SU(3) structures on S2 bundles over four-manifolds

We construct globally-defined SU(3) structures on smooth compact toric varieties (SCTV) in the class of ℂℙ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{C}{\mathrm{\mathbb{P}}}^1 $$\end{document} bundles over M , where M is an arbitrary SCTV of complex dimension two. The construction can be extended to the case where the base is Kähler-Einstein of positive curvature, but not necessarily toric, and admits a parameter space which includes SU(3) structures of LT type.


Introduction
In many cases the problem of supersymmetric compactification to four-dimensional Minkowski or AdS space can be reformulated as the problem of existence of SU(3) structures with appropriate torsion classes parameterized by the "fluxes". Although this approach has lead to great progress in the construction of string vacua, the search for manifolds with suitable SU(3) structures has been far less systematic than the construction of Calabi-Yau manifolds, for which powerful algebro-geometric tools are available.

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In [1] it was proposed to use smooth compact toric varieties (SCTV) as a class of manifolds for which tools from both algebraic and differential geometry can be used, and develop a formalism suitable for the description of SU(3) structures on SCTV. The idea is to use the canonical structure that comes with the symplectic quotient description of the SCTV (metric, complex structure, set of coordinates), and construct on it a different (nonintegrable in general) almost complex structure associated with a globally-defined SU(3) structure.
The construction of SU(3) structures on SCTV proposed in [1] relies on the existence of a one-form K in the parent space of the symplectic quotient, satisfying certain requirements. Thus the problem of constructing SU(3) structures on SCTV is reduced to the problem of constructing one-forms K satisfying the requirements of [1]. Although that reference gave some examples of suitable one-forms, and many more were subsequently constructed in [2], no general formula for K exists satisfying the requirements of [1]. As a result, the search for SU(3) structures on SCTV had up to now proceeded on a case-by-case basis.
In the present paper we extend the formalism of [1] for SCTV to construct globallydefined SU(3) structures on the class CP 1 over M , where M is an arbitrary two-dimensional SCTV. As in [1], our construction is based on the existence of a one-form K which, in our case, is naturally distinguished by the structure of the bundle. This one-form does not have the right U(1) charge (in symplectic-quotient terminology) for the procedure of [1] to go through. A different procedure is used instead, which can be thought of as a generalization of the supersymmetric ansatz of [3,4] to the toric case, exploiting the local SU(2) structure of the base M of the fibration.
More specifically we give a general formula, eq. (4.11) below, for globally-defined SU(3) structures on all CP 1 bundles whose U(1) charges satisfy eq. (4.1). The latter equation can always be satisfied for any two-dimensional SCTV base, and amounts to choosing a specific twisting of the CP 1 bundle. The SU(3) structures thus constructed admit a space of deformations parameterized by certain functions, described below eq. (4.12). The associated torsion classes depend on these functions, and are nonvanishing in general.
This method can also be used to construct SU(3) structures on S 2 bundles over B 4 , where B 4 is Kähler-Einstein but not necessarily toric, in which case we have an SU(3)structure ansatz of the general form of [3,4]. Provided B 4 has positive scalar curvature, i.e. if it is CP 1 ×CP 1 , CP 2 , or one of the del Pezzo surfaces dP 3 , . . . , dP 8 [5], the total space of the S 2 bundle is complete and the associated metric is regular. We show that the parameter space includes SU(3) structures of LT type, suitable for N = 1 AdS 4 compactifications of massive IIA.
The outline of the remainder of paper is as follows. In section 2 we review the formalism of [1] for SCTV, and introduce the tools that will be used in the rest of the paper. The toric CP 1 bundles are described in section 2.4. In section 3 we work out in detail the example of CP 1 over CP 2 . This is the simplest example in the class of toric CP 1 bundles over M , but it already captures the main idea of the construction. The SU(3) structure is constructed in section 3.2. Section 4 discusses the construction of SU(3) structures on toric CP 1 bundles over general two-dimensional SCTV. Section 5 discusses the construction of SU(3) structures on S 2 bundles over four-dimensional Kähler-Einstein bases. We conclude JHEP09(2017)133 in section 6. For ease of presentation, many technical details have been moved to the appendices.

Review of the formalism
In order to fix the notation and make the paper self-contained, in this section we give a review of the SCTV formalism developed in [1]. Along the way we introduce the tools that will be useful in the rest of the paper. The description of the toric CP 1 bundles is given at the end of the section.
There are various equivalent ways to define a toric variety see e.g. [6], or [7] for an introduction for physicists. In the following we will use the symplectic quotient description, which turns out to be the best suited for the explicit construction of G-structures and the associated differential calculus. The starting point of the symplectic quotient description is a parent space C k , with coordinates {z i , i = 1, . . . , k}, and a set of s linearly-independent integer k-vectors Q a i , {a = 1, · · · , s} called the charges. LetM be the real submanifold defined by the following set of moment map equations, The real parameters ξ a are the so-called, Fayet-Iliopoulos parameters: they correspond to Kähler moduli, parametrizing the sizes of cycles of the toric variety. On the other hand the topology of the variety is independent of the ξ a as long as we stay inside the Kähler cone, defined by the conditions ξ a > 0. In the following we will always assume this to be the case. The associated toric variety M is given by the quotient M =M /U(1) s where the phase vector φ a ∈ U(1) s acts on the coordinates z i ∈M through the following gauge transformations, Hence M is a manifold of complex dimension d = k − s: the equations (2.1) can be thought of as removing s real "radial" directions, whereas the action of (2.2) removes s real 'angular' directions. In total the equations (2.1), (2.2) remove s pairs consisting of one radial and one angular variable, which may be thought of as s complex variables.
Since the Q a are independent as k-vectors, one may choose a set S of s indices such that Q a b , b ∈ S, is invertible. The open set {z b = 0, b ∈ S} ⊂ C k then descends to a welldefined open set in M , denoted by U S . On this patch one can then use the z b coordinates to compensate the U(1) s action on the z α coordinates, where the index α takes values in the complement of S, α ∈ S. One may then define the following gauge-invariant quantities, where we have set,

