Coupled SU(3)-Structures and Supersymmetry

Abstract

We review coupled 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 behavior of flows of SU(3)-structures with respect to the coupled condition.

1. Introduction

In the physical literature, manifolds endowed with SU(3)-structures have been frequently considered to construct string vacua [1–10].

In this paper, we are mainly interested in the class of SU(3)-structures that are relevant for $\mathcal{N}=1$ compactifications of Type IIA string theory on spaces of the form AdS₄ × N, where AdS₄ is the four-dimensional anti-de Sitter space and N is a six-dimensional compact smooth manifold. The requirement of $\mathcal{N}=1$ supersymmetry implies the existence of a globally-defined complex spinor on the internal six-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 [3], in the case where the two SU(3)-structures are proportional, imposing the Killing spinor equations for four-dimensional $\mathcal{N}=1$ string vacua of Type IIA on AdS₄ constrains the intrinsic torsion of the SU(3)-structure to lie in ${\mathcal{W}}_1^{-}\oplus {\mathcal{W}}_2^{-}$. Further constraints on the torsion forms are implied by the Bianchi identities for the background fluxes in the absence of sources. Moreover, all of 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 [3,6,11,12].

SU(3)-structures whose torsion class is ${\mathcal{W}}_1^{-}\oplus {\mathcal{W}}_2^{-}$ are known as coupled SU(3)-structures [13] 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 [14–16]. 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 ${\psi }_{-}:=\Im (\mathrm{\Psi })$. 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 [17], while new complete inhomogeneous examples were recently found on S⁶ and S³ × S³ in [18]. Among the remarkable properties of nearly Kähler structures in dimension six, 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 [16], where the existence of invariant coupled Einstein structures was excluded on the compact manifold S³ × S³ for Ad(S¹)-invariant Einstein metrics and on all of the six-dimensional Einstein solvmanifolds. While writing this paper, we found out that the work [6], which provides a family of AdS₄ vacua in IIA string theory, contains an example of a coupled Einstein structure. This answers the question and can be used to construct examples of G₂-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 G₂. More in detail, by a result of Hitchin [19], on a six-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 the 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 one to define a torsionless G₂-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 G₂-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 G₂-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 [20], 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 $\mathcal{N}=\frac{1}{2}$ 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 six-manifold and observed that it is possible to define on it a G₂-structure whose non-vanishing intrinsic torsion forms can be recovered using the results of [8,9].

Furthermore, homogeneous spaces admitting coupled structures were used to provide examples of heterotic $\mathcal{N}=\frac{1}{2}$ domain wall solutions with vanishing fluxes in [7], and an attempt to generalize this result in a more general case was done in [9].

The present paper is organized as follows. In Section 2, we review some definitions and properties regarding SU(3)- and G₂-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 behavior of flows of SU(3)-structures with respect to the coupled condition.

2. Review of SU(3)-Structures and G₂-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 two-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.,

$$\omega \wedge \mathrm{\Psi }=0,$$

and satisfies the normalization condition:

$$\frac{i}{2}(\mathrm{\Psi }\wedge \overline{\mathrm{\Psi }})=\frac{2}{3}{\omega }^3=4d{V}_h,$$

where dV_h is the Riemannian volume form of h. At each point p ∈ N, there exists an h-orthonormal frame (e¹,…, e⁶) of $T_p^{*}N$, called the adapted frame for the SU(3)-structure, such that:

$$\begin{array}{l}\omega =e^{12}+e^{34}+e^{56},\\ \mathrm{\Psi }=(e^1+i{e}^2)\wedge (e^3+i{e}^4)\wedge (e^5+i{e}^6),\end{array}$$

and whose dual frame (e₁,…, e₆) is adapted for J, i.e.,

$$J{e}_i=e_{i+1},i=1,3,5.$$

Remark 1. Here and hereafter, the notation e^ijk⋯ is a shortening for the wedge product eⁱ∧e^j ∧e^k ∧⋯. Moreover, we will also use the notation θⁿ as a shortening for the wedge product of a differential form θ by itself for n-times.

Using the results of [21,22], one can show that an SU(3)-structure actually depends only on the pair (ω, ψ₊); let us recall briefly how. For each p ∈ N, let V := T_pN; denote by A : Λ⁵ (V^*) → V ⊗ Λ⁶(V^*) the canonical isomorphism given by A(ξ) = v ⊗ Ω, where i_vΩ = ξ, and define for a fixed ρ ∈ Λ³(V^*):

$$K_{\rho }:V\to V\otimes {\mathrm{\Lambda }}^6(V*),K_{\rho }(v)=A((i_v\rho )\wedge \rho )$$

and

$$\lambda :{\mathrm{\Lambda }}^3(V*)\to {({\mathrm{\Lambda }}^6(V*))}^{\otimes 2},\lambda (\rho )=\frac{1}{6}\mathrm{tr}{K}_{\rho }^2.$$

If λ(ρ) ≠ 0, $\sqrt{\left|\lambda (\rho )\right|}\in {\mathrm{\Lambda }}^6(V*)$ defines a volume form by choosing the orientation of V for which ω³ is positively oriented. Moreover, whenever λ(ρ) < 0, the following endomorphism defines an almost complex structure:

$$J_{\rho }:=-\frac{1}{\sqrt{-\lambda (\rho )}}{K}_{\rho }.$$

An SU(3)-structure on N can then be defined as a pair (ω, ψ₊), such that the two-form ω is non degenerate, i.e., ω³ ≠ 0, the three-form ψ₊ is compatible with ω and satisfies λ(ψ₊(p)) < 0 for each p ∈ N, the almost complex structure is J = J_ψ+, the imaginary part of Ψ is given by ψ₋ := Jψ₊, the normalization condition holds and h(·, ·) := ω(·, J·) defines a Riemannian metric.

The intrinsic torsion τ of an SU(3)-structure is completely determined by the exterior derivatives of ω, ψ₊, ψ₋, as shown in [23]. More in detail, we have:

$$\begin{array}{l}d\omega =-\frac{3}{2}{w}_1^{-}{\psi }_{+}+\frac{3}{2}{w}_1^{+}{\psi }_{-}+w_3+w_4\wedge \omega ,\\ d{\psi }_{+}=w_1^{+}{\omega }^2-w_2^{+}\wedge \omega +w_5\wedge {\psi }_{+},\\ d{\psi }_{-}=w_1^{-}{\omega }^2-w_2^{-}\wedge \omega +J{w}_5\wedge {\psi }_{+},\end{array}$$

(1)

where $w_1^{\pm }\in C^{\infty }(N)$, $w_2^{\pm }\in {\mathrm{\Lambda }}_0^{1,1}(N)$, $w_3\in {\mathrm{\Lambda }}_0^{2,1}(N)$, w₄, w₅ ∈ Λ¹ (N) are the intrinsic torsion forms of the SU(3)-structure. It is then possible to divide the SU(3)-structures into classes by seeing which torsion forms vanish. For example, if ω, ψ₊ and ψ₋ are all closed, then all of the torsion forms vanish, and the manifold N is Calabi–Yau. If all of the torsion forms, but $w_1^{-}$, vanish, the SU(3)-structure is said to be nearly Kähler, and we write $\tau \in W_1^{-}$. If both ψ₊ and ω² are closed, then the torsion forms $w_1^{+}$, $w_2^{+}$, w₄, w₅ vanish, the SU(3)-structure is said to be half-flat, and we write $\tau \in W_1^{-}\oplus W_2^{-}\oplus W_3$. As recently shown in [24], the SU(3)-structures can also be described in terms of a characterizing spinor and the spinorial field equations it satisfies.

