degree 3 in the unknowns ω ij appearing as coefficient of e 123456 in the expression of ω 3 has to be non-vanishing, this gives a first constraint for (ω ij ). Since we want a coupled structure, we consider a 3-form ψ + on N given by ψ + = cdω, for some nonzero real number c. Assuming λ(ψ + ) = −4c 4 ω 2 56 (ω 36 ω 56 − ω 45 ω 56 − ω 2 46 + ω 2 56 ) < 0, that is ω 56 = 0 and B := ω 36 ω 56 −ω 45 ω 56 −ω 2 46 +ω 2 56 > 0, we can compute the almost complex structure J induced by the stable form ψ + . Now, we change the basis from (e 1 , . . . , e 6 ) to a basis (E 1 , . . . , E 6 ) which is adapted for J. To do this, it suffices to define E i = e i and E i+1 = Je i for i = 1, 3, 5. With respect to (E 1 , . . . , E 6 ), the matrix associated to J is skew-symmetric with non-vanishing entries given by J 2 1 = 1 = J 4 3 = J 6 5 . We can then compute the new structure equations with respect to the dual basis (E 1 , . . . , E 6 ), obtaining Moreover, we have We can write ω with respect to the new basis and impose it is of type (1, 1) with respect to 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 1 , . . . , E 6 ) and denote by P ⊂ A 15 R the set on which it is positive definite. 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 + ) < 0}, we have all the possible non-normalized coupled SU(3)-structures on N. The intrinsic torsion form w − 1 is always − 2 3c , while w − 2 can be computed from its defining properties and the expression of dψ − . We are interested in the coupled structures having w − 2 such that dw − 2 is proportional to ψ + . Thus, we can start with a generic 2-form w of type (1, 1) with respect to J and write it as where w ij are real numbers. Then, we have to impose that w is primitive (w ∧ ω 2 = 0) and fulfills dψ − = − 2 3c ω 2 − w ∧ ω and that dw is proportional to ψ + . The last condition gives rise to a set of polynomial equations 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 3456 , E 2356 , E 1256 , E 2345 , but then we get that some of the remaining equations can be solved only if c = 0 or λ(ψ + ) = 0. The assertion is then proved.
The previous results can be summarized as follows Proposition 4.10. Let g be a six-dimensional, non-abelian, nilpotent Lie algebra endowed with a coupled SU(3)-structure (ω, ψ + ) having dw − 2 proportional to ψ + . Then g is isomorphic to the Iwasawa Lie algebra.

4.2.
Twistor spaces. In the work [40], it was observed that on the twistor space Z over a self-dual Einstein 4-manifold (M 4 , g) there exists a coupled structure. Moreover, for a suitable 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 4 , g), the set of positive, orthogonal almost complex structures on M 4 forms a smooth manifold Z called the twistor space of M 4 , which can be viewed as the 2-sphere bundle π : Z → M 4 consisting of the unit −1 eigenvectors of the Hodge operator acting on Λ 2 T M 4 [34].
On Z, it is possible to define two almost complex structures (see for example [3]), one of which 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 exists a basis of (1, 0)-forms ε 1 , ε 2 , ε 3 such that the first structure equations are: where α is a 2 × 2 skew-Hermitian matrix of 1-forms and σ := Scal(g) 24 . Using these, it is easy to show that the following pair of forms defines a coupled SU(3)-structure on Z [40] Observe that J is the almost complex structure induced by ℜ(Ψ) and that the metric induced by ω and J takes the following form Moreover, the non-vanishing intrinsic torsion forms have the following expressions We can consider a local frame (e 1 , . . . , e 6 ) for Λ 1 (Z) such that ε 1 = e 1 + ie 2 , ε 2 = e 3 + ie 4 , ε 3 = e 5 +ie 6 and compute the Ricci curvature of the metric induced by the coupled structure using the results of [4]. What we get is that the scalar curvature of h is Scal(h) = −2σ 2 + 24σ + 8 and the traceless part of the Ricci tensor of h with respect to the considered frame has the following form 1, 1, 1, −2, −2).
Thus, the metric h is Einstein if and only if σ = 1 or σ = 2, that is if and only if the scalar curvature of g is 24 or 48 respectively. In the first case the coupled structure is actually nearly Kähler since w − 2 = 0, while in the second case we get an example of a coupled SU(3)-structure inducing an Einstein metric. More in detail, the latter has the following non-vanishing intrinsic torsion forms 3 e 12 + e 34 − 2e 56 . In particular, the coupled constant is c = −4 and the scalar curvature is Scal(h) = 48. Moreover, the characterization given in Proposition 3.5 is satisfied by this example.

