Instantons on conical half-flat 6-manifolds

We present a general procedure to construct 6-dimensional manifolds with SU(3)-structure from SU(2)-structure 5-manifolds. We thereby obtain half-flat cylinders and sine-cones over 5-manifolds with Sasaki-Einstein SU(2)-structure. They are nearly Kahler in the special case of sine-cones over Sasaki-Einstein 5-manifolds. Both half-flat and nearly Kahler 6-manifolds are prominent in flux compactifications of string theory. Subsequently, we investigate instanton equations for connections on vector bundles over these half-flat manifolds. A suitable ansatz for gauge fields on these 6-manifolds reduces the instanton equation to a set of matrix equations. We finally present some of its solutions and discuss the instanton configurations obtained this way.

The outline of the paper is as follows: Section 2 is devoted to a review of various SU(2)structures, focusing on hypo geometry and investigating the 5-sphere as an example. Section 3 then provides several cone constructions that link Sasaki-Einstein 5-manifolds to particular SU(3) 6-manifolds. In Section 4 the instanton equations on these 6-dimensional SU(3)-manifolds are derived, and utilizing a certain ansatz for the gauge connection these equations are reduced to matrix equations. We derive some particular solutions for these matrix equations by the choice of a suitable matrix ansatz and discuss their corresponding gauge field configurations.

Sasakian structures
We begin by introducing several geometric structures that will become important in the constructions of this paper. As in [48], an almost contact metric manifold is an odd-dimensional Riemannian manifold (M 2m+1 , g) such that there exists a reduction of the structure group SO(2m + 1) of the bundle of orthonormal frames on T M to U(m). For such manifolds there exist a 1-form η and a 2-form ω such that η ∧ (ω) m = 0. Contact metric structures are characterized by dη = 2 ω in our sign convention.
An almost contact structure is characterized by the Nijenhuis torsion tensor [49] N = (N σ µν ). A quasi-Sasakian structure is given by N = 0 and dω = 0. In particular, if dη = α ω with α ∈ R, then the almost contact structure is called α-Sasakian. If α = 2 the structure is called Sasakian.
Nearly hypo: An SU(2)-structure on a 5-manifold is called nearly hypo if it satisfies Note that any SU(2) structure which satisfies the first two identities of (2.5) is a nearly hypo structure.
Double hypo: An SU(2)-structure on a 5-manifold is called double hypo if it is hypo and nearly hypo simultaneously, i.e. if it satisfies (2.6) and (2.7). Thus, the Sasaki-Einstein 5-manifolds are a subset of the double hypo manifolds.
As shown in [51], SU(2)-structures in 5 dimensions always induce a nowhere-vanishing spinor on M 5 . This will be generalized Killing if and only if the SU(2)-structure is hypo, and Killing if and only if the SU(2)-structure is Sasaki-Einstein. In [22] it has been argued that in the latter case there exists a one-parameter family of metrics g M 5 = e 2h δ ab e a ⊗ e b + e 5 ⊗ e 5 (2.8) which is compatible with an su(2)-valued connection on M 5 for which the Killing spinor is parallel. For the special value exp(2h) = 4/3 the torsion of that connection is totally antisymmetric and parallel with respect to that connection, i.e. there exists a canonical su(2) connection. For all values of h however, this connection is an su(2) instanton on T M 5 for the respective SU(2)-structure. For h = 0, M 5 is a Sasaki-Einstein manifold and the torsion components of the canonical connection read T a = 3 4 P aµν e µν and T 5 = P 5µν e µν . (2.9)

Example: the 5-sphere
We illustrate how different types of SU(2)-structures are embedded into each other with the example of the 5-sphere written as the homogeneous space S 5 = SU(3)/SU(2).
The SU(3)-structure constants can be chosen as The use of left-invariant objects on SU(3) enables us to explicitly compute connection components from the Maurer-Cartan equation. The connection 1-forms Γ µ ν and the torsion 2-forms T µ are then given as such that (2.14) With the Cartan-Killing metric (2.12) one obtains the totally antisymmetric components Note that 16) and that SU(3)/SU(2) is endowed with an SU(2)-structure given by e 5 and ω α as defined in (2.3). As a canonical connection on SU(3)/SU(2) we have with the above choices Now, we introduce a two-parameter family of SU(2)-structures on S 5 by a rescaling of the su(3) generators. Consider Consequently, the structure constants are changed as follows, A rescaling of the generators of su(3) rescales the left-invariant vector fields and 1-forms accordingly, and this is propagated to the coset via the pullback as used before. In particular, the rescaled structure constants have to be used in the Maurer-Cartan equation in order to compute the differentials of the rescaledẽ µ . We can use (2.3) with respect to the new coframesẽ µ to define a rescaled SU(2)-structure on S 5 . The differentials of the defining forms then read Thus, (η,ω 1 ,ω 2 ,ω 3 ) is a two-parameter family of hypo SU(2)-structures on S 5 , as the conditions (2.6) are satisfied for all values of β and γ. For the value (γ, β) = (− 1 3 , 1 2 √ 3 ) this turns out to be nearly hypo additionally, and, as a consequence, at this value the SU(2)-structure is double hypo. Furthermore, this particular SU(2)-structure is even Sasaki-Einstein, as we also show by a direct calculation of the Ricci tensor below. Therefore, the family of SU(2)-structures on S 5 does not discriminate between the double hypo and Sasaki-Einstein property. However, it shows how, by a simple rescaling of the generators of su(3), one can induce different SU(2)-structure geometries on S 5 . Note that there are many possible choices of a Riemannian metric on the coset space. Among them are the Cartan-Killing metric and the round metric on S 5 , which we consider in the following: Cartan-Killing metric: From the definition (2.11) we obtain g CK = δ ab e a ⊗ e b + e 5 ⊗ e 5 . (2.21) We express this with respect to local framesẽ adapted to the Sasaki-Einstein SU(2)-structure (i.e. for (γ, β) ). Thus, we arrive at By means of the Maurer-Cartan equations and demanding the torsion 2-form T µ to vanish, one obtains (2.24) for the connection 1-forms of the Levi-Civita connection induced by the Cartan-Killing metric on S 5 = SU(3)/SU (2). The curvature 2-form can be computed, and all 2-form contributions proportional toẽ j ∧ẽ k orẽ j ∧ẽ µ vanish due to the Jacobi identity [52]. Thus, the Ricci tensor reads (c.f. equation (2.20) for γ = β = 1), but not an Einstein space.
Round metric: Using again the local coframesẽ adapted to the Sasaki-Einstein structure, the metric induced by stereographic projection from the ambient R 6 reads Employing the Koszul formula for the round metric and the coframesẽ µ , one can calculate the Christoffel symbols of the Levi-Civita connection to be As before, the computation of the Ricci tensor is straightforward, and the result for this case is rnd Ric µν = 4 (g rnd ) µν = 4 δ µν . (2.30) Hence, the 5-sphere endowed with the round metric is an Einstein space with Einstein constant 4, just as expected.