In [25], 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 $E_1:{\mathrm{\Lambda }}^2(N)\to {\mathrm{\Lambda }}_0^{1,1}(N)$ and $E_2:{\mathrm{\Lambda }}^3(N)\to {\mathrm{\Lambda }}_0^{2,1}(N)$given by:

$$\begin{array}{l}{E}_1(\beta )=\frac{1}{2}(\beta +J\beta )-\frac{1}{18}*((*(\beta +J\beta )+(\beta +J\beta )\wedge \omega )\wedge \omega )\omega ,\\ E_2(\rho )=\rho -\frac{1}{2}*(J\rho \wedge \omega )\wedge \omega -\frac{1}{4}*(\rho \wedge {\psi }_{-}){\psi }_{+}-\frac{1}{4}*({\psi }_{+}\wedge \rho ){\psi }_{-},\end{array}$$

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:

$${\mathrm{Ric}}^0(h)={\iota }^{-1}(E_1({\varphi }_1))+{\gamma }^{-1}(E_2({\varphi }_2)),$$

where the two-form ϕ₁ and the three-form ϕ₂ depend on the intrinsic torsion forms and their derivatives and the maps $\iota :S_{+}^2(N)\to {\mathrm{\Lambda }}_0^{1,1}(N)$ and $\gamma :S_{-}^2(N)\to {\mathrm{\Lambda }}_0^{2,1}(N)$ are (pointwise) $\mathrm{su}(3)\text{-module}$ isomorphisms (see [25] for details). The Ricci tensor of h can then be recovered from the identity:

$$\mathrm{Ric}(h)=\frac{1}{6}\mathrm{Scal}(h)h+{\mathrm{Ric}}^0(h).$$

Starting from an SU(3)-structure (ω, ψ₊) on a six-manifold N, it is possible to construct a G₂-structure on the seven-manifold I × N, where I ⊆ ℝ is a connected open interval. Before describing how, we recall that a G₂-structure on a seven-dimensional manifold M is characterized by the existence of a globally-defined three-form φ inducing a Riemannian metric g_φ and a volume form dV_gφ given by:

$$g_{\phi }(X,Y)d{V}_{g_{\phi }}=\frac{1}{6}{i}_X\phi \wedge i_Y\phi \wedge \phi ,$$

(2)

for any pair of vector fields $X,Y\in \mathfrak{X}(M)$. The intrinsic torsion of a G₂-structure φ is completely determined by the exterior derivatives of φ and ∗_φφ, where ∗_φ is the Hodge operator defined using the metric g_φ and the volume form dV_gφ. More in detail, it holds [26]:

$$\begin{array}{l}d\phi ={\tau }_0{\ast }_{\phi }\phi +3{\tau }_1\wedge \phi +{\ast }_{\phi }{\tau }_3,\\ d{\ast }_{\phi }\phi =4{\tau }_1\wedge {\ast }_{\phi }\phi +{\tau }_2\wedge \phi ,\end{array}$$

(3)

where τ₀ ∈ C^∞(M), τ₁ ∈ Λ¹(M), ${\tau }_2\in {\mathrm{\Lambda }}_{14}^2(M)=\{\beta \in {\mathrm{\Lambda }}^2(M)\}:{\ast }_{\phi }(\phi \wedge \beta )=-\beta \}$, ${\tau }_3\in {\mathrm{\Lambda }}_{27}^3(M)=\{\rho \in {\mathrm{\Lambda }}^3(M):\phi \wedge \rho =0\mathrm{and}{\ast }_{\phi }\phi \wedge \rho =0\}$ are the intrinsic torsion forms of the G₂-structure. Furthermore, in this case, it is possible to classify the G₂-structures in terms of the non-vanishing torsion forms. For example, if φ is both closed and co-closed, then all of the torsion forms vanish, Hol(g_φ) ⊆ G₂ and the G₂-structure is called parallel. If φ is a closed form, then all of the torsion forms, but τ₂, vanish, and the G₂-structure is said to be calibrated. If the only non-vanishing torsion forms are τ₁ and τ₂, then at least locally, the metric g_φ is conformally equivalent to the metric induced by a calibrated G₂-structure, and the G₂-structure is called locally-conformal calibrated. If the only vanishing torsion form is τ₂, then the G₂-structure is said to be integrable. In this case, there exists a unique affine connection with totally skew-symmetric torsion preserving the G₂-structure by [27].

Consider now (ω, ψ₊) and two smooth functions F : I → ℂ − {0} and G : I → ℝ⁺; the following three-form defines a G₂-structure on I × N [28]:

$$\phi =\Re (F^3\mathrm{\Psi })+G{\left|F\right|}^2\omega \wedge dt,$$

where t is the coordinate on I. Moreover, we have:

$$\begin{array}{l}{g}_{\phi }=G^{2}d{t}^2+|F{|}^{2}h,\\ d{V}_{g\phi }=G|F{|}^{6}dt\wedge d{V}_h,\\ {\ast }_{\phi }\phi =G\Im (F^3\mathrm{\Psi })\wedge dt+\frac{1}{2}|F{|}^4{\omega }^2.\end{array}$$

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 dt² + h, if I = ℝ and G, F ≡ 1,

• the cone C(N) with the metric dt² + t²h, if I = ℝ⁺, G ≡ 1 and F (t) = t,

• the sin-cone SC(N) with the metric dt² + sin²(t)h, if I = (0, π), G ≡ 1 and $F(t)=\sin (t)e^{i\frac{1}{3}}$.

Observe that with the choice G ≡ 1, the manifold I × N with metric dt² + |F|²h is the warped product of I and N with warping function |F|. Using the expression of the Ricci tensor of the warped product metric [29], it is possible to show the following general properties (see also [30]).

Proposition 1. Let (M^m, g) be a Riemannian manifold of dimension m. Then, the cone metric dt² + t²g is Ricci-flat if and only if the metric g is Einstein with Ric(g) = (m − 1)g.

Proposition 2. 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 dt² + sin²(t)h is Einstein with Einstein constant m.

3. Coupled Structures and Supersymmetry

In [3], the authors considered the problem of finding necessary and sufficient conditions for $\mathcal{N}=1$ 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 that 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 ${\mathcal{W}}_1^{-}\oplus {\mathcal{W}}_2^{-}$, 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 $w_2^{-}$:

$$d{w}_2^{-}\propto {\psi }_{+},$$

(4)

and the norms of $w_1^{-}$ and $w_2^{-}$ have to satisfy the following inequality [12]:

$$3{(w_1^{-})}^2\ge |w_2^{-}{|}^2,$$

(5)

where |·| denotes the norm with respect to the metric h induced by the SU(3)-structure. In the massless limit, the solutions reduce to AdS₄ × N, N being a compact six-manifold endowed with an SU(3)-structure with torsion in ${\mathcal{W}}_1^{-}\oplus {\mathcal{W}}_2^{-}$ and for which Equation (4) holds. Moreover, it was observed in [12] that Equations (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 ${\mathcal{W}}_1^{-}\oplus {\mathcal{W}}_2^{-}$ are known as coupled structures [13] or restricted half-flat structures [10] in the literature. They can be defined as the subclass of half-flat structures having w₃ ≡ 0. In this case, dω is proportional to ψ₊; the intrinsic torsion form $w_1^{-}$ is constant [16] and has to be nonzero if we want the intrinsic torsion τ to belong to the class ${\mathcal{W}}_1^{-}\oplus {\mathcal{W}}_2^{-}$. Thus, if we let $c:\equiv -\frac{3}{2}{w}_1^{-}$, we have: c := −

$$\begin{array}{l}dw=c{\psi }_{+},\\ d{\psi }_{+}=0,\\ d{\psi }_{-}=-\frac{2}{3}c{\omega }^2-w_2^{-}\wedge \omega .\end{array}$$

(6)

The two-form $w_2^{-}$ lies in the space ${\mathrm{\Lambda }}_0^{1,1}(N)$; therefore, it satisfies the following properties:

$$w_2^{-}\wedge {\omega }^2=0,$$

(7)

$$w_2^{-}\wedge {\psi }_{\pm }=0,$$

(8)

$$w_2^{-}\wedge \omega =-\ast w_2^{-}.$$

(9)

Using Equation (9) and the expression of dψ₋, it is easy to show that the two-form $w_2^{-}$ is co-closed, that is $\delta w_2^{-}=\ast d\ast w_2^{-}=0$.

Remark 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, define$\tilde{\omega }:=r^2\omega$, ${\tilde{\psi }}_{+}:=r^3{\psi }_{+}$and obtain a new coupled structure$(\tilde{\omega },{\tilde{\psi }}_{+})$ with coupled constant$\tilde{c}=\frac{c}{r}$. In particular, it is always possible to find a coupled structure having a positive coupled constant.