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where d := k − s, is a well-defined homeomorphism, while the transition functions ϕ S • ϕ −1 S are biholomorphic and rational. The charts (U S , ϕ S ) form a holomorphic atlas on M = |Q S |=1 U S : the t α , α ∈ S, define d gauge-invariant local holomorphic coordinates on U S . Note that for i = c ∈ S, we thus find t c = 1. The existence of this covering of M is related to the condition of smoothness for general toric varieties, and can be checked explicitly for the cases we consider here.
To take a simple example, consider the case s = 1 and Q = (1, · · · , 1). The corresponding toric variety is the complex projective space CP k−1 . Indeed (2.1) gives z 2 = ξ, i.e.M = S 2k−1 . Taking the U(1) quotient, M can be written as M = (C k \{0})/C * , the set of complex lines in C k . On the patch U j = {z j = 0}, the local coordinates take the form, which we recognize as the set of canonical coordinates of CP k−1 . The z i on the other hand correspond to homogeneous coordinates of CP k−1 .

Differential forms
We have seen that toric varieties are equipped with systems of complex coordinates which can easily be made explicit. Moreover it is often advantageous to work directly in the parent space C k using the homogeneous coordinates z i . We will be interested in particular in globally-defined differential forms on the manifold M . One way to construct a differential form on M is to start from its local expression on a patch, and make sure a regular global extension exists by checking its compatibility with the transition functions of the cotangent bundle. Working directly in C k drastically simplifies this problem: since the topology of the parent space is trivial, a single expression suffices to define differential forms globally. From this point of view the key question is to identify the differential forms of C k which descend to well-defined forms on M . In the following we review how the formalism of [1] can be used to treat this question. Let Φ be a differential form on C k . In order for Φ to descend to a well defined form on M , it should be well-defined onM . Hence it should be compatible with the moment map equations (2.1) which imply, Consequently Φ should not have any components along the η a , where we have defined, In other words, we require, where V a is the dual of η a (with respect to the canonical metric of C k ), These conditions have a natural interpretation: first note that a form Φ has charge q a if it is an eigenvector of the Lie derivative L (V a ) , We then see that the first of the two conditions above is simply the gauge invariance of Φ, i.e. the condition that the total charge of Φ vanishes. Moreover the second condition combined with (2.9) gives, which is equivalent to Φ being vertical with respect to V a . Thus in order to construct a well-defined form on M descending from a form Φ on C k , the gauge invariance of Φ must be imposed from the outset. On the other hand, the verticality condition is purely algebraic and can be imposed by projecting out the components along η a . Let us now come to the explicit construction of the vertical projector. We introduce the real symmetric matrix, The projection P of a (1, 0)-form Φ is then given by, This definition of P can be readily extended to all (k, l)-forms [1]. In the following it will be useful to define the vertical projections, Dz i , of the one-forms dz i , The Dz i are the building blocks that we will use to construct global forms on M . Note however that since they are not gauge invariant, one must compensate their charge by appropriate (charged) coefficients. On the other hand the (singular) form Dz i /z i is both gauge invariant and vertical and therefore admits an expression in terms of the local coordinates t i . On the patch U S we have, where we took (2.3) into account. Setting i = c ∈ S then gives dt c = 0, cf. (2.4). This leaves us with d linearly-independent one-forms dt α , α ∈ S. We can then compute, where we took into account that: As expected, given that the form on the left-hand side of (2.18) is vertical and gauge invariant, the result can be expressed in terms of the local coordinates alone. Note also that the gauge-invariant |z j | 2 can be expressed as a function of t i using (2.1).
Conversely, (2.17) can be used to express dt i as a function of Dz i , since dt i is vertical by definition. We now have all the necessary tools to translate back and forth between the local coordinate system {t i } on M and the global coordinate system {z i } on C k .

The hermitian metric
A useful object on an almost complex manifold M is the hermitian metric, where h ij can be thought of as a hermitian positive-definite matrix. The real and imaginary parts of h are real bilinear forms, so that, where g is symmetric, positive definite and can be identified with the Riemannian metric, while J is antisymmetric and can be identified with the almost symplectic form -which is a (1, 1)-form with respect to the almost complex structure. In other words the hermitian metric contains both the metric and the almost symplectic form of M . On toric varieties there is a canonical hermitian metric, As we have already noted, the Dz i above are not linearly independent and do not form a basis of the cotangent bundle of M . Using (2.18) it is not difficult to see that the hermitian metric takes the following form in local coordinates, where we have made use of the identity h ij h ik |z i | 2 = h jk which can be shown by taking into account the various definitions.
In this case J is in fact the Kähler form of the toric manifold, Although Dz i are not closed, it can readily be verified that dJ vanishes as it should. Let us illustrate the above with the example of CP k−1 : on the patch U k we have g = ξ and h ij = 1 ξ . Moreover (2.1) gives |z k | 2 = |zα| 2 |tα| 2 = ξ 1+t 2 , where t 2 := α |t α | 2 ; α = 1, . . . , k − 1. Hence,
A hermitian metric also gives rise to a scalar product "·" on forms on M . Since P 2 = P , the calculation of the scalar product on vertical forms can be done in the parent space C k using the flat metric. Then, using (2.15) we find: which shows that the Dz i are not orthogonal.