SU(3)-structures in d = 6
As pointed out in the introduction, one of our goals is the construction of SU(3)-structures on 6-dimensional manifolds. Therefore, we introduce these structures and their characterization via intrinsic torsion classes. In a manner similar to Subsection 2.2, an SU(3)-structure on a 6-manifold M 6 is given by a reduction of the frame bundle to an SU (3) subbundle. An SU(3)-structure on a 6-dimensional manifold M 6 is characterized in terms of a triple (J, ω, Ω), where J is an almost complex structure, ω a (1, 1)-form, and Ω a (3, 0)-form with respect to J. These are subject to the algebraic relations The compatible Riemannian metric is determined by ω(·, ·) = g(J(·), ·), and the (3, 0)-form can be split into its real and imaginary part, i.e. Ω = Ω + + i Ω − . By an appropriate choice of a local frame, these forms can always be brought into the form ω = e 1 ∧ e 2 + e 3 ∧ e 4 + e 5 ∧ e 6 and Ω = (e 1 + ie 2 ) ∧ (e 3 + ie 4 ) ∧ (e 5 + ie 6 ). (2.32) For SU(3)-structures in 6 dimensions, there exist several types of such structures with different geometric behavior, which is mostly governed by the differentials dω and dΩ. SU(3)-structures in 6 dimensions have been classified in terms of their five intrinsic torsion classes [53]. These are encoded in the differential of the defining forms in the following manner: Here W ± 1 are real functions, W 4 and W 5 are real 1-forms, W ± 2 are the real and imaginary part of a (1, 1)-form, respectively, and W 3 is the real part of a (2, 1)-form. Note that both W 2 and W 3 are primitive forms [8], i.e. ω W 2 = 0 and ω W 3 = 0. (2.33c) The Nijenhuis tensor gives rise to the components W 1 and W 2 ; thus, the almost complex structure J of any SU(3)-structure with non-vanishing W 1 or W 2 is non-integrable.
To finish this section, let us list the structures of particular relevance to us.

Kähler-torsion:
On any almost Hermitian manifold (M, g, J) there exists a unique connection preserving this structure and having totally antisymmetric torsion [54]. This connection is called the KT connection or Bismut connection [55]. Kähler-torsion (KT) 6-manifolds are characterized by its torsion, which is given by and which is the real part of a (2, 1)-form. From [54] one can see that KT manifolds are complex manifolds, i.e. they enjoy Note that in general their structure group is U(3) rather than SU(3), as they are a subclass of almost Hermitian structures. However, they may reduce to an SU(3)-structure that is contained in the U(3)-structure.
Calabi-Yau-torsion: If the KT connection is traceless, its holonomy is SU(3) instead of U(3) and, therefore, the structure group is reduced to SU (3). Conversely, if one is given an SU(3)structure (g, ω, Ω) on M 6 , this is always contained in the almost Hermitian structure defined by (g, ω). The KT connection of the latter then comprises an SU (3) Note that, in general, one does not need a vanishing W + 1 , but this can be achieved by suitable phase-transformation in Ω.
Half-flat: An SU(3)-structure on a 6-manifold which satisfies is called half-flat.
Note that generic nearly Kähler and half-flat 6-manifolds have a non-integrable almost complex structure J and that nearly Kähler manifolds are a subclass of half-flat manifolds.