From the results of [25], we have that the scalar curvature of the metric h induced by a coupled structure is given by:

$$\mathrm{Scal}(h)=\frac{15}{2}{(w_1^{-})}^2-\frac{1}{2}|w_2^{-}{|}^2.$$

(10)

Moreover, the forms ϕ₁ and ϕ₂ appearing in the traceless part of the Ricci tensor are:

$$\begin{array}{l}{\varphi }_1=\frac{1}{4}\ast (w_2^{-}\wedge w_2^{-})+\frac{1}{4}\delta (w_1^{-}{\psi }_{+}),\\ {\varphi }_2=-2\ast J(d{w}_2^{-}).\end{array}$$

(11)

Let us now focus on Equation (4). It forces the proportionality constant between $d{w}_2^{-}$ and ψ₊ to satisfy the following result.

Proposition 3. Let (ω, ψ₊) be a coupled SU(3)-structure and suppose that$d{w}_2^{-}$ is proportional to ψ₊. Then it holds:

$$d{w}_2^{-}=-\frac{|w_2^{-}{|}^2}{4}{\psi }_{+}.$$

Moreover, the norm of$w_2^{-}$ is constant.

Proof. First of all, observe that if $d{w}_2^{-}=k{\psi }_{+}$ for some function k ∈ C^∞(N), then k has to be constant. Indeed, taking the exterior derivatives of both sides, we get:

$$dk\wedge {\psi }_{+}=0,$$

which implies dk = 0. Now, suppose that $d{w}_2^{-}=k{\psi }_{+}$. Then, starting from $w_2^{-}\wedge {\psi }_{-}=0$, taking the exterior derivatives of both sides and using the previous identities, we have:

$$\begin{array}{l}0=d{w}_2^{-}\wedge {\psi }_{-}+w_2^{-}\wedge d{\psi }_{-}=k{\psi }_{+}\wedge {\psi }_{-}-w_2^{-}\wedge w_2^{-}\wedge \omega \\ =\frac{2}{3}k{\omega }^3+w_2^{-}\wedge \ast w_2^{-}=\frac{2}{3}k{\omega }^3+|w_2^{-}{|}^2-1\\ =\frac{2}{3}k{\omega }^3+|w_2^{-}{|}^2\frac{1}{6}{\omega }^3.\end{array}$$

Thus, $k=-\frac{|w_2^{-}{|}^2}{4}$. From the observation made at the beginning of the proof, we also get that $\left|w_2^{-}\right|$ is constant. □

From Proposition 3 and the fact that $w_1^{-}$ is constant, we obtain the following constraint.

Proposition 4. Let (ω, ψ₊) be a coupled SU(3)-structure, such that$d{w}_2^{-}$ is proportional to ψ₊. Then, the scalar curvature of the metric induced by the coupled structure is constant.

Proof. Consider Equation (10) of the scalar curvature of h, and conclude using the fact that both $w_1^{-}$ and $\left|w_2^{-}\right|$ are constant.

Consider now Equation (5); this implies a further constraint on the scalar curvature.

Proposition 5. Let (ω, ψ₊) be a coupled SU(3)-structure whose non-vanishing intrinsic torsion forms satisfy$3{(w_1^{-})}^2\ge |w_2^{-}{|}^2$. Then, the scalar curvature of the metric induced by the coupled structure is positive. Moreover, it is also constant if$d{w}_2^{-}$is proportional to ψ₊.

Proof. Using the expression of the scalar curvature of a coupled structure and the inequality $3{(w_1^{-})}^2\ge {\left|w_2^{-}\right|}^2$, we get:

$$\mathrm{Scal}(h)=\frac{15}{2}{(w_1^{-})}^2-\frac{1}{2}{\left|w_1^{-}\right|}^2\ge 2{\left|w_1^{-}\right|}^2>0.$$

Moreover, if $d{w}_2^{-}$ is proportional to ψ₊, then the scalar curvature is constant by Proposition 4. □

It is also easy to characterize the coupled structures having $d{w}_2^{-}$ proportional to ψ₊ and inducing an Einstein metric:

Proposition 6. Let (ω, ψ₊) be a coupled SU(3)-structure, such that$d{w}_2^{-}$ is proportional to ψ₊. Then, the induced metric h is Einstein if and only if the following identity holds:

$$\ast (w_2^{-}\wedge w_2^{-})=w_1^{-}{w}_2^{-}-\frac{{\left|w_2^{-}\right|}^2}{3}\omega .$$

Proof. Recall that a Riemannian metric h is Einstein if and only if Ric⁰(h) = 0. We know that:

$${\mathrm{Ric}}^0(h)={\iota }^{-1}(E_1({\varphi }_1))+{\gamma }^{-1}(E_2({\varphi }_2)),$$

where ϕ₁ and ϕ₂ for a coupled structure are given in Equation (11). Now, using the fact that $d{w}_2^{-}$ is proportional to ψ₊, one gets that also ϕ₂ is proportional to ψ₊. Thus, E₂(ϕ₂) = 0, since ψ₊ belongs to a subspace of Λ³(N), which is disjoint from ${\mathrm{\Lambda }}_0^{2,1}$ (N). Moreover,

$${\varphi }_1=\frac{1}{4}*(w_2^{-}\wedge w_2^{-})-\frac{1}{2}{(w_2^{-})}^2\omega -\frac{1}{4}{w}_1^{-}{w}_2^{-}$$

and

$$E_1({\varphi }_1)=\frac{1}{4}*(w_2^{-}\wedge w_2^{-})-\frac{1}{4}{w}_1^{-}{w}_2^{-}+\frac{1}{12}{\left|w_2^{-}\right|}^2\omega .$$

Therefore, Ric⁰(h) = ι⁻¹(E₁(ϕ₁)) is zero if and only if E₁(ϕ₁) is zero, and from this, the assertion follows.□

4. Examples

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

4.1. Nilmanifolds

We recall here the definition of a nilmanifold and some useful properties.

Definition 7. 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 of the 34 six-dimensional nilpotent Lie algebras existing up to isomorphisms [31] satisfy the following result.

Proposition 8 ([32]). Let$\mathfrak{g}$ 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, if G denotes the simply connected nilpotent Lie group whose Lie algebra is, $\mathfrak{g}$ 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 one 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.

Theorem 9 ([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 5, it is immediate to show that:

Proposition 10. There are no six-dimensional nilmanifolds admitting an invariant coupled structure satisfying the condition$3{(w_1^{-})}^2\ge {\left|w_2^{-}\right|}^2$.

Thus, we can only look for nilpotent Lie algebras endowed with a coupled structure (ω, ψ₊) having $d{w}_2^{-}$ proportional to ψ₊. In [14], we showed that among the 34 non-isomorphic six-dimensional nilpotent Lie algebras, there are only two of them admitting a coupled structure; we recall the result here.

Proposition 11 ([14]). Let$\mathfrak{g}$ be a six-dimensional, non-abelian, nilpotent Lie algebra endowed with a coupled SU(3)-structure (ω, ψ₊). Then, $\mathfrak{g}$ is isomorphic to one of the following nilpotent Lie algebras:

$$\begin{array}{l}\mathfrak{J}=(0,0,0,0,e^{14}+e^{23},e^{13}-e^{24}),\\ \mathfrak{N}=(0,0,0,e^{13},e^{14}+e^{23},e^{13},-e^{15}-e^{24}).\end{array}$$

Remark 3. Recall that the notation$\mathfrak{J}=(0,0,0,0,e^{14}+e^{23},e^{13}-e^{24})$ means that there exists a basis of one-forms (e¹,…,e⁶) for$\mathfrak{J}*$, such that de¹ = 0, de² = 0, de³ = 0, de⁴ = 0, de⁵ = e¹⁴ + e²³, de⁶ = e¹³ − e²⁴, where d is the Chevalley–Eilenberg differential.

Observe that the two Lie algebras $\mathfrak{J}$ and $\mathfrak{N}$ are isomorphic respectively to the Lie algebras labeled by ${\mathfrak{N}}_{28}$ and ${\mathfrak{N}}_9$ in the work [14]. Here, they are given with different structure equations, since in both cases, the frame (e¹,…,e⁶) is an adapted frame for the coupled SU(3)-structure. We emphasize some properties of these coupled structures in the following examples.