SU(d) structures
A Riemannian 2d-dimensional manifold M with metric g and associated (g-compatible) almost complex structure I admits a reduction of its structure group to U(d). At each point over M , the almost complex structure I, which need not be integrable, splits the cotangent space of M into a holomorphic and an antiholomorphic subspace, corresponding to the spaces of (1,0)-forms and (0,1)-forms with respect to I. Furthermore a holomorphic top form can be defined, i.e. a (d, 0)-form with respect to I, which transforms as a section of the canonical bundle of I. The canonical bundle of I is trivial, and so has vanishing Chern class: c 1 (I) = 0, precisely when it has a non vanishing global section, i.e. when there is a nowhere-vanishing holomorphic top form. In that case the structure of the manifold is further reduced to SU(d). An equivalent description of an SU(d) structure on M is given by a complex decomposable d-form Ω and a real two-form J such that, In this formulation the Riemannian metric on M is constructed from the pair (J, Ω). In six real dimensions (d = 3) it is well-known that the topological obstruction for the existence of an SU(3) structure is that the manifold should be spin. We can make contact with the discussion of the previous paragraph by noting that c 1 (I) modulo 2 is a topological invariant, and c 1 (I) is even in cohomology iff M is spin. 1 Moreover the torsion classes characterizing the SU(3) structure are given by the decomposition of (dJ, dΩ), As follows from the previous discussion, it is not always possible to construct an SU(d) structure on an arbitrary toric variety M . Although C k has a canonical SU(k) structure 1 Note however that c1(I) itself is not a topological invariant. A well-known counterexample is CP 3 which admits both a non-integrable almost complex structure (with c1(I) = 0) and an integrable one (with c1(I) = 0). In both cases c1(I) is even in cohomology, as of course it should, since CP 3 is spin.

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given by,J Ω does not in general descend to M . One can always define, which is vertical and gauge invariant: it is the almost symplectic form associated with the hermitian metric (2.22). However P (Ω) vanishes trivially since there are no (k, 0) forms on M . To obtain a (d := k − s, 0)-form on M we must contractΩ with each of the V a vectors, so that,Ω But thenΩ has the same charge asΩ, i.e. q a = i Q a i , and so it is not gauge invariant. On the other hand the pair (Ĵ,Ω) does satisfy the compatibility equations (2.26), thus defining a local SU(d) structure on M . MoreoverΩ admits a simple expression in terms of local coordinates 2 on U S . After some straightforward manipulations we obtain, where a ∈ S, α ∈ S and we have defined, (2.33) In [1] a prescription was given for the construction of global SU(d) structures on M . 3 It relies on the existence of a one-form K on C k with the following properties: 1. It is vertical and (1,0) with respect to the complex structure of C k .
2. It has half the charge ofΩ.
Given a one-form K on C k satisfying the conditions above, [1] showed that a global SU(3) structure on M can be constructed, and provided explicit examples of such a K for certain toric CP 1 bundles. Many more examples of K were provided for other toric varieties in [2], which also provided explicit computations of the torsion classes of the associated SU(3) structures. However there is no known construction for K that would be applicable in JHEP09(2017)133 general, even for a subclass of SCTV, and the search for SU(3) structures on SCTV had so far proceeded in a case by case fashion. In the following we will present a construction of SU(3) structures valid for toric CP 1 bundles over any 2d SCTV. As we will see, our method is not equivalent to the prescription of [1], although it also makes use of a certain (1,0)-form on C k .

Toric CP 1 bundles over SCTV
In [8], the classification of SCTV in three (complex) dimensions was shown to reduce to the classification of certain weighted triangulations of the two-dimensional sphere. In [1] it was shown how to systematically translate the results of [8] into the symplectic quotient language reviewed previously. In the following we will be interested in the subclass of the classification of [8] corresponding to CP 1 bundles over a two-dimensional SCTV base. However the formalism applies generally to the case of CP 1 bundles over SCTV, so in this subsection we will keep the dimension of the base arbitrary.
The U(1) charges of these bundles are given by the following set of (s + 1) × (k + 2) matrices, where A = 1, . . . , s + 1, I = 1, . . . , k + 2; n a ∈ N, a = 1, . . . , s, are integers specifying the twisting of the CP 1 bundle over a SCTV M ; q a i , a = 1, . . . , s, i = 1, · · · , k, are the U(1) charges of the symplectic quotient description of M , which is therefore of complex dimension d = k − s. (In subsequent subsections we will specialize to the case d = 2.) The total space of the bundle is constructed by appending two coordinates and one new charge to those of M (given by the q a i ), as in (2.34), thus obtaining a space of complex dimension d + 1. We will use the following notation for the data related to the fiber, (2.35) The last charge Q s+1 i defines a CP 1 fiber over M , while the integers n a determine the twisting of the bundle. Indeed the moment map equations for the total space read, Thus the last two coordinates define a sphere of radius √ ξ, while the first n coordinates define locally an M ρ whose "radii" (ρ a ) 2 := ξ a + n a |u| 2 depend on the fiber. The twisting can be thought of as a consequence of the modified U(1) s+1 action.
We would now like to construct a metric that exhibits the bundle structure, i.e. a metric of the form h d+1 = h d + h CP 1 , where h d is a metric on M and h CP 1 is a metric on the fiber CP 1 , possibly modified by a connexion on the base. By expanding the canonical metric (2.21) we find,