Cylinders and sine-cones over 5-manifolds with SU(2)-structure
Cylinders, metric cones, and sine-cones represent a tool for constructing (n+1)-dimensional Gstructure manifolds starting from n-dimensional G-structure manifolds with G ⊂ G . At first, we review the well-known Calabi-Yau cone and the previously presented Kähler-torsion sine-cone [47] for completeness. Next, we focus on the nearly Kähler sine-cone and the half-flat cylinder, which will provide the stage for the instanton equations considered in this paper.
First, let us assume we are given a 5-dimensional manifold M 5 with an SU(2)-structure defined by (η, ω α ) and a Riemannian metric if r is the natural coordinate on the interval I. Next, we can apply transformations to these local frames; for example, perform a transformation like e µ → φ(r) e µ and e 6 → e 6 , (3.2a) changing the metric on M 5 × I to the warped-product metric Still, the forms (φ η, φ 2 ω α , dr) will have the same components as in (3.1) with respect to the altered frames.
Afterwards, one still has the freedom of further transformations. These need to map one SU(2)-structure to another, which means that the defining forms need to have the standard components (2.3) with respect to the new frame. In addition, those transformations can be chosen to preserve the warped-product metric. In other words, these admissible transformations are given by maps from M 5 × I to the normalizer subgroup of SU(2) in GL(6, R) (or SO(6) if one wants to preserve g), i.e.
L : The crucial statement is that if we are given a set of forms (η, ω α ) on M 5 × I such that around every point in M 5 × I there is a local frame with respect to which (3.1) holds true, the forms defined by take the standard components (2.32) with respect to these local frames and, therefore, define an SU(3)-structure on M 5 × I. Note that ω and Ω are globally well-defined, simply because η and the ω α are.
This provides us with a general way to construct SU(3)-structure manifolds in 6 dimensions. Namely we push a given SU(2)-structure on M 5 forward to M 5 × I and apply transformations such that we still are given forms with components (3.1). Then we know that there exists an extension to an SU(3)-structure given by (3.4). In the following subsections we apply this procedure in several cases.

Calabi-Yau metric cones
One result that makes Sasaki-Einstein manifolds interesting for string theorists as well as mathematicians is that their metric cones are Calabi-Yau. Here we demonstrate this explicitly for the 5-dimensional case. Consider a Sasaki-Einstein 5-manifold M 5 with local coframes e µ , where µ = (a, 5) and a = 1, 2, 3, 4. The metric on its metric cone reads g = r 2 δ ab e a ⊗ e b + e 5 ⊗ e 5 + dr ⊗ dr = r 2 δ ab e a ⊗ e b + e 5 ⊗ e 5 + e 6 ⊗ e 6 (3.5) with The last equality in (3.5) displays the conformal equivalence to the cylinder over M 5 with the metric g cyl = δ ab e a ⊗ e b + e 5 ⊗ e 5 + e 6 ⊗ e 6 . (3.7) We can introduce an almost complex structure J on the metric cone via and we setêμ = reμ forμ = 1, . . . , 6. The SU(3)-structure forms (ω,Ω) have the local expressionŝ for which a direct computation yields dω = 0 and dΩ = 0 .
The explicit solution of τ = τ (ϕ) is computed to τ = ln tan ϕ 2 + constant , (3.13) and the integration constant can be chosen such that the sine-cone becomes the metric cone in the limit Λ → ∞. Hence, the computation yields (3.14) Next, we introduce an almost complex structure J and the associated fundamental (1, 1)-form ω on the sine-cone as follows (α = 1, 2, 3): where ω 3 is defined in (2.3). As shown in [47], the above structure comprises a Kähler-torsion structure on the sine-cone. That is, there exists the uniquely defined Bismut ∇ B connection, which preserves g and J, and has torsion given by Remarks: One can also introduce a globally well-defined complex (3, 0)-formΩ defined as Applying the exterior differential yields thus rendering the sine-cone over M 5 an SU(3)-structure manifold as defined in Section 2.4. From (3.18) we immediately see that J is integrable and whence the Bismut connection does not preserve the SU(3)-structure unless Λ = ∞. Nevertheless, the condition 3 W 4 + 2 W 5 = 0 is satisfied, which is in agreement with the conformal equivalence between the sine-cone over a Sasaki-Eintein 5-manifold and the Calabi-Yau metric cone over M 5 [53,57]. That is, the conformal equivalence of the Calabi-Yau cone and the Kähler torsion sine-cone also maps their two SU(3)-structures onto one another. We also note that 2W 4 + W 5 → 0 as Λ → ∞, and the KT sine-cone becomes the Calabi-Yau metric cone. Recall from section 2.4 that Kähler-torsion structures are U(3)-structures, whence one has to distinguish between this and the additional SU(3)-structure.

Nearly Kähler sine-cones
In [28] a nearly Kähler structure on the sine-cone over a Sasaki-Einstein 5-manifold has been obtained by means of flow equations. Here, in contrast, we show that this structure can be constructed by means of a combined rotation and rescaling of the coframes of the cylinder over the Sasaki-Einstein 5-manifold. We will carry this construction out in the following three steps: 1. An SU(3)-structure on the cylinder over a Sasaki-Einstein 5-manifold M 5 can be introduced via a metric (3.7), an almost complex structure J or the equivalent (1, 1)-form ω, and a (3, 0)-form Ω. These objects are (3.20c) 2. Next, we consider an SO(5)-rotation of the SU(2)-structure (η, ω α ) on M 5 . Let η 2 be the matrix of the 't Hooft symbols η 2 ab and perform a rotation of the basis 1-forms e 1 , . . . , e 4 , In the rotated frame (e a ϕ , e 5 ) we define the SU(3)-structure forms to have the same components as in the unrotated frame (3.20), i.e.
where ω α ϕ = 1 2 η α µν e µν ϕ . Note that this is still an SU(3)-structure on the cylinder, because the defining forms still have the standard components (3.20) with respect to the coframes e µ ϕ .
3. Last, the pullback to the sine-cone C s (M 5 ) along the map establishing the conformal equivalence to the cylinder yields e a s = Λ e a ϕ sin ϕ , e 5 s = Λ e 5 sin ϕ , e 6 s = Λ e 6 sin ϕ = Λ dϕ = dr , as an SU (3)-structure on the sine-cone. By a direct calculation we obtain which confirms that (3.23) induces a nearly Kähler structure on the sine-cone.