Example 1.$\mathfrak{J}=(0,0,0,0,e^{14}+e^{23},e^{13}-e^{24}),$ is (isomorphic to) the well-known Iwasawa–Lie algebra, which is the Lie algebra of the six-dimensional nilmanifold known in the literature as the Iwasawa manifold (see, for instance, [34] for the definition). Since the frame (e¹,…,e⁶) is adapted, we have that the pair

$$\begin{array}{l}\omega =e^{12}+e^{34}+e^{56},\\ {\psi }_{+}=e^{135}-e^{146}-e^{236}-e^{245},\end{array}$$

defines an SU(3)-structure on$\mathfrak{J}$. In this case dω = − ψ₊, and the non-vanishing intrinsic torsion forms are:

$$\begin{array}{l}{w}_1^{-}=\frac{2}{3},\\ w_2^{-}=-\frac{4}{3}{e}^{12}-\frac{4}{3}{e}^{34}+\frac{8}{3}{e}^{56}.\end{array}$$

It is easy to check that Equation (4) is satisfied:

$$d{w}_2^{-}=-\frac{8}{3}{\psi }_{+},$$

and that$-\frac{1}{4}{\left|w_2^{-}\right|}^2=-\frac{8}{3}$, as we expected from Proposition 3. Finally, the scalar curvature of the metric h induced by the coupled structure is Scal(h) = −2.

Example 2. Consider the Lie algebra$\mathfrak{N}=(0,0,0,e^{13},e^{14}+e^{23},e^{13},-e^{15}-e^{24})$. Since the frame (e¹,…,e⁶) is adapted, we have that the pair

$$\begin{array}{l}\omega =e^{12}+e^{34}+e^{56},\\ {\psi }_{+}=e^{135}-e^{146}-e^{236}-e^{245},\end{array}$$

defines an SU(3)-structure on$\mathfrak{N}$. Moreover, dω = − ψ₊, and the non-vanishing intrinsic torsion forms are:

$$\begin{array}{l}{w}_1^{-}=\frac{2}{3},\\ w_2^{-}=-\frac{4}{3}{e}^{12}-\frac{4}{3}{e}^{34}+e^{36}-e^{45}+\frac{8}{3}{e}^{56}.\end{array}$$

In this case$d{w}_2^{-}$ is not proportional to ψ₊, and the scalar curvature of the metric h induced by the coupled structure is Scal(h) = −3.

The fact that the Iwasawa manifold admits an invariant coupled structure was also observed in [3], where the authors wrote that it was the unique nilmanifold admitting a coupled structure they knew. Proposition 11 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 1, the coupled structure on the Iwasawa–Lie algebra satisfies Equation (4), i.e., $d{w}_2^{-}$ is proportional to ψ₊. Thus, it is a natural question to ask whether $\mathfrak{N}$ admits a coupled structure satisfying Equation (4) or not. In [11], the authors looked for the possible nilmanifolds admitting an invariant coupled structure satisfying Equation (4) and concluded that a systematic scan of all of the possible six-dimensional nilmanifolds yields two possibilities: the six-torus and the Iwasawa manifold. The six-torus has abelian Lie algebra, so it is not considered in Proposition 11. 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.

Proposition 12. There are no coupled SU(3)-structures on$\mathfrak{N}$ for which the exterior derivative of the intrinsic torsion form$w_2^{-}$ is proportional to ψ₊.

Proof. The idea is to describe all of the possible coupled structures on $\mathfrak{N}$ and to see if there exists one of these whose intrinsic torsion form $w_2^{-}$ satisfies the required condition. Let us start considering a generic two-form ω on $\mathfrak{N}$; we can write it as:

$$\omega ={\displaystyle \sum _{1\le i<j\le 6}{\omega }_{ij}{e}^{ij}},$$

where ω_ij are real numbers. We may think of the 15-tuple (ω₁₂,…,ω₅₆) =: (ω_ij) as a point in the affine space ${\mathbb{A}}_{ℝ}^{15}-\left\{0\right\}$. The homogeneous polynomial P of degree three in the unknowns ω_ij appearing as a coefficient of e¹²³⁴⁵⁶ in the expression of ω³ has to be non-vanishing; this gives a first constraint for (ω_ij). Since we want a coupled structure, we consider a three-form ψ₊ on $\mathfrak{N}$given by ψ₊ = cdω, for some nonzero real number c. Assuming:

$$\lambda ({\psi }_{+})=-4c^4{\omega }_{56}^2({\omega }_{36}{\omega }_{56}-{\omega }_{45}{\omega }_{56}-{\omega }_{46}^2+{\omega }_{56}^2)<0,$$

that is ω₅₆ ≠ 0 and $B:={\omega }_{36}{\omega }_{56}-{\omega }_{45}{\omega }_{56}-{\omega }_{46}^2+{\omega }_{56}^2>0$, we can compute the almost complex structure J induced by the stable form ψ₊. Now, we change the basis from (e₁,…,e₆) to a basis (E₁,…,E₆), 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 (E₁,…,E₆), the matrix associated to J is skew-symmetric with non-vanishing entries given by J²₁ = 1 = J⁴₃ = J⁶₅. We can then compute the new structure equations with respect to the dual basis (E¹,…,E⁶), obtaining:

$$\begin{array}{l}d{E}^i=0,i=1,2,3\\ d{E}^4=\frac{{\omega }_{56}}{\sqrt{B}}{E}^{13},\\ d{E}^5=-\frac{{\omega }_{46}}{{\omega }_{56}}{E}^{13}+\frac{\sqrt{B}}{{\omega }_{56}}(E^{14}+E^{23}),\\ d{E}^6=-\frac{{\omega }_{26}}{{\omega }_{56}}{E}^{12}-\frac{{\omega }_{46}}{{\omega }_{56}}{E}^{14}-\frac{{\omega }_{36}{\omega }_{56}{\omega }_{45}{\omega }_{56}-{\omega }_{46}^2-{\omega }_{56}^2}{{\omega }_{56}\sqrt{B}}{E}^{13}-\frac{{\omega }_{56}}{\sqrt{B}}{E}^{15}-\frac{\sqrt{B}}{{\omega }_{56}}{E}^{24}.\end{array}$$

Moreover, we have:

$$\begin{array}{l}{\psi }_{+}=-c\frac{B}{{\omega }_{56}}(E^{135}-E^{146}-E^{236}-E^{245}),\\ {\psi }_{-}=-c\frac{B}{{\omega }_{56}}(E^{136}-E^{145}-E^{235}-E^{246}).\end{array}$$

We can write ω with respect to the new basis and impose it to be of type (1, 1) with respect to J, obtaining three 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 with h(·,·) = ω(·, J·) with respect to the basis (E₁,…,E₆) and denote by $\mathcal{P}\subset {\mathbb{A}}_{ℝ}^{15}$ the set on which it is positive definite. One can check that P ≠ 0 when (ω_ij) ∈ $\mathcal{P}$. Now, if we let (ω_ij) vary in the (non-empty) set $\mathcal{Q}:=\mathcal{P}\cap \{({\omega }_{ij}):\lambda ({\psi }_{+})<0\}$, we have all of the possible non-normalized coupled SU(3)-structures on N. The intrinsic torsion form $w_1^{-}$ is always $-\frac{2}{3c}$, 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 $d{w}_2^{-}$ is proportional to ψ₊. Thus, we can start with a generic two-form ω of type (1, 1) with respect to J and write it as:

$$\begin{array}{c}w=w_{12}{E}^{12}+w_{34}{E}^{34}+w_{56}{E}^{56}+w_{13}(E^{13}+E^{14})+w_{14}(E^{14}-E^{23})+w_{15}(E^{15}+E^{26})\\ w_{16}(E^{16}-E^{25})+w_{35}(E^{35}+E^{46})+w_{36}(E^{36}-E^{45}),\end{array}$$

where w_ij are real numbers. Then, we have to impose that w is primitive (w ∧ ω² = 0) and fulfills 2

$$d{\psi }_{-}=-\frac{2}{3c}{\omega }^2-w^\omega$$

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 $\mathcal{Q}$. The condition on dψ₋ gives 13 equations of the same kind as before; we can solve four of them, namely those obtained comparing the coefficients of E³⁴⁵⁶, E²³⁵⁶, E¹²⁵⁶, E²³⁴⁵, 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 proven. □

The previous results can be summarized as follows.