G 2 -structures with special metrics induced by coupled Einstein structures.
We can now use the coupled Einstein structure on Z to construct a G 2 -structure with full intrinsic torsion inducing an Einstein metric and a locally conformal calibrated G 2 -structure inducing a Ricci-flat metric.
First of all, we rescale the coupled Einstein structure on Z it in the following waỹ Then, (ω,ψ + ) is a coupled structure with coupled constant c = − √ 10 and inducing the metrich = 8 5 h. Moreover, Scal(h) = 30 and Ric(h) = 5h. As we observed in Section 2, starting from the coupled Einstein structure (ω,ψ + ), we can construct a G 2 -structure ϕ on the sin-cone S(Z) inducing the sin-cone metric g ϕ = dt 2 + sin 2 (t)h. By Proposition 2.3, we then have that g ϕ is Einstein with Einstein constant 6. Moreover, it is not difficult to show that the intrinsic torsion forms of the G 2 -structure induced on the sin-cone by a coupled structure with coupled constant c are Thus, the coupled Einstein structure (ω,ψ + ) induces a G 2 -structure with full intrinsic torsion and Einstein metric on the sin-cone S(Z).
If we consider the G 2 -structure ϕ induced on the cone C(Z) by (ω,ψ + ), then the metric g ϕ = dt 2 + t 2h is Ricci-flat by Proposition 2.2. Moreover, the non-vanishing intrinsic torsion forms of the G 2 -structure constructed on the cone from a coupled structure with coupled constant c are Therefore, the coupled Einstein structure (ω,ψ + ) induces a locally conformal calibrated G 2 -structure on the cone C(Z) whose associated metric is Ricci-flat.
Remark 4.11. It is worth observing here that calibrated G 2 -structures inducing a Ricci-flat metric are actually parallel [7]. The previous example shows that a result of this kind is not true anymore for locally conformal calibrated G 2 -structures.

Flows
In this section, we study the behaviour of coupled structures with respect to known evolution 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 2 starting from a suitable SU(3)-structure. The idea is to consider a 6-manifold N endowed with an SU(3)-structure (ω, ψ + ) and define a G 2 -structure on M := I × N for some interval I ⊆ R by where ω and ψ + depend on the coordinate t on I. If we require the G 2 -structure to be parallel, 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 These equations are the so called Hitchin flow equations. A solution of them with initial condition a given SU(3)-structure (ω(0), ψ + (0)) exists when the latter is half-flat and analytic, but may not exist when the analytic hypothesis is dropped [8]. Moreover, it is easy to show that an SU(3)-structure (ω(t), ψ + (t)) which is half-flat for t = 0 and evolves as prescribed 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 moduli space of SU(3)-structure manifolds. The starting point to define this flow is to consider the embedding of an SU(3)-structure into a non-compact manifold endowed with an integrable G 2 -structure. This is motivated by the subject the authors are interested in, namely four-dimensional domain wall solutions of heterotic string theory that preserve N = 1 2 supersymmetry (see also [19]). In this case, the internal six-dimensional manifold is endowed with an SU(3)-structure and one can combine it with the direction perpendicular to the domain wall in the four-dimensional non-compact space time to get a seven-dimensional non-compact manifold endowed with a G 2 -structure. The physical setting provides further constraints on the intrinsic torsion forms of the G 2 -structure, which we will recall later. One can then study under which conditions a certain class of SU(3)-structures is preserved by this generalized flow.
It is then a natural question to ask whether the coupled condition is preserved by the Hitchin flow and, more in general, which constraints arise requiring that a solution to the generalized Hitchin flow is coupled as long as it exists. We begin giving the following definition.
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)) is a 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 almost complex structure not depending on t.
Proposition 5.2. Let N be a six-dimensional manifold and suppose there exists on it a solution (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 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 from the flow equation ∂ ∂t ψ + (t) = dω(t) we obtain Thus the solution is a coupled structure with c(t) = d dt f (t). Suppose now that the solution is coupled, dω(t) = c(t)ψ + (t). Then from the flow equation we obtain Working in coordinates on N , it is easy to show that Corollary 5.3. Let (ω(t), ψ + (t)) be a coupled solution for the Hitchin flow on a sixdimensional 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 since the almost complex structure induced by ψ + does not change if we rescale ψ + by a real constant. Proof. Let us start with I, it admits a coupled solution for the Hitchin flow already described in [11]. We recover it here starting from a suitable pair (ω(t), ψ + (t)) and requiring it satisfies the Hitchin flow equations. From Proposition 5.2, we know that (ω(t), ψ + (t)) is a coupled solution if and only if 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 From the first Hitchin flow equation in (15) we obtain while from the second one we have From (19), (20) and the starting conditions at t = 0 we deduce that Using this result and (17), it holds necessarily Thus the ODE (18) becomes d dt and solving this we get . It is then easy to check that also (21) is satisfied. Then the pair We consider now N, we will show that there are no coupled solutions starting from (16). Also in this case we need First of all, we impose that the equations resulting from the compatibility condition ω(t) ∧ ψ + (t) = 0 are satisfied, then we consider the first and the second Hitchin flow equations and we compute the ODEs deriving from them. What we obtain after solving some of these differential equations is that f ≡ 0, which can not be possible.