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where the details of the computation, which are somewhat involved, can be found in appendix A together with the definitions of the various quantities in the second term of the right-hand side above. The one-form ε can be thought of as an analogue of the vertical displacement along the fiber.
3 CP 1 over CP 2 Let us now examine in detail the construction of an SU(3) structure on the CP 1 bundle over CP 2 . This is the simplest example in the class of 3d SCTV of the form CP 1 bundle over M , where M is a 2d SCTV, but it already captures the main idea of the construction. We will treat the general case in section 4. The toric data in this case are: k = 5 (the complex dimension of the parent space), s = 2 (the number of charges), d = 3 (the complex dimension of the toric variety). Explicitly the charges are given by, where n ∈ N. The corresponding moment map equations read, using the notation introduced in section 2.4, This is a CP 1 bundle over CP 2 , with twisting parameterized by n. We can make this more explicit in local coordinates: on the patch U 1,5 := {z 1 , v = 0} we define, Hence t 2 , t 3 are local coordinates parameterizing a CP 2 whereas, for z 1 fixed, t 4 is a local coordinate on a CP 1 . For n = 0, the bundle becomes trivial and we obtain the direct product CP 2 × CP 1 . We can also see explicitly that the toric variety can be covered with patches of the form U S , as in (2.3): in the present case S is given by the pair (i, j) where i = 1, 2, 3 and j = 4, 5, and the moment map equations (3.2) exclude the simultaneous vanishing of z 1 , z 2 , z 3 or that of u, v. To make contact with our previous discussion of local coordinates, we can check here that |Q S | = 1 for all the S defined above. On the other hand for the patch U 4,5 we do not get compatible local coordinates in general, since Q S ={4,5} = −n, however this patch is not used in the covering of the toric variety by U S . Let us now calculate explicitly the various objects introduced in section 2. Since the base is defined by only one charge q i = (1, 1, 1), the calculations are rather simple. We have,ĝ

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We thus find, where we have set, If we now introduce, the decomposition of the metric (2.37) can be written, where h CP 2 is the hermitian Fubini-Study metric of CP 2 with unit radius, cf. eq. (2.24). We see the fibration structure appearing naturally in (3.5): the displacement along t 4 is modified by a connection, proportional to η, depending on the variables of the CP 2 base, whereĵ is the Kähler form of CP 2 , cf. eq. (2.24). For vanishing n the connection piece drops out from the vertical displacement and the metric becomes that of a direct product as excpected.

Comparison with the literature
Endowed with the hermitian metric (2.24), the base CP 2 of the CP 1 fibration is a Kähler-Einstein manifold obeying, dĵ = 0 ; R mn = λ g mn . i.e.ĵ is closed and the Ricci tensor is proportional to the metric. With our conventions, setting ξ = 1 gives λ = 6. Identifying the CP 1 fiber with S 2 (by forgetting the complex structure), M can be thought of as an S 2 fibration over a Kähler-Einstein base B 4 , denoted by S 2 (B 4 ). These spaces appear naturally in the context of supersymmetric AdS 4 compactifications of M-theory on the so-called Y p,q (B 4 ) spaces [9,10], which can be thought of as S 1 fibrations over S 2 (B 4 ). Compactifying M-theory on an appropriately chosen S 1 then leads to N = 2 type IIA solutions of the form AdS 4 × S 2 (B 4 ) [3]. The latter can be deformed to solutions of massive IIA for any Kähler-Einstein base B 4 [4], although regularity requires B 4 to have positive curvature.
In the conventions of [10] the S 2 (B 4 ) metric reads, At the endpoints of theρ interval the ψ circle contracts to a point, thus resulting in a total space with the topology of S 2 . The period of ψ is fixed by requiring the metric to be smooth at the endpoints, i.e. that, where u is a function ofρ that vanishes at the endpointsρ 1 ,ρ 2 , and we have defined an angular variableψ := λψ/2 with period 2π. Moreover the ψ coordinate parameterizes an S 1 fibration in the canonical bundle of B 4 . To see this, note that the connection of the canonical bundle of a Kähler-Einstein space with curvature normalized as in (3.7) obeys, dP = λĵ , (3.11) cf. appendix B. Comparing with (3.9) we see that P = λA/2, and so the vertical displacement along the S 1 fiber, cf. the last term in (3.8), is proportional to (dψ + P), as required for the canonical bundle. The fact that λ is positive for CP 2 guarantees that the total space of the S 1 fibration, written in local coordinates in (3.8), extends globally to a smooth five-dimensional (squashed) Sasaki-Einstein space.
To make contact with the coordinates of (3.5), we must rewrite the CP 1 fiber coordinate t 4 in terms of a pair of real coordinates. Using the formulas of section C we can rewrite the Riemannian metric g and Kähler form J associated with (3.5) for n = 0. The result reads, and, where we are using local coordinates on the patch U 1,5 , and ϕ ∈ [0, 2π] denotes the phase of t 4 . The CP 1 fiber is parameterized by the (ρ, ϕ) coordinates: ϕ parameterizes a circle, fibered over the interval ρ ∈ [ρ 1 , ρ 2 ] = [ √ ξ 1 , √ ξ 1 + nξ], whose radius vanishes at the endpoints. Indeed Γ vanishes for u = 0 or v = 0 which correponds respectively to ρ = ρ 1 and ρ = ρ 2 , following from the moment map equations (3.2). Moreover it can be checked that the metric is smooth there.
Furthermore we need to deform the canonical hermitian metric of the toric variety by introducing two warp factors F (ρ), G(ρ) along the base and fiber respectively, (3.14) It can then be seen that the functions F (ρ), G(ρ) together with a change of variables ρ =ρ(ρ) may be chosen so that the real and imaginary parts of (3.14) reduce to the metric JHEP09(2017)133 in (3.8) and the form J + of [10] respectively, provided we set n = 3. The details of this exercise can be found in appendix C. The condition n = 3 is also important for the existence of a globally-defined SU(3) structure. We turn to the construction of this structure in section 3.2. Note however that the canonical metric of the SCTV, eq. (2.21), is smooth by construction for all n ∈ N. This can also be verified explicitly by examination of the local form of the metric in terms of the coordinates (2.3) in each patch U S .