Remarks:
In the limit Λ → ∞, in which the sine-cone becomes the metric cone, this nearly Kähler structure on the sine-cone is smoothly deformed to the Calabi-Yau structure on the metric cone since lim Λ→∞ dω s = 0 and lim Generically, the sine-cone, as a conifold, has two singularities at ϕ = 0 and ϕ = π. As we see from (3.23), the SU(3)-structure cannot be extended to the tips, because all defining forms vanish at these points. Hence, the sine-cone is a nearly Kähler manifold only for ϕ ∈ (0, π), and one cannot add the singular points.

Half-flat cylinders
Consider a 5-dimensional manifold M 5 endowed with a Sasaki-Einstein SU(2)-structure defined by (η, ω 1 , ω 2 , ω 3 ) as in Section 2. For an arbitrary coframe e µ belonging to the SU(2)-structure, consider the transformation Here ζ ∈ [0, 2π] and ρ ∈ R + are two constant parameters. For = 1 this can be seen to be an SO(5)-transformation of the coframe, such that the metric on M 5 is unchanged. Nevertheless, we obtain a two-parameter family of SU(2)-structures on M 5 by defining These are globally well-defined as can be seen from and, thus, yield a two-parameter family of SU(2)-structures on M 5 . Note that these structures are neither hypo nor nearly hypo any more.
With these SU(2)-structures on M 5 at hand we define a two-parameter family of SU(3)structures on the metric cylinder (M 5 × R,ḡ z = g z + dr ⊗ dr) by which yields a two-parameter family of half-flat SU(3)-structures. The non-vanishing torsion classes can be computed to read

Summary of cone constructions
The different cone constructions linking Sasaki-Einstein to U(3) or SU(3) 6-manifolds, which have been presented in [47] and this paper, are summarized in the following

Definition and reduction of instanton equations on conical 6-manifolds
Having constructed several 6-dimensional SU(3) manifolds in the last section, we now turn our attention to instanton equations on such spaces. Thus, let M 6 be a 6-manifold with a connection A on the tangent bundle. The curvature 2-form F associated to A is given by where D A is the covariant differential associated to A, and the Bianci identity D A F = 0 holds true. As before, we can perform the type-decomposition of a form with respect to any almost complex structure J, yielding For a given SU(3)-structure (ω, Ω) on a 6-manifold and a curvature 2-form F, the instanton equation can be defined in two steps: first, the pseudo-holomorphicity condition reads and, second, applying the covariant differential to (4.3a), and using the Bianchi identity as well as (4.3a) yields The last equation, although a mere consequence of (4.3a), depends strongly on the type of SU(3)manifold under consideration. For example, on nearly Kähler manifolds one has whereas on half-flat SU(3)-manifolds this is not true as dΩ = κ ω ∧ ω. For Calabi-Yau spaces, on the other hand, (4.3b) is trivial as dΩ = 0, and the condition ω F = 0 is added as an additional stability condition for the holomorphic instanton bundle [31][32][33].
Following [58], one considers a complex vector bundle V → M 6 of rank k on which we are given an instanton Γ with curvature R Γ . Here this vector bundle will be the tangent bundle of 6-manifolds arising as certain conical extensions of SU(2) 5-manifolds M 5 , just as we considered in the previous section. We then generalize this instanton Γ by extending it to a connection A with curvature F by the ansatz where µ = 1, . . . , 5 and HereÎ i is a representation of the SU(2)-generators I i on the fibres R 6 of the bundle, and Γ i are the components of an su(2)-connection on the tangent bundle of M 6 . Furthermore, X µ are matrices from End(R 6 ).
The computation of F with the ansatz for A yields (4.7) Herein, T denotes the torsion of the connection Γ.
In order to simplify this further, we investigate the matrices X µ and their transformation behavior under a change of e. By construction, X µ e µ is the local representation of an Ad-equivariant 1-form X on the gauge principal bundle, which here coincides with the SU(3)-subbundle P of the frame bundle of M 6 that constitutes the SU(3)-structure. Note that, in the aforementioned cases, P contains a principal SU(2)-subbundle Q; the latter is the principal bundle for the connection Γ. Now let e and e be two local sections of Q ⊂ P over some U ⊂ M 6 related by an SU(2)transformation L : U → SU (2). The components X µ and X µ of X with respect to e and e are related via Here ρ is the representation of SU (2)  Since we will search for su(3)-valued connections A, we consider the su(3)-generator algebra The generators with indices i, j, k belong to the su(2) subalgebra, and the indices µ, ν, σ correspond to its orthogonal complement m in the SU(2)-invariant splitting Generically, only X is well-defined globally, rather than the component maps X µ . The latter strongly depend on the choice of the local frame e and, therefore, we have no control over their behavior in general. That would be different, if the components X µ were independent of the trivialization of the involved bundles, that is, if the X µ were invariant under the aforementioned transformations (4.8) that change the local frames. Furthermore, since SU(2) is connected, this is equivalent to the infinitesimal version of the invariance, i.e.
Note that this simplification implies that the X µ are independent of the choice of frame adapted to the SU(2)-structure Q; hence, we can choose them to vary with the cone direction only. Condition (4.11) appeared, for example, in [59,60] on coset spaces, where equivariant connections have been constructed. We will in the following refer to (4.11) as the equivariance condition, despite its different origin in this context.
Inserting this simplification and the accompanying consistency condition (4.11) into (4.7), we are left with Here the dot denotes the derivation in the cone direction. In any case, the instanton condition is the requirement that the 2-form part of F takes values in a certain subbundle of Λ 2 T * M 6 , which we call the instanton bundle. Anticipating that 2-forms of the general form e 6 ∧ e σ + 1 2 N σ µν e µ ∧ e ν , with N to be determined from the geometry under consideration, are local sections of this instanton bundle, we add a zero to the above expression and obtain (4.13) As argued above, R Γ and the second term already are instantons. Thus, we are left to require that the last term satisfies the instanton equation; this leads us to 14) where N has to be an instanton on M 6 that compensates for the su(2)-component of the lefthand-side commutator. Hence, N can only be a linear combination of the three instantons [22] f i µν e µ ∧ e ν for i = 6, 7, 8, which depends on the cone coordinate. That is, In summary, we are searching for m-valued matrices X µ that solve equations (4.11) and (4.15), as these will give rise to new instantons on the considered manifolds.