Proposition 13. Let$\mathfrak{g}$ be a six-dimensional, non-abelian, nilpotent Lie algebra endowed with a coupled SU(3)-structure (ω, ψ₊) having$d{w}_2^{-}$ proportional to ψ₊. Then, $\mathfrak{g}$ is isomorphic to the Iwasawa–Lie algebra.

4.2. Twistor Spaces

In the work [6], it was observed that on the twistor space Z over a self-dual Einstein four-manifold (M⁴, 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 (M⁴, g), the set of positive, orthogonal almost complex structures on M⁴ forms a smooth manifold Z called the twistor space of M⁴, which can be viewed as the two-sphere bundle π: Z → M⁴ consisting of the unit −1 eigenvectors of the Hodge operator acting on Λ²TM⁴ [35].

On Z, it is possible to define two almost complex structures (see, for example, [36]), one of which is never integrable, as shown in [37]. Let us denote it by J.

When the metric g is self-dual and Einstein, Xu showed in [38] that on (Z, J), there exists a basis of (1, 0)-forms ε¹, ε², ε³, such that the first structure equations are:

$$d\left(\begin{array}{c}{\epsilon }^1\\ {\epsilon }^2\\ {\epsilon }^3\end{array}\right)=-\left(\begin{array}{cc}\alpha & 0\\ 0& -\mathrm{tr}(\alpha )\end{array}\right)\wedge \left(\begin{array}{c}{\epsilon }^1\\ {\epsilon }^2\\ {\epsilon }^3\end{array}\right)+\left(\begin{array}{c}\overline{{\epsilon }^2\wedge {\epsilon }^3}\\ \overline{{\epsilon }^3\wedge {\epsilon }^1}\\ \sigma \overline{{\epsilon }^1\wedge {\epsilon }^2}\end{array}\right),$$

(12)

where α is a 2 × 2 skew-Hermitian matrix of one-forms and $\sigma :=\frac{\mathrm{Scal}(g)}{24}$. Using these, it is easy to show that the following pair of forms defines a coupled SU(3)-structure on Z [6]:

$$\begin{array}{l}\omega =\frac{i}{2}\left({\epsilon }^1\wedge \overline{{\epsilon }^1}+{\epsilon }^2\wedge \overline{{\epsilon }^2}+{\epsilon }^3\wedge \overline{{\epsilon }^3}\right),\\ \mathrm{\Psi }=i\left({\epsilon }^1\wedge {\epsilon }^2\wedge {\epsilon }^3\right).\end{array}$$

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:

$$h={\epsilon }^1\odot \overline{{\epsilon }^1}+{\epsilon }^2\odot \overline{{\epsilon }^2}+{\epsilon }^3\odot \overline{{\epsilon }^3}.$$

Moreover, the non-vanishing intrinsic torsion forms have the following expressions:

$$\begin{array}{l}{w}_1^{-}=\frac{2}{3}\left(\sigma +2\right),\\ w_2^{-}=\frac{2}{3}i\left(\sigma -1\right)\left({\epsilon }^1\wedge \overline{{\epsilon }^1}+{\epsilon }^2\wedge \overline{{\epsilon }^2}-2{\epsilon }^3\wedge \overline{{\epsilon }^3}\right),\end{array}$$

$d{w}_2^{-}$ is proportional to ψ₊

$$d{w}_2^{-}=-\frac{8}{3}{(\sigma -1)}^2{\psi }_{+},$$

and $3{(w_1^{-})}^2\ge {\left|w_2^{-}\right|}^2$ if and only $\frac{10-6\sqrt{2}}{7}\le \sigma \le \frac{10-6\sqrt{2}}{7}$.

We can consider a local frame (e¹,…,e⁶) for Λ¹(Z), such that ε¹ = e¹ + ie², ε² = e³ + ie⁴, ε³ = e⁵ + ie⁶, and compute the Ricci curvature of the metric induced by the coupled structure using the results of [25]. What we get is that the scalar curvature of h is:

$$\mathrm{Scal}(h)=-2{\sigma }^2+24\sigma +8$$

and the traceless part of the Ricci tensor of h with respect to the considered frame has the following form:

$${\mathrm{Ric}}^0(h)=-\frac{2}{3}(\sigma -1)(\sigma -2)\mathrm{diag}(1,1,1,1,-2,-2).$$

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:

$$\begin{array}{l}{w}_1^{-}=\frac{8}{3},\\ w_2^{-}=-\frac{4}{3}(e^{12}+e^{34}-2e^{56}).\end{array}$$

In particular, the coupled constant is c = −4, and the scalar curvature is Scal(h) = 48. Moreover, the characterization given in Proposition 6 is satisfied by this example.

Recall that when Scal(g) > 0, a compact, self-dual, Einstein four-manifold (M⁴, g) is isometric either to S⁴ or to ℂℙ² with their canonical metrics (Theorem 13.30 in [39]); thus, Z is either ℂℙ³ or the flag manifold SU(3)/T².

4.3. G₂-Structures with Special Metrics Induced by Coupled Einstein Structures

We can now use the coupled Einstein structure on Z to construct a G₂-structure with full intrinsic torsion inducing an Einstein metric and a locally-conformal calibrated G₂-structure inducing a Ricci-flat metric.

First of all, we rescale the coupled Einstein structure on Z in the following way:

$$\begin{array}{l}\tilde{\omega }=\frac{8}{5}\omega ,\\ {\tilde{\psi }}_{+}={\left(\frac{8}{5}\right)}^{\frac{3}{2}}{\psi }_{+}.\end{array}$$

Then $(\tilde{\omega },{\tilde{\psi }}_{+})$ is a coupled structure with coupled constant $c=-\sqrt{10}$ and inducing the metric $\tilde{h}=\frac{8}{5}h$. Moreover, Scal( $\tilde{h}$) = 30 and Ric( $\tilde{h}$) = 5 $\tilde{h}$.

As we observed in Section 2, starting from the coupled Einstein structure $(\tilde{\omega },{\tilde{\psi }}_{+})$, we can construct a G₂-structure φ on the sin-cone S(Z) inducing the sin-cone metric $g_{\phi }=d{t}^2+{\sin }^2(t)\tilde{h}$. By Proposition 2, 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 G₂-structure induced on the sin-cone by a coupled structure with coupled constant c are:

$$\begin{array}{l}{\tau }_0=\frac{8c+4}{7},\\ {\tau }_1=\left(1-\frac{c}{3}\right)\cot (t)dt,\\ {\tau }_2=-\frac{\sin (2t)}{2}{w}_2^{-},\\ {\tau }_3=\frac{c-3}{7}\left({\sin }^4(t){\psi }_{-}-{\sin }^3(t)\cos (t){\psi }_{+}+\frac{4}{3}{\sin }^2(t)dt\wedge \omega \right)-{\sin }^2(t)dt\wedge w_2^{-}.\end{array}$$

Thus, the coupled Einstein structure $(\tilde{\omega },{\tilde{\psi }}_{+})$ induces a G₂-structure with full intrinsic torsion and Einstein metric on the sin-cone S(Z).

If we consider the G₂-structure φ induced on the cone C(Z) by $(\tilde{\omega },{\tilde{\psi }}_{+})$, then the metric $g_{\phi }=d{t}^2+t^2\tilde{h}$ is Ricci-flat by Proposition 1. Moreover, the non-vanishing intrinsic torsion forms of the G₂-structure constructed on the cone from a coupled structure with coupled constant c are:

$$\begin{array}{l}{\tau }_1=\left(\frac{1}{t}-\frac{1}{3t}c\right)dt,\\ {\tau }_2=-t{w}_2^{-}.\end{array}$$

Therefore, the coupled Einstein structure $(\tilde{\omega },{\tilde{\psi }}_{+})$ induces a locally-conformal calibrated G2-structure on the cone C(Z) whose associated metric is Ricci-flat.