The SU(3) structure
In this section we will set F = G = 1 for simplicity of presentation: the two warp factors F (ρ), G(ρ) discussed in section 3.1 can be easily reinstated without changing any of the conclusions.
Specializing the formalism of section 2.3 to the present example we obtain a local SU(3) structure (Ĵ,Ω), whereĴ is obtained from (3.13) by setting n = 3. On the other hand we have,Ω which is not gauge invariant, so this SU(3) structure is not globally defined. In fact neither of the two local SU (3) structures (J ± , Ω ± ) of [10] can be globally extended: in the following we will see how to make contact with their results. Let us first define a local SU(2) structure (ĵ, ω) on CP 2 , where, andĵ is the Kähler form of CP 2 , cf. eq. (2.24), so that, This SU(2) structure is only locally defined since ω has a singularity at z 1 = 0, as can be seen by using the transition functions to rewrite ω in a patch where z 1 is allowed to vanish. The SU(3) structures of [10] are then obtained by appending the contribution of the fiber coordinate, where, We see that exchanging K ↔ K * is equivalent to (J + , To better understand the global properties of the Ω ± , let us start from their local expression on the patch U 1,5 ,

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We can see that the singularity in ω has been compensated by wedging with K, K * . On the other hand, we can rewrite Ω ± in the patch U 1,4 by using the transition function t 5 = 1/t 4 , We see that Ω ± has singularities of the form e iϕ = t 4 /|t 4 | = |t 5 |/t 5 at t 4 = 0 and t 5 = 0: indeed the phase of a complex number z is ambiguous at z = 0. It is always possible to soak up one of the two singularities by multiplying or dividing by e iϕ , but never both at the same time. Hence e ±iϕ Ω ± are well-defined at t 4 = 0 but singular at t 5 = 0, whereas e ∓iϕ Ω ± are well defined at t 5 = 0 but singular at t 4 = 0. This problem does not arise for J ± , since K ∧ K * does not suffer from any phase ambiguities. The way out is then to construct an Ω which combines both e ±iϕ Ω ± and e ∓iϕ Ω ± . We can take a hint from the supersymmetric SU(3) structure of [4] which we know is globally well-defined. We use a new coordinate θ instead of ρ, defined by |u| 2 = ξ sin 2 θ 2 . Thus we see that |v| 2 = ξ cos 2 θ 2 and ρ 2 = ξ 1 + n ξ sin 2 θ 2 , which means that θ = 0 or π for ρ = ρ 1 (corresponding to t 4 = 0) or ρ = ρ 2 (corresponding to t 5 = 0), respectively. The idea is then to modify ω →ω by including the problematic phase e iϕ , then define another form ω with the property thatω varies fromω toω * as θ varies from 0 to π. More specifically we define,ω := e iϕ ω j := sin θ ω + cos θĵ ω := cos θ ω − sin θĵ + i ω , (3.20) so that the SU(3) structure is given by, (3.21) The relations (3.17) ensure that (2.26) is satisfied. Moreover at θ = 0 we have Ω = e iϕ Ω + , whereas at θ = π we have Ω = − e iϕ Ω − * . The two singularities have thus been regularized and Ω is globally defined. Thus the pair (J, Ω) is a globally-defined structure SU(3) on the manifold. Let us make one final comment: the prescription of [1] for constructing global SU(3) structures, reviewed at the end of section 2.3, gives a form Ω which is of type (2,1) with respect to the integrable complex structure of the toric variety. We see that the prescription used here can never coincide with that of [1]: the form Ω defined in eq. (3.21) is of mixed type, varying from (3,0) at θ = 0 to (1,2) at θ = π, with respect to the integrable complex structure.