Remarks on the instanton equation
Before proceeding with the particular cases of the nearly Kähler sine-cone and the half-flat cylinder, one needs to clarify an important point regarding the transformations of coframes mentioned in Section 3.
The SU(2)-structure on the Sasaki-Einstein 5-manifold is understood as an SU(2)-principal bundle Q, a subbundle of the frame bundle F (T M 5 ). The warped product M 5 × φ I (c.f. (3.2)) is equipped with an SU(3)-structure via (3.4) and the corresponding principal bundle is denoted with P ⊂ F (T (M 5 × φ I)) (c.f. Fig. 1). However, P is not the SU(3)-structure one is interested in, i.e. in our cases it is neither nearly Kähler nor half-flat. The constructions of Subsections 3.3 and 3.4 rely on transformations of the coframes on M 5 : they generate a different SU(2)-structure Q that can be extended to the desired SU(3)-structure P on the warped product. An important observation is the following: for a G-structure Q the bundle Q defined via Q = R L Q is a G-structure if and only if L is a map from the base to the normalizer N GL(6,R) (G), c.f. (3.3).
The crux of the instanton equation is the following: the defining forms (ω , Ω ) stem from P , whereas the canonical connection Γ P belongs to Q and is trivially lifted to an instanton on P. Let us denote by e ∈ Γ(U, Q) an adapted frame for Q. Then by construction e =: (R L • e) ∈ Γ(U, Q ) The employed extension A = Γ P + X relies on the splitting (4.10) such that X corresponds to m-valued 1-forms. However, this only holds in the frame e, due to the following: Starting with Γ P on Q, one has a purely su(2)-valued connection. Applying any transformation L to Q, Γ P is generically not an su(2)-valued connection on Q . This is due to the fact that L −1 dL, in general, takes values in the Lie-algebra of N GL(6,R) (SU(2)) instead of su(2). Therefore, one cannot simply take e * Γ P as an starting point for some ansatz like e * A = e * Γ P + X µ e µ .
For the cases under consideration, L depends (at most) on the cone direction r. Hence, one has that Ad(L −1 ) • e * A is su(2)-valued and L −1 dL ∝ dr, but generically not su(2)-valued. The immediate consequences are the following: • For instance, on the nearly Kähler sine-cone one has to perform all calculations in the frame e, because for the derivation of Subsection 4.1 we employed a section of the bundle on which Γ P is an su(2)-valued connection. We will, however, compute e * Γ P explicitly in Subsection 4.3.2 and demonstrate that it yields an su(3)-valued instanton on the sine-cone.
• In contrast, the transformation for the half-flat cylinder (3.26) is, although a 2-parameter fam-ily, base-point independent. Therefore, one is allowed to consider the frames e as well as e for this instanton equation, as e * Γ P and e * Γ P are su(2)-valued connection 1-forms. However, this raises the question whether the two extensions X µ e µ and X µ e µ are in any sense comparable. Unfortunately, the coframe-transformations are only required to be N GL(6,R) (SU(2))-valued, which implies that the m-piece will, in general, not be mapped into m or even su(3). Hence, one cannot simply compare both extensions, but it is admissible to consider both cases.
In summary, these remarks were not relevant for the cases studied for example in [22,58] or our earlier results [47], because the construction of the G-structures on the warped product M 5 × φ I followed immediately from the chosen frame on M 5 . In other words, no (base-point dependent) transformation of coframes was necessary. Even on our KT-and HKT-sine cones of [47], the relevant rescaling (3.15) does not affect the computations due to conformal equivalence to the cylinder. However, here the situation is more involved and a careful analysis is mandatory.