Remark 4. It is worth observing here that calibrated G₂-structures inducing a Ricci-flat metric are actually parallel [26]. The previous example shows that a result of this kind is not true anymore for locally-conformal calibrated G₂-structures.

5. Flows

In this section, we study the behavior of coupled structures with respect to known evolution equations (flows) of SU(3)-structures. The Hitchin flow, introduced in [19] as the Hamiltonian flow of a certain functional, allows one to construct (non-complete) metrics with holonomy in G2 starting from a suitable SU(3)-structure. The idea is to consider a six-manifold $\mathfrak{N}$ endowed with an SU(3)-structure $(\tilde{\omega },{\tilde{\psi }}_{+})$ and to define a G₂-structure on M := I × N for some interval I ⊆ ℝ by:

$$\phi =dt\wedge \omega +{\psi }_{+},$$

where ω and ψ₊ depend on the coordinate t on I. If we require the G₂-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:

$$\{\begin{array}{l}\frac{\partial }{\partial t}{\psi }_{+}=dw\\ \frac{\partial }{\partial t}\omega \wedge \omega =-d{\psi }_{-}\end{array}.$$

(13)

These equations are the so-called Hitchin flow equations. A solution of them with the 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 [40]. 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 Equation (13), stays half-flat as long as it exists.

In the work [20], 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 G₂-structure. This is motivated by the subject the authors are interested in, namely four-dimensional domain wall solutions of heterotic string theory that preserve $\mathfrak{N}=\frac{1}{2}$ supersymmetry (see also [9]). 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 G₂-structure. The physical setting provides further constraints on the intrinsic torsion forms of the G₂-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 generally, which constraints arise requiring that a solution of the generalized Hitchin flow is coupled as long as it exists. We begin giving the following definition.

Definition 14. Let (ω(t), ψ₊(t)) be a solution of 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 of the Hitchin flow.

Coupled solutions of the Hitchin flow can be easily characterized and induce an almost complex structure not depending on t.

Proposition 15. Let N be a six-dimensional manifold, and suppose there exists on it a solution (ω(t), ψ₊(t)) of 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

$${\psi }_{+}(t)=f(t){\psi }_{+}(0).$$

Conversely, if the pair (ω(t), ψ₊(t)) is a solution of 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 of the Hitchin flow with ψ₊(t) = f(t) ψ₊(0), then from the flow equation $\frac{\partial }{\partial t}{\psi }_{+}(t)=d\omega (t)$, we obtain:

$$d\omega (t)=\frac{\partial }{\partial t}(f(t){\psi }_{+}(0))=\left(\frac{d}{dt}f(t)\right){\psi }_{+}(0).$$

Thus, the solution is a coupled structure with $c(t)=\frac{d}{dt}f(t)$. Suppose now that the solution is coupled, dω(t) = c(t) ψ₊(t). Then, from the flow equation, we obtain:

$$\frac{\partial }{\partial t}{\psi }_{+}(t)=c(t){\psi }_{+}(t).$$

Working in local coordinates on N, it is easy to show that

$${\psi }_{+}(t)=f(t){\psi }_{+}(0),$$

where

$$f(t)=e^{{\displaystyle {\int }_0^{t}c(s)ds}}.$$

□

Corollary 16. Let (ω(t), ψ₊(t)) be a coupled solution of 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:

$$J(t)=J_{{\psi }_{+}(t)}=J_{f(t){\psi }_{+}(0)}=J_{{\psi }_{+}(0)}=J(0),$$

since the almost complex structure induced by ψ₊ does not change if we rescale ψ₊ by a real constant. □

5.1. Coupled Solutions on Six-Dimensional Nilpotent Lie Algebras

Working on six-dimensional nilpotent Lie algebras, it is possible to show that a coupled solution of the Hitchin flow may not exist. As we recalled in Proposition 11, the only six-dimensional nilpotent Lie algebras admitting a coupled structure are, up to isomorphisms, $\mathfrak{J}$ and $\mathfrak{N}$. In each case, with respect to the frame we considered, the pair (ω, ψ₊), where:

$$\begin{array}{l}\omega =e^{12}+e^{34}+e^{56},\\ {\psi }_{+}=e^{135}-e^{146}-e^{236}-e^{245},\end{array}$$

(14)

is a coupled structure with dω = − ψ₊. For completeness, we observe also that:

$${\psi }_{-}=J{\psi }_{+}=e^{136}+e^{145}+e^{235}-e^{246}.$$

The following result shows our claim.

Proposition 17. Consider the Hitchin flow on the six-dimensional nilpotent Lie algebras$\mathfrak{N}$ and$\mathfrak{J}$. Then, on$\mathfrak{J}$, there exists a coupled solution starting from Equation (14) at t = 0, while on$\mathfrak{N}$, there are no coupled solutions of the Hitchin flow starting from Equation (14).

Proof. Let us start with $\mathfrak{J}$; it admits a coupled solution of the Hitchin flow already described in [41]. We recover it here starting from a suitable pair (ω(t), ψ₊(t)) and requiring that it satisfies the Hitchin flow equations. From Proposition 15, we know that (ω(t), ψ₊(t)) is a coupled solution if and only if:

$${\psi }_{+}(t)=f(t){\psi }_{+}(0)=f(t)(e^{135}-e^{146}-e^{236}-e^{245}),$$

with f(0) = 1. It is also clear that ψ−(t) = f(t) (e¹³⁶ + e¹⁴⁵ + e²³⁵ − e²⁴⁶). Moreover, since we already know the form of the solution, we consider three smooth functions a₁(t), a₂(t), a₃(t) with a_i (0) = 1 and such that:

$$\omega (t)=a_1(t)e^{12}+a_2(t)e^{34}+a_3(t)e^{56}.$$

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:

$$f^2=a_1{a}_2{a}_3.$$

(15)

From the first Hitchin flow equation in Equation (13), we obtain:

$$\frac{d}{dt}f=-a_3,$$

(16)

while from the second one, we have:

$$\frac{d}{dt}(a_1{a}_3)=0,$$

(17)

$$\frac{d}{dt}(a_2{a}_3)=0,$$

(18)

$$\frac{d}{dt}(a_1{a}_2)=-4f.$$

(19)

From Equations (17) and (18) and the starting conditions at t = 0, we deduce that:

$$a_1=a_2=\frac{1}{a_3}.$$

Using this result and Equation (15), it holds necessarily:

$$f=\frac{1}{\sqrt{a_3}}.$$

Thus, the ODE Equation (16) becomes:

$$\frac{d}{dt}{a}_3=2a_3^2\sqrt{a_3}$$

and solving this, we get:

$$a_3={(1-3t)}^{-\frac{2}{3}}.$$

It is then easy to check that also Equation (19) is satisfied. Then, the pair:

$$\begin{array}{l}\omega (t)={(1-3t)}^{\frac{2}{3}}{e}^{12}+{(1-3t)}^{\frac{2}{3}}{e}^{34}+{(1-3t)}^{-\frac{2}{3}}{e}^{56},\\ {\psi }_{+}(t)={(1-3t)}^{\frac{1}{3}}(e^{135}-e^{146}-e^{236}-e^{245})\end{array}$$

is a coupled solution of the Hitchin flow.