CP 1 over general SCTV
We will now show how to construct a globally-defined SU(3) structure on a canonical (defined in eq. (4.1) below) CP 1 bundle over a SCTV of complex dimension d = 2. This is a generalization, to any SCTV base, of our construction of a globally-defined SU(3) structure on CP 1 over CP 2 , discussed in section 3.2.
As we saw explicitly in the special case of CP 1 over CP 2 , the canonical metric of the SCTV, eq. (2.21), is smooth for any twisting of the bundle parameterized by n a ∈ N. On the other hand the existence of a globally-defined SU(3) structure imposes a topological constraint and hence a constraint on the n a , as we explain in the following. This constraint is automatically satisfied for the canonical CP 1 bundle. 4 We start with a (d + 1)-dimensional toric CP 1 bundle over a d-dimensional base M , whose charges were given in (2.34). The CP 1 bundle will be called canonical if the charge of z k+1 , defining the twisting of the bundle, is taken to compensate exactly for the charges of the base, i.e., As emphasized in [2], the topological condition for the existence of an SU(3) structure on the total space of the SCTV is that its first Chern class should be even. Condition (4.1) guarantees that there is no topological obstruction for the existence of an SU(3) structure. This can be seen as follows: the first Chern class of the SCTV is given by, where we have denoted by D I the divisors corresponding to {z I = 0}. On the other hand on a toric variety there are as many linearly-independent divisors as there are U(1) charges [7]. In our case the fact that the local coordinates defined by S in (2.3) are gauge-invariant is equivalent to the linear relations, Taking the charges (2.34) into account, and inserting into (4.2) then leads to, which, as advertised, is even if the bundle is canonical. More generally, we see that a globally-defined SU(3) structure exists provided ( k i=1 q a i − n a ) are even for all a [2].

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We define the usual toric coordinates and a local SU(d+1) structure (Ĵ,Ω) as explained in section 2.3. We recall thatΩ is not gauge-invariant: for the canonical CP 1 bundle it has charge, where we took (4.1) into account. Following the strategy of section 3.2 we would like to define the analogue of the local SU(2) structure (ĵ,ω) on the base M , cf. (3.20). As in that case we first note that the CP 1 fiber distinguishes a one-form K, which we normalize such that K * · K = 2, (4.6) Note that K is not globally defined since it is not gauge-invariant. This can be seen explicitly by taking the u → 0 limit, in which Dv vanishes. Indeed in this limit we have, where ϕ u , ϕ v denote the phases of u, v. However K ∧ K * does not suffer from any phase ambiguity, so that,ĵ is globally well-defined. Furthermore a somewhat tedious calculation which can be found in appendix D shows thatΩ can be simplified to, (4.8) Its contraction with K is given by, which is not gauge-invariant. A gauge-invariant local holomorphic formω on the base can be constructed as follows,ω := 1 2 e −iϕv K * ·Ω . Let us now specialize to d = 2. We can apply the procedure of section 3.2 and modify the local SU(2) structure (ĵ,ω) in order to construct a global SU(3) structure. Since we have |u| 2 + |v| 2 = ξ, we can define a parameter θ ∈ [0, π] such that |u| = √ ξ s sin θ 2 and JHEP09(2017)133 |v| = √ ξ s cos θ 2 . By the same argument as in section 3.2, the SU(3) structure (J, Ω) given by, where, j := sin θ ω + cos θĵ ω := cos θ ω − sin θĵ + i ω , can be seen to be globally-defined. Its associated metric is the canonical metric of the SCTV, given in (2.21), (2.37). The expressions above can be thought of as a natural extension of the ansatz of [3,4] to the toric case. Note however that, contrary to the simple case discussed in section 3, j,ω depend in general on the S 2 -fiber coordinates and should be thought of as forms on the total space. Moreover, except for simple bases such as CP 2 or CP 1 × CP 1 , there is not a simple relation between the exterior differential of the connection one-form in K and the two-forms on the base, cf. appendix D.1 for more details. As a result, the associated torsion classes are all nonvanishing in general.
This structure could be easily modified by multiplying (j, ω) and K by functions of the coordinates of the S 2 fiber. The associated metric will be modified accordingly to, for some functions of the fiber coordinates, f , h. Indeed modifying the local SU(2) structure via ω → h 2 ω, j → |h| 2 ω, K → f K results in the metric (4.12). More generally, an orthogonal transformation can be applied on the triplet (j, ω, ω), without changing the metric h 2 of the base. Provided f , h are smooth and nowhere-vanishing, the topology of the total space is that of the SCTV CP 1 over M . The metric (4.12) is smooth, since it is a smooth deformation of the canonical metric (2.37) of the SCTV. In some cases allowing f , h to have singularities or zeros can lead to a smooth metric on a total space of different topology. We will see an example of this phenomenon in section 5 where an apparently singular metric on S 2 over CP 2 is in fact the local form of the round metric on S 6 .