Matrix equations -part I
The set-up for the nearly Kähler sine-cone has been described in Section 3.3. In particular, we are investigating extensions of the connection Γ P on the sine-cone in this subsection. M 6 being a nearly Kähler manifold, the instanton equation with respect to the coframe e µ is equivalent to  This can be seen either by direct computation or by the explicit form of the projectors from so (6) to su(3) of [45]. Here we have used the Riemannian metric to pull up one of the indices of η 3 , and from here on we use e 6 = dr.
A 6-dimensional representation of m can be chosen as in [22,58], from which one obtains the structure constants The torsion components of the canonical su(2)-connection Γ P in the unrotated frame e µ read With the chosen representation and by inserting the ansatz into (4.17), one obtains the non-vanishing components N ρ µν of the parametrization (4.15) as follows: Finally, the matrix equations for X µ read where the first line is just the equivariance condition (4.11). The dot-notation meansẎ ≡ d dr Y . An obvious solution to (4.24) is X µ ≡ 0, which yields the instanton solution A = Γ P that is the lift of the instanton Γ P from M 5 to the sine-cone C s (M 5 ). as well as the following differential equations Λ 2χ (r) sin(2ϕ) = 4 ψ 2 (r) − χ(r) and Λ 2ψ (r) sin(2ϕ) = 3 2 ψ(r) (χ(r) − 1) , (4.27a) which are subject to the constraints

Consider the ansatz
As a matter of fact, these equations (4.27) hold for any value of ξ ∈ [0, 2π). The solutions to (4.27) are readily obtained to be the following: • (ψ, χ) = (0, 0): This is, of course, the trivial solution of (4.24), but is still required for consistency as it confirms that Γ P satisfies the Ω s -instanton condition on M 6 .
• (ψ, χ) = (1, 1): Here we obtain an extension of the original instanton Γ P . Despite being an Ω s -instanton, this newly obtain instanton is a mere lift of an instanton in M 5 as it does not have any dependence on the cone direction.
Note that the existence of this solutions follows from ξ → ξ + π, as exp(π Hence, we have a one-parameter family of su(3)-valued instantons given by Remarks: First, the family solutions (4.28) can be seen to be gauge orbit if we recall that (η 3 ) ν µ ∝ f ν 5µ = ad(I 5 ) ν µ and then exp(ξ η 3 ) ∝ Ad(exp(I 5 )). Nevertheless, this gauge symmetry clarifies the origin of the ψ-reflection symmetry of the solutions.
Second, in the same manner as in our previous studies [47] we can equivalently provide the matrix equations on the conformally equivalent cylinder with coordinate τ as follows: Further, the limit Λ → ∞ (with ϕ= r Λ → 0 and keeping r fixed) transforms the sine-cone into the Calabi-Yau cone, as mentioned in Subsection 3.3. In this limit, the matrix equations (4.29) take the following form: which are exactly the same equations as on the Kähler-torsion sine-cone of our early results [47]. Applying the τ -dependent version of the ansatz (4.25) yieldṡ Obviously, all constant solutions found above are still instantons on the CY-cone, but the reduced equations do not automatically enforce constant χ and ψ. Finally, note that (4.31) is, of course, equivalent to (4.27) in the limit Λ → ∞ as the constraint on the derivatives vanishes.
Third, the sine-cone is a conifold with two conical singularities, here at ϕ = 0 and ϕ = π. One observes that the coefficient functions, i.e. cos ϕ and sin ϕ, of (4.24) as well as our solutions are well-behaved at the singular points. However, recall the remark from Subsection 3.3 that the defining sections of the SU(3)-structure become trivial at these singular points; hence, the instanton condition is not well-defined there. Yet, in principal one could continue the gauge field to these points.