We consider now $\mathfrak{N}$; we will show that there are no coupled solutions starting from Equation (14). In this case, we need:

$${\psi }_{+}(t)=f(t){\psi }_{+}(0)=f(t)(e^{135}-e^{146}-e^{236}-e^{245}),$$

with f(0) = 1, while we consider 15 smooth real valued functions b_ij(t), 1 ≤ i < j ≤ 6, such that:

$$\omega (t)={\displaystyle \sum _{1\le i\le j\le 6}{b}_{ij}(t)e^{ij}},$$

b₁₂(0) = b₃₄(0) = b₅₆(0) = 1 and b_ij(0) = 0 for the remaining functions. 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 equation, and we compute the ODEs derived from them. What we obtain after solving some of these differential equations is that f ≡ 0, which cannot be possible. □

5.2. Generalized Hitchin Flow

Since coupled solutions of the Hitchin flow may not exist in general, as Proposition 17 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 G₂-structure on M := I × N by:

$$\phi =v_{t}dt\mathrm{\Lambda }\omega +\Re (F\mathrm{\Psi }),$$

where ν_t ∈ C^∞(M) and F is a complex valued smooth function defined on M and having constant module one. Observe that the Riemannian metric defined by φ is:

$$g_{\phi }=v_t^{2}d{t}^2+h.$$

As we already recalled, in the case of $\mathfrak{N}=\frac{1}{2}$ domain wall solutions, the non-vanishing intrinsic torsion forms of the G₂-structure are τ₀, τ₁, τ₃. On M = I × N, τ₁ and τ₃ can be decomposed as:

$$\begin{array}{l}{\tau }_1=u_{t}dt+{\tau }_1^N,\\ {\tau }_3=dt\mathrm{\Lambda }{\eta }_t+{\tau }_3^N,\end{array}$$

where u_t is a smooth function on M, ${\tau }_1^N$ is a one-form on N, η_t is a two-form on N depending on t and ${\tau }_3^N$ is a three-form on N. Moreover, the following constraints hold:

$$\begin{array}{l}{u}_t=\frac{1}{2}\frac{\partial }{\partial t}\varphi ,\\ {\tau }_1^N=\frac{1}{2}d\varphi ,\\ d_7{\tau }_0=0,\end{array}$$

where ϕ is the ten-dimensional dilaton, d₇ denotes the exterior derivative on M and d denotes it on N.

A general argument allows one to write down the equations of the SU(3)-structure flow associated with the embedding and some relations between the torsion forms of the SU(3)-structure and the G₂-structure. In particular:

$$w_4=2{\tau }_1^N.$$

Therefore, if we have an SU(3)-structure with vanishing w₄, we get $d\varphi =2{\tau }_1^N=0$. Following [20], we work in the gauge F = 1, in this case:

$$\phi =v_{t}dt\mathrm{\Lambda }\omega +{\psi }_{+}.$$

If we suppose that the structure is coupled for each t, i.e.,

$$\begin{array}{l}d\omega (t)=c(t){\psi }_{+}(t),\\ d{\psi }_{+}(t)=0,\\ d{\psi }_{-}(t)=-\frac{2}{3}c(t){(\omega (t))}^2-w_2^{-}(t)\mathrm{\Lambda }\omega (t),\end{array}$$

(20)

where c : I → ℝ is a smooth function, such that $w_1^{-}(t)=-\frac{2}{3}c(t)$, then the two-form ω(t) evolves as:

$$\frac{\partial }{\partial t}\omega (t)={\lambda }_t\omega (t)+h_t,$$

(21)

where

$${\lambda }_t=2u_t-{\nu }_t{w}_1^{-}(t),$$

(22)

$$h_t=v_t{w}_2^{-}(t)-*(d{v}_t\mathrm{\Lambda }*{\psi }_{+}(t)).$$

(23)

Moreover, it follows from a general argument involving the flow equations that:

$$d{\lambda }_t=0$$

and using one of the constraints recalled earlier, we get:

$$d{u}_t=\frac{1}{2}d\left(\frac{\partial }{\partial t}\varphi \right)=\frac{1}{2}\frac{\partial }{\partial t}(d\varphi )=0.$$

Taking the exterior derivative of both sides of Equation (22), we then have:

$$d{\nu }_t=0.$$

Thus, ν_t is actually a function of t, and Equation (23) becomes $h_t=v_t{w}_2^{-}$. Remark 5. With our convention, $w_2^{-}$ here is$-w_2^{-}$ in the work [20].

The flow equations for ψ₊(t) and ψ₋(t) determined in [20] reduce to the following in the coupled case:

$$\frac{\partial }{\partial t}{\psi }_{+}(t)=\frac{3}{2}{\lambda }_t{\psi }_{+}(t)-\frac{7}{4}{\tau }_0{v}_t{\psi }_{-}(t)-v_t\gamma ,$$

(24)

$$\frac{\partial }{\partial t}{\psi }_{-}(t)=\frac{7}{4}{\tau }_0{v}_t{\psi }_{+}(t)+\frac{3}{2}{\lambda }_t{\psi }_{-}(t)+v_t{J}_{\gamma },$$

(25)

where γ is a primitive three-form of type (2, 1) + (1, 2) appearing in the expression of the Hodge dual of ${\tau }_3^N$ on N.

We derive now all of the conditions that arise requiring these flow equations to preserve the coupled 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:

$$d\left(\frac{\partial }{\partial t}\omega \right)=\dot{c}{\psi }_{+}+c\left(\frac{3}{2}{\lambda }_t{\psi }_{+}-\frac{7}{4}{\tau }_0{v}_t{\psi }_{-}-v_t\gamma \right).$$

Moreover, taking the exterior derivative of both sides of Equation (21), using dν_t = 0 and the hypothesis on the coupled condition, we obtain:

$$d\left(\frac{\partial }{\partial t}\omega \right)={\lambda }_{t}c{\psi }_{+}+v_{t}d{w}_2^{-}.$$

Comparing the two equations, it follows:

$$v_{t}d{w}_2^{-}=\dot{c}{\psi }_{+}+\frac{1}{2}c{\lambda }_t{\psi }_{+}-\frac{7}{4}c{\tau }_0{v}_t{\psi }_{-}-c{v}_t\gamma .$$

Wedging both sides by ψ₋ and using the fact that γ ∧ ψ₋ = 0, since γ ∈ Λ^2,1 ⊕ Λ^1,2, we get:

$$v_{t}d{w}_2^{-}\wedge {\psi }_{-}=\frac{2}{3}\dot{c}{\omega }^2+\frac{1}{3}c{\lambda }_t{\omega }^3.$$

(26)

Since for each t it holds $d{w}_2^{-}\wedge {\psi }_{-}=-|w_2^{-}{|}^2\frac{{\omega }^2}{6}$, where the norm is induced by h(t), Equation (26) becomes:

$$-v_t|w_2^{-}{|}^2\frac{{\omega }^3}{6}=\frac{2}{3}\dot{c}{\omega }^3+\frac{1}{3}c{\lambda }_t{\omega }^3$$

and the following result is proven.

Proposition 18. 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:

$$\frac{\partial }{\partial t}c(t)=-\frac{1}{2}c(t){\lambda }_t-\frac{1}{4}{v}_t|w_2^{-}(t){|}_{h(t)}^2.$$

Moreover, for each t, it must hold:

$$d{w}_2^{-}=-\frac{1}{4}|w_2^{-}{|}^2{\psi }_{+}-\frac{7}{4}c{\tau }_0{\psi }_{-}-c\gamma .$$

In order to preserve the closedness of ψ₊(t), we need:

$$d\left(\frac{\partial }{\partial t}{\psi }_{+}\right)=0.$$

Moreover, taking the exterior derivative of both sides of the flow Equation (24) of ψ₊, we have:

$$d\left(\frac{\partial }{\partial t}{\psi }_{+}\right)=-\frac{7}{4}{\tau }_0{v}_{t}d{\psi }_{-}-v_{t}d\gamma .$$

Comparing the two equations, it then follows:

$$d\gamma =-\frac{7}{4}{\tau }_0{v}_{t}d{\psi }_{-}.$$

(27)

Observe now that dγ ∧ ω = 0, since γ is a primitive form of type (2, 1) + (1, 2). Therefore, wedging both sides of Equation (27) by ω and recalling that $d{\psi }_{-}\wedge \omega =-\frac{2}{3}c{\omega }^3$, we get:

$${\tau }_0{\nu }_{t}c=0,$$

and then τ₀ = 0, since both c and ν_t cannot be zero. In particular:

$$d\gamma =0.$$

We can summarize the results in the following:

Proposition 19. If the closedness of ψ₊ is preserved by the generalized Hitchin flow, then the intrinsic torsion form τ₀ vanishes, and the thee-form γ is closed.