Generalized Hitchin flow.
Since coupled solutions for the Hitchin flow may not exist in general as Proposition 5.4 states, we can consider the generalized Hitchin flow and investigate which properties the intrinsic torsion forms have to satisfy in order to preserve the 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 2 -structure on M := I × N by where ν t ∈ C ∞ (M ) and F is a complex valued smooth function defined on M and having constant module 1. Observe that the Riemannian metric defined by ϕ is g ϕ = ν 2 t dt 2 + h. As we already recalled, in the case of N = 1 2 domain wall solutions the non-vanishing intrinsic torsion forms of the G 2 -structure are τ 0 , τ 1 , τ 3 . On M = I × N , τ 1 and τ 3 can be decomposed as where u t is a smooth function on M , τ N 1 is a 1-form on N , η t is a 2-form on N depending on t and τ N 3 is a 3-form on N . Moreover, the following constraints hold where φ is the ten-dimensional dilaton, d 7 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 associated to the embedding and some relations between the torsion forms of the SU(3)-structure and the G 2 -structure. In particular w 4 = 2τ N 1 , therefore, if we have an SU(3)-structure with vanishing w 4 , we get dφ = 2τ N 1 = 0. Following [14], we work in the gauge F = 1, in this case If we suppose that the structure is coupled for each t, i.e., (22) where c : I → R is a smooth function such that w − 1 (t) = − 2 3 c(t), then the 2-form ω(t) evolves as Moreover, it follows from a general argument involving the flow equations that dλ t = 0 and using one of the constraints recalled earlier we get 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 − 2 . Remark 5.5. With our convention, w − 2 here is −w − 2 in the work [14].
A solution of these equations which is coupled for each t is also a coupled solution for the Hitchin flow equations and vice-versa. For example, the coupled solution for the Hitchin flow on the Iwasawa Lie algebra I obtained in the proof of Proposition 5.4 satisfies (33) and the conditions (34), (35). In the general case, the presence of w − 2 (t) in the flow equations makes rather complicated any attempt to solve them. However, we can show that a solution of 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 − 2 (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.

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 string theory 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 implied by 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 of SU(3)-structures. In the case of nilmanifolds, we already knew that up to isomorphisms there 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 be used to construct examples of the considered type of compactification. This was done for the Iwasawa manifold in [10], thus it would be interesting to see what happens for the nilmanifold corresponding to N. We also recalled an example firstly described in [40], this is of particular interest not only because it answers a question arising from [36], but also because allows to construct examples of G 2 -structures with torsion inducing remarkable metrics.
In the last section, we considered the behaviour of the coupled condition with respect to the Hitchin flow and one of its possible generalizations determined starting from fourdimensional domain wall solutions of heterotic string theory preserving N = 1 2 supersymmetry. We observed that it is not always possible to find coupled solutions for the Hitchin flow by working on explicit examples and derived the conditions implied by requiring that the coupled condition is preserved by the generalized Hitchin flow. An interesting open question 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 not for all t.