LT structures on S 2 (B 4 )
We will now show that the sphere bundles of the form S 2 (B 4 ), where B 4 is any fourdimensional Kähler-Einstein space of positive curvature, admit regular globally-defined SU(3) structures of LT type, i.e. such that all torsion classes vanish except for W 1 and W 2 . This is the generic type of SU(3) structure that appears in supersymmetric AdS 4 compactifications of massive IIA supergravity [11].
For H 0 > 0 the functions f , h are nowhere vanishing. Moreover the θ → 0, π limit gives a regular metric, provided the period of ψ is 2π. Then by the same argument as in [4,9], the SU(3) structure (5.2) is globally-defined and the associated metric (5.4) is regular and complete: the (ψ, x µ ) space, where x µ are the coordinates of B 4 , parametrizes a circle fibration in the canonical bundle L over B 4 ; it extends to a complete, regular five-dimensional Sasaki-Einstein manifold provided B 4 is Kähler-Einstein of positive curvature [12]. The (ψ, θ) space parameterizes a smooth S 2 , so that the total space has the same topology as L × U(1) CP 1 , in the notation of [9]. The nonvanishing torsion classes read, Therefore the S 2 (B 4 ) bundles admit SU(3) structures of LT type, rendering them suitable as compactification spaces for supersymmetric AdS 4 solutions of massive IIA [11]. Note that unlike the LT SU(3) structures on S 2 (CP 2 ) discussed in [13] from the point of view of twistor spaces (cf. appendix F) or in [14] from the point of view of cosets, the structure (5.9) does not obey dW 2 ∈ (3, 0) ⊕ (0, 3). 5 Indeed a direct calculation gives, (5.10) As a consequence, if these manifolds are to be used as compactification spaces for massive IIA, the Bianchi identity for the RR two-form will require the introduction of (smeared) sixbrane sources. Another difference from the LT structures of [13,14] is that the discussion of this section applies to any S 2 (B 4 ) bundle with Kähler-Einstein base, not only to B 4 = CP 2 .
In the case H 0 = 0, on the other hand, one obtains the solution, This corresponds to the nearly Kähler limit, in which also W 2 vanishes. Moreover the θ → 0, π limit results in a conical metric of the form, g ∼ dθ 2 + θ 2 ds 2 5 , (5.12)

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where, ds 2 5 := g 4 + 1 9 (dψ + P) 2 , (5.13) is the canonically normalized metric of a five-dimensional Sasaki-Einstein base written as a circle fibration on the canonical bundle over B 4 ; the normalization is such that the cone metric (5.12) is Ricci-flat. Hence for H 0 = 0 the metric presents conical singularities in general, unless B 4 is CP 2 , in which case the associated Sasaki-Einstein metric (5.13) is that of the round sphere, and the associated cone (5.12) is not only Ricci-flat but also flat. Going back to the metric (5.4) we obtain, g = 9α 2 dθ 2 + sin 2 θ ds 2 5 . (5.14) We thus see that in the smooth case, B 4 = CP 2 , we obtain a round S 6 of radius 3α. We thus recover the well-known result that the round S 6 admits an associated nearly-Kähler structure.
Let us finally note that we may relax the condition on B 4 , so that B 4 is any fourdimensional Kähler manifold (not necessarily toric, or Einstein). In this case the torsion classes can also be explicitly calculated, cf. appendix G, however we do not expect the structure to admit a global extension to a complete space with a regular metric.

Conclusions
The construction of SU(3) structures on SCTV had up to now proceeded on a case-by-case basis. In the present paper we gave a formula for a globally-defined SU(3) structure valid on all canonical CP 1 bundles over two-dimensional SCTV. This SU(3) structure admits a space of deformations parameterized by certain functions, on which the associated torsion classes depend. The construction is genuinely different from that in [1]: as opposed to the construction in that reference, it produces a holomorphic three-form of varying type (with respect to the integrable complex structure of the SCTV).
Having a general formula for the SU(3) structure opens up the possibility of a systematic (possibly automatized) scan for flux vacua. Such a procedure has been successfully carried out in the case of solvmanifolds [15] and cosets [14], and would be interesting to undertake also in the class CP 1 over SCTV considered here. It could be extended to CP 1 fibrations over noncompact toric varieties, as the formalism does not rely on compactness other than in the input of the U(1) charges specifying the toric variety [16].
The construction of the SU(3) structure was also applied to the case of S 2 (B 4 ) bundles. These spaces first appeared as six-dimensional bases of seven-dimensional Sasaki-Einstein spaces in the context of N = 2 AdS 4 vacua of M-theory [9]. It was subsequently realized [10] that reducing along the so-called α-circle produces a (warped) N = 2 AdS 4 × S 2 (B 4 ) vacuum of IIA. The relevant supersymmetric SU(3) structure, whose existence was implicitly inferred in [10], was first constructed explicitly in [3] for the case B 4 = CP 2 . The generalization to arbitrary B 4 was given in [4].
In the present paper we showed that the S 2 (B 4 ) spaces also admit a different SU(3) structure of LT type, thus making them suitable for N = 1 compactifications of massive JHEP09(2017)133 IIA. However, these solutions require (smeared) six-brane sources, complicating their physical interpretation. It is possible that a more general orthogonal rotation of the local SU(2) structure acting on the triplet (j, ω, ω) may produce a sourceless LT structure, although the analysis becomes rather cumbersome in this case and we have been unable to obtain a conclusive result. 6