Let us now consider the expression of dψ₋ in Equation (20) and differentiate it with respect to t, having in mind the results already obtained:

$$d\left(\frac{\partial }{\partial t}{\psi }_{-}\right)=\left(-\frac{2}{3}\dot{c}-\frac{4}{3}c{\lambda }_t\right){\omega }^2+\left(-\frac{4}{3}c{v}_t-{\lambda }_t\right)w_2^{-}\wedge \omega -\frac{\partial }{\partial t}{w}_2^{-}\wedge \omega -v_t{w}_2^{-}\wedge w_2^{-}.$$

Taking the exterior derivative of both sides of the flow Equation (25) of ψ₋(t), we get:

$$d\left(\frac{\partial }{\partial t}{\psi }_{-}\right)=-{\lambda }_{t}c{\omega }^2-\frac{3}{2}{\lambda }_t{w}_2^{-}\wedge \omega +v_{t}d(J\gamma ).$$

Comparing the two equations, we obtain that the flow of $w_2^{-}$ must obey the following equation:

$$\frac{\partial }{\partial t}{w}_2^{-}\wedge \omega =\frac{1}{6}{v}_t|w_2^{-}{|}^2{\omega }^2+\left(-\frac{4}{3}c{v}_t+\frac{1}{2}{\lambda }_t\right)w_2^{-}\wedge \omega -v_t{w}_2^{-}\wedge w_2^{-}-v_{t}d(J\gamma ).$$

We also know that the following necessary conditions deriving from the Bianchi identity d₇Ĥ = 0 must hold:

$$d{S}^X=0,$$

(28)

$$d{S}_t=\frac{\partial }{\partial t}{S}^X,$$

(29)

where Ĥ = dt ∧ S_t + S^X is the component of the ten-dimensional flux along M.

Remark 6. The other constraint obtained from the Bianchi identities was recalled earlier; it is d₇τ₀ = 0.

Using the previous results, it follows from [20] that for a coupled structure:

$$S^X={\nu }_t^{-1}{u}_t{\psi }_{-}+J\gamma ,S_t=0.$$

From the first identity Equation (28), we then get:

$$d(J\gamma )=-{\nu }_t^{-1}{u}_{t}d{\psi }_{-}.$$

(30)

Observe that d(Jγ) ∧ ω = 0. Thus, if we wedge both sides of Equation (30) by ω, we obtain:

$${\nu }_t^{-1}{u}_{t}c=0,$$

from which follows u_t = 0 and, as a consequence, d(Jγ) = 0. The second identity Equation (29) then reads:

$$\frac{\partial }{\partial t}(J\gamma )=0.$$

We can summarize here some of the results obtained:

• The only non-vanishing intrinsic torsion form of the G₂-structure after imposing all conditions is τ₃. Moreover, $*{\tau }_3^X=\gamma$ and η_t = 0.

• dν_t = 0.

• dγ and d(Jγ) = 0; thus, γ is harmonic.

• ${\lambda }_t=\frac{2}{3}{v}_{t}c(t).$

In particular, the evolution equations of the differential forms defining the coupled structure become:

$$\begin{array}{l}\frac{\partial }{\partial t}\omega (t)=\frac{2}{3}{\nu }_{t}c(t)\omega (t)+{\nu }_t{w}_2^{-}(t),\\ \frac{\partial }{\partial t}{\psi }_{+}(t)={\nu }_{t}c(t){\psi }_{+}(t)-{\nu }_t\gamma ,\\ \frac{\partial }{\partial t}{\psi }_{-}(t)={\nu }_{t}c(t){\psi }_{-}(t)+{\nu }_{t}J\gamma .\end{array}$$

Moreover, the intrinsic torsion forms of the coupled structure must evolve as:

$$\begin{array}{l}\frac{\partial }{\partial t}c(t)=-\frac{1}{3}{\nu }_t{(c(t))}^2-\frac{1}{4}{\nu }_t|w_2^{-}(t){|}_{h(t)}^2,\\ \frac{\partial }{\partial t}{w}_2^{-}(t)\wedge \omega (t)=\frac{1}{6}{\nu }_t|w_2^{-}(t){|}_{h(t)}^2{(\omega (t))}^2-{\nu }_{t}c(t)w_2^{-}(t)\wedge \omega (t)-v_t{(w_2^{-}(t))}^2,\end{array}$$

and for each t, it must hold:

$$d{w}_2^{-}=\frac{1}{4}|w_2^{-}{|}^2{\psi }_{+}-c\gamma .$$

5.3. The Hitchin Flow as a Particular Case of the Generalized Hitchin Flow

If we suppose that ν_t = 1 and γ = 0, then:

$$\phi =dt\wedge \omega +{\psi }_{+}$$

is a parallel G₂-structure. In this case, the evolution equations of the differential forms ω(t), ψ₊(t), ψ₋(t) read:

$$\begin{array}{l}\frac{\partial }{\partial t}\omega (t)=\frac{2}{3}c(t)\omega (t)+w_2^{-}(t),\\ \frac{\partial }{\partial t}{\psi }_{+}(t)=c(t){\psi }_{+}(t),\\ \frac{\partial }{\partial t}{\psi }_{-}(t)=c(t){\psi }_{-}(t),\end{array}$$

(31)

the evolution equations of the intrinsic torsion forms of the coupled structure must be:

$$\begin{array}{l}\frac{\partial }{\partial t}c(t)=-\frac{1}{3}{(c(t))}^2-\frac{1}{4}|w_2^{-}(t){|}_{h(t)}^2,\\ \frac{\partial }{\partial t}{w}_2^{-}(t)\wedge \omega (t)=\frac{1}{6}|w_2^{-}(t){|}_{h(t)}^2{(\omega (t))}^2-c(t)w_2^{-}(t)\wedge \omega (t)-{(w_2^{-}(t))}^2,\end{array}$$

(32)

and for each t, the two-form $w_2^{-}$ has to satisfy the following property:

$$d{w}_2^{-}=-\frac{1}{4}|w_2^{-}{|}^2{\psi }_{+},$$

(33)

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 of the Hitchin flow equations and vice versa. For example, the coupled solution of the Hitchin flow on the Iwasawa–Lie algebra I obtained in the proof of Proposition 17 satisfies Equations (31), (32) and (33). 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 20. Let$(\omega (t),{\psi }_{+}(t),c(t),w_2^{-}(t))$ be a solution of Equations (31), (32) and (33), with the 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 hypothesis, we get (omitting the t-dependence for brevity):

$$\begin{array}{l}\frac{\partial }{\partial t}(d\omega -c{\psi }_{+})=d\left(\frac{\partial }{\partial t}\omega \right)-\dot{c}{\psi }_{+}-c\frac{\partial }{\partial t}{\psi }_{+}\\ =\frac{2}{3}cd\omega +d{w}_2^{-}+\frac{1}{3}{c}^2{\psi }_{+}+\frac{1}{4}|w_2^{-}{|}^2{\psi }_{+}-c^2{\psi }_{+}\\ =\frac{2}{3}c(d\omega -c{\psi }_{+}).\end{array}$$

Thus, if we denote by ρ(t) = dω(t) − c(t)ψ₊(t), we have that $\frac{\partial }{\partial t}\rho (t)=\frac{2}{3}c(t)\rho (t)$. Therefore, ρ(t) = q(t)ρ(0), where $q(t)=e^{{\displaystyle {\int }_0^t\frac{2}{3}c(s)ds}}$. However, 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. □

6. 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 $\mathfrak{N}=1$ compactifications of Type IIA string theory to four-dimensional anti-de Sitter space on six-manifolds endowed with an SU(3)-structure, namely coupled structures satisfying (all or in part) the constraints given in Equations (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 six-manifolds endowed with this kind of SU(3)-structure. 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 [14]. Here, we showed that for only one of these, Equation (4) is satisfied, while Equation (5) cannot be ever satisfied. However, since in the physical setting, Equations (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 [11]; thus, it would be interesting to see what happens for the nilmanifold corresponding to $\mathfrak{N}$. We also recalled an example firstly described in [6]; this is of particular interest, not only because it answers a question arising from [16], but also because it allows one to construct examples of G₂-structures with torsion inducing remarkable metrics.

In the last section, we considered the behavior of the coupled condition with respect to the Hitchin flow and one of its possible generalizations determined starting from four-dimensional domain wall solutions of heterotic string theory preserving $\mathfrak{N}=\frac{1}{2}$ supersymmetry. We observed that it is not always possible to find coupled solutions of 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 one-parameter family of SU(3)-structures that solves the Hitchin flow equations and is coupled for at least one, but not for all t.