Nearly Kähler canonical connection
In this section we construct the canonical su(3)-connection of the nearly Kähler sine-cone. It turns out that we obtain an instanton for the SU(3)-structure that is not the lift of an instanton on M 5 ; furthermore, this instanton is of the form (4.22) presented above. On the 5-manifold M 5 the Maurer-Cartan equations read where the torsion components are given by (cf. [22,58]) In particular, the last identity implies (Γ P ) 5 5 = 0 due to the Sasaki-Einstein relation de 5 = −2 ω 3 . Next, we are interested in the Maurer-Cartan equations for the frame e µ s resulting from the rotation (3.21) and rescaling (3.23) of the SU(2)-structure. With respect to coframes e adapted to Q, the canonical su(2)-connection Γ P has components where α(i) = i−5 andη α are the anti-self-dual 't Hooft tensors. Noting that [η α ,η β ] = 0 for all α, β, we see that the components of the canonical su(2)-connection are unaffected by the homogeneous part of the transformation (4.16) with which realizes the rotation (3.21) and the rescaling (3.23). In detail, the transformation reads no longer comprises the canonical su(2)-connection; however, it forms a different su(2)-valued connection Γ su (2) . This is because the inhomogeneous term in (4.16), which results from the change of basis, has been split off.
The connection 1-formsΓ su (2) β α with α, β = 1, 2 are defined via the components (Γ P ) b a by employing (4.32) and (4.36) as well as the change to the complex basis (4.37). We use the hat to indicate that we are considering the connection forms with respect to the complex basis Θ s rather than the real basis e s . Thus, the corresponding Maurer-Cartan equations read (3) is indeed a connection on T M 6 , which can be seen from (4.39) and the fact thatT transforms as a tensor. Furthermore, Γ su (3) is an instanton because it satisfies the conditions of proposition 3.1 of [22].
The above result (4.38) can be brought into a more suggestive form by rewriting it aŝ which reflects exactly the X µ -ansatz from (4.22). One can check that the matrices B µ satisfy the equivariance condition (4.11). Thus, as Γ su (3) is a connection on T M 6 , one can infer by the same arguments as in Section 4.1 that Γ su(2) is a well-defined connection on T M 6 . An alternative way to see that is to check that the inhomogeneous part, which has been split off in the transformation law (4.16) for the components of Γ P , glues to globally well-defined 1-forms with values in the adjoint bundle of P. This, however, holds due to the fact that the transformation L given in (4.35) commutes with the SU(2) subgroup of GL(6, R), i.e. takes values in centralizer C GL(6,R) (SU(2)).
Note that in the limit Λ → ∞ (i.e. ϕ = r Λ → 0) the torsion on C(M 5 ) vanishes, andΓ su(3) coincides with the connection corresponding to the χ = ψ = 1 case of [22], which has been stated to be the Levi-Civita connection of the cone. Furthermore, this is consistent with the observation that asΓ su(3) preserves the metric and as in the above limit its torsion vanishes,Γ su (3) has to converges to the Levi-Civita connection of the CY-cone.

Matrix equations -part II
As pointed out above, there are two different su(2)-valued connections on the nearly Kähler sinecone. On the one hand, there is the lift of the canonical connection Γ P of the Sasaki-Einstein 5manifold; on the other hand, there is Γ su (2) . Remarkably, the respective curvature 2-forms coincide, i.e. R Γ P = R Γ su (2) . (4.41) This stems from the fact that the generators of the two transformations (3.21) and (3.23), which lead from the cylinder to the sine-cone, commute with su (2). In other words, the inhomogeneous part of (4.16) yields an abelian flat part proportional to e 6 s . As a consequence, Γ su(2) is another su(2)-valued instanton on the sine-cone, since Γ P is an instanton itself 2 . Therefore, we can use Γ su (2) in the procedure described in Section 4.1: One extends Γ su(2) by some suitable 1-form X µ e µ s and investigates the conditions on X µ such that the new connection is an instanton on the sine-cone.
However, we have to adjust the equations (4.24) due to the different torsion of Γ su (2) . Denoting by T the torsion of Γ P , the torsion of Γ su (2) reads where we defined η 2 µν = η 2 ab forμ,ν = a, b ∈ {1, . . . , 4} and η 2 µν = 0 wheneverμ ≥ 5 orν ≥ 5. The components of N are the same as in Subsection 4.3.1 and, by inserting everything into (4.13), we obtain the matrix equations with the notationẎ = d dr Y . Next, we use the matrices in (4.40) for the extension of Γ su (2) . Recall that we had defined auxiliary matrices B µ that solve the equivariance condition (4.11) by writing (4.40) in the formΓ su(3) =Γ su(2) + B µ e µ s , (4.44) and that the B µ explicitly depend on ϕ = r Λ . Hence, we may set Let us now comment on the three solutions to this system: • (ψ, χ) = (0, 0): To start with, there is the obvious trivial solution of (4.43). This is required for consistency, since Γ su (2) is an instanton.
• (ψ, χ) = (1, 1): This second solution is very important because it reproduces Γ su(3) from Subsection 4.3.2. We already knew from proposition 3.1 of [22] that this particular connection is an instanton on the nearly Kähler sine-cone, but here we confirmed it directly, using techniques completely different than those employed in [22]. In addition, this provides us with another way of constructing the canonical connection of the nearly Kähler sine-cone than the one we followed in Subsection 4.3.2, namely as the extension of an su(2)-valued instanton.
In summary, the solutions we obtained here are isolated su(3)and su(2)-valued connections on M 6 that cannot be traced back to lifts of connections on M 5 . In contrast to e.g. [29], there are no instanton solutions that interpolate between these isolated instantons.
Remarks: First, the CY-limit Λ → ∞ of (4.43) is given by wherein one requires the rescaling X µ → 1 r X µ , which can be seen from X µ e µ s → X µ re µ for Λ → ∞. Further, recall that in the limit Λ → ∞ we have dτ = 1 r dr. The above matrix equations coincide with the ones obtained in Kähler-torsion case of [47] as well as with the limit (4.30) of Subsection 4.3.1. Remarkably, the two reductions of Subsections 4.3.1 and 4.3.3 used the different su(2)-instantons Γ P and Γ su(2) as starting point; however, in the above limit the difference ∈ Ω 1 (M 6 , End(R 6 )) (4.50) becomes an abelian flat part, which contributes to the instanton equation via the altered torsion.
Second, note the explicit impact of the conical singularities at ϕ = 0 or ϕ = π in the matrix equations (4.43) as well as the B µ -matrices of (4.40). However, we do not have to consider these singularities, as there is no well-defined instanton equation.

Transfer of solutions
The previous subsections considered the nearly Kähler sine-cone from two perspectives: in Subsection 4.3.1 we extended the instanton Γ P , which is a connection on Q; whereas, Subsection 4.3.3 was concerned with Γ su (2) , being an su(2)-valued connection on Q , as a starting point for our ansatz (4.5). The local representations of these are related via a transformation L as considered in (4.35). Due to the properties of L we arrive at the following statement (c.f. Subsection 4.2): implying that Γ su (2) and Γ P have the same components with respect to their adapted coframes e and e. Observe that the inhomogeneous part that is split off in the connection 1-form enters in the torsion (4.42) of Γ su (2) , thus altering the matrix equations. However, from (4.13) one can check that the local expressions of the respective field strengths of the extension of both Γ P and Γ su (2) by X µ ⊗ e µ = X µ L µ ν ⊗ e ν coincide. Consequently, every instanton extension X µ of Γ su (2) gives rise to an instanton extension X ν L ν µ of Γ P and vice versa. In other words, we have the relation The benefit from observation (4.52) is that we can generate further instanton solutions from our previous ones.
On the one hand, we can apply the above to (4.28) and obtain the ansatz which inserted into (4.43) has precisely the solutions (ψ, χ) = (0, 0), (±1, 1), just as one would expect from the above arguments. This is another non-constant instanton extension for Γ su (2) .
On the other hand, the same can be done for (4.45) in the other direction. There one derives the ansatz Rewritten in a linear combination of theÎ µ , the ansatz (4.54) is given as (4.55) One can check that this, again, produces the solutions (ψ, χ) = (0, 0), (±1, 1). Remarkably, the two non-trivial instanton solutions correspond to non-constant extensions of Γ P .

Instantons on half-flat cylinders
Let us now return to the half-flat 6-manifolds constructed in Section 3.4 and apply the ansatz developed above to the instanton equation on these spaces. The instanton equation on spaces with non-vanishing W 2 was introduced in (4.3). In a local coframe adapted to the SU(3)-structure imposing the pseudo-holomorphicity condition yields the set of six equations, precisely as it has been in the nearly Kähler case. But the additional equation implied by the pseudo-holomorphicity condition reads in the rotated frame e z . Note that for = ± It is important to recall that the lift of the canonical connection of the Sasaki-Einstein M 5 provides an instanton on the cylinder that one can extend by some X in our ansatz to su(3)-valued connections, being defined either on P or P . We will do so in two set-ups: first, we formulate the matrix equations in the frame e µ and, second, the analogous computation is performed in the adapted frame e µ z for the half-flat SU (3)-structure.
Identically to the nearly Kähler case, one obtains the one-parameter family (4.28) as a solution.
As a matter of fact, these instanton solutions are identical to the ones obtained in Subsection 4.3.1. The explanation is as follows: first, note that nearly Kähler 6-manifolds are a subset of half-flat 6-manifolds; thus, any nearly Kähler instanton solution must necessarily appear in the half-flat scenario. Second, the matrix equations (4.24) and (4.60) differ only in their derivative parts, i.e. in the coefficients ofẊ µ , which implies that both sets have coinciding constant solutions.

Matrix equations -part II
In particular, we constructed a nearly Kähler 6-manifold as a sine-cone over an arbitrary Sasaki-Einstein 5-manifold by means of a rotation of the SU(2)-structures on the slices. Employing the ansatz (4.22), the instanton equation was reduced to the set (4.24) of matrix equations, for which we found a family of non-trivial, but constant solutions. All of these correspond to lifts of M 5instantons to C s (M 5 ). In addition, in Subsection 4.2.2 we obtained an instanton solution on the manifold C s (M 5 ) by the construction of its su(3)-valued canonical connection. We decomposed this connection Γ su(3) into another su(2)-valued instanton Γ su (2) plus an additional part resembling the ansatz used before. Using this decomposition and, again, carrying the reduction of the instanton equation out, we obtained a set of four equations for two functions which parametrize the ansatz. Its three solutions, for which the scalar functions take certain constant values, correspond to three instantons on the nearly Kähler sine-cone that cannot be constructed as lifts of instanton connections on M 5 . As a by-product, we explicitly confirmed the nearly Kähler canonical connection to be an instanton. In addition, observing a correspondence between the solutions, we transferred the solutions of the two cases to new r-dependent instanton extensions of Γ P as well as Γ su (2) . Remarkably, the extension found for Γ P does not seem to correspond to a lift of an instanton from M 5 .
Furthermore, we introduced a two-parameter family of half-flat structures on the cylinder over a generic Sasaki-Einstein 5-manifold. Again employing the ansatz (4.5) on these cylindrical half-flat 6-manifolds, we were able to deduce the matrix equations (4.67) on the two local frames eμ and eμ z . Moreover, we provided families of constant, but non-trivial solutions. In that case, the instantons obtained this way do correspond to lifts of instantons on M 5 .
It would be interesting to extend the methods presented here, i.e. the reduction of the instanton equation to matrix equations and the construction of higher-dimensional G-structure manifolds from lower-dimensional ones, to other scenarios that appear in sting theory. For example, in Mtheory desirable manifolds are 7-dimensional and are endowed with a G 2 -structure. Therefore, the study of certain SU(3)-structures seems to be promising, as one could hope to a obtain interesting G 2 -geometries as well as explicit instanton solutions via the procedures employed here.