A Decomposition of the metric
In this section we fill in some of the details leading up to eq. (2.37). We start by defining the vertical one-forms using the formalism introduced in section 2.1, for the total bundle. The various objects are thus given in terms of the charges Q A I . Expressing them in terms of the charges of the base q a i we have, where hatted symbols are used to denote objects relative to the base, in order to distinguish from the objects constructed in (2.1). Note thatĝ,ĥ,Dz i live on the same space as their non-hatted counterparts, which are the relevant objects for the definition of forms in the symplectic quotient description. This means that they do not have any a priori interpretation as objects on the base. For example, the |z i | 2 do not verify the moment map equations of the base but those of the total bundle, and thusĝ,ĥ andDz i depend on the radii. A quick calculation confirms that theDz i do obey the expected algebraic relations, Recall the form of the canonical metric on a SCTV (the generalization of the Fubiny-Study metric of CP 1 ), We will now decompose this metric into base and fiber components. Since the Dz depends on the matrix g AB , the key here will be to decompose it and its inverse along the different bundle directions. The definition of Q A I leads to: g AB = ĝ ab + n a n b |u| 2 −n a |u| 2 −n b |u| 2 ξ . 6 If the existence of a sourceless LT structure could be established within the context of the ansatz of the present paper, it would exist for all S 2 (B4) spaces, not only for B4 = CP 2 , as our ansatz only relies on the Kähler-Einstein property of the base. As already remarked, a sourceless LT structure does exist on S 2 (CP 2 ) [13,14], but seems to rely on the special properties of CP 2 as a selfdual Einstein manifold [17]. This is not taken into account by our ansatz, and would not be applicable to the other Kähler-Einstein bases B4: CP 2 is the only Kähler-Einstein four-manifold of positive curvature that is also selfdual.
We now use the property of multilinearity of the determinant to expand this expression. We then get all different terms of order s−m in g and m in n n T . But since rank n n T = 1, only the terms of order zero or one remain. The terms of order one are merely the determinant ofĝ where the column a has been replaced by the vector n a ξ |u| 2 |v| 2 n. By expanding along this same column, we exhibit the cofactors ofĝ which are independent of this exact column, and are related to the inverse matrix, Thus we have: g = ξ ĝ + n a ξ |u| 2 |v| 2ĝĝ ab n b =ĝ(ξ +ĝ ab n a n b |u| 2 |v| 2 ) .
The same trick can be used to compute the inverse matrix: (1 +ĝ ab n a n b |u| 2 ) . Moreover, The last cofactors are somewhat more complicated, since they involve double cofactors. Eventually we get: g ab =ĝ ab −ĝ g |u| 2 |v| 2ĝ ac n cĝ bd n d .
It is now possible to compute the h µν . Let us introduce the objects V :=ĝ ab n a n b , V i :=ĝ ac q a i n c , (A.1) in terms of which we obtain,

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We can now compute the Dz I , The last two coordinates correspond to colinear one-forms, Finally the canonical metric reads, On the other hand we have, so that the metric simplifies to, Note that this decomposition remains valid in the complex local coordinates t i , t k+1 , on the chart U S defined by S =Ŝ ∪ {k + 2}, in which ε can be written as, TheDz i happen to be the projections on the space generated by the dt i , in fact they are related to the dt i by the relations (2.18) where we takeĥ ij instead of h IJ . This justifies that in the decomposition (A.6), the metric on the base is exactly the canonical metric whose radii vary along the fiber.

B Kähler-Einstein manifolds
where P := 2 W 5 is a real one-form. (Note that the existence of the complex structure allows us to reconstruct the torsion W 5 from its imaginary part alone.) Moreover (B.2) can be inverted to obtain P from the covariant spinor derivative, where R is the Ricci form. Hence P can be identified with the connection of the canonical bundle of M . On the other hand, the Ricci tensor is obtained from the Riemann tensor via, On a Kähler manifold the Ricci form, the Ricci tensor and the Ricci scalar obey, Furthermore for a Kähler-Einstein manifold such that, eqs. (B.7), (B.6) imply, R = λJ , (B.8) but in general the Ricci form need not be proportional to the Kähler form. The above relations are valid for arbitrary dimension. Specializing to four real dimensions we adopt the notation (J, Ω) → (ĵ,ω), in accordance with the main text. We may decompose any two-form Φ on the basis of a local SU(2) structure (ĵ,ω) as follows: where ϕ := 1 4ĵ mn Φ mn is the trace of Φ, and Φ is (1,1)-traceless:ĵ mn Φ mn = 0. Equivalently, It is also straightforward to show that (ĵ,ω) are selfdual forms while (1,1)-traceless forms are anti-selfdual, (ĵ,ω) = (ĵ,ω) ; In particular for the Ricci form the expansion reads, Moreover the above properties can be used to calculate, where the volume is given by, ∧ĵ . (B.14)

JHEP09(2017)133 C Real coordinates
In this section we explain in detail how to rewrite the hermitian metric (3.5) in terms of real coordinates, and make contact with the metric (3.8). Let us start by rewriting the CP 1 fiber coordinate t 4 in terms of a pair of real coordinates. It is not necessary to do the same for t 2 , t 3 , since the coordinates of the CP 2 base do not appear explicitly in (3.8).
Using eq. (2.36), |t 4 | can be written in terms of ρ and the base coordinates, Let ϕ ∈ [0, 2π] be the phase of t 4 , so that t 4 becomes a function of t 2 , t 3 , ρ, ϕ, Moreover we set, The term |ε| 2 := ε ⊗ε appears naturally in (3.5) through the contribution, The last term on the right-hand side above contributes to the Kähler form, while the rest contributes to the metric. Setting ψ := ϕ/n and A := η, we recover the terms appearing in (3.8), provided we set n = 3. Moreover the relative coefficient between the dρ 2 and the (dψ + A) 2 term is fixed in the expression of |ε| 2 , and this determines the change of variables ρ →ρ(ρ) by comparing with (3.8). However, performing this change of variables in (3.5) does not directly bring us to the metric of (3.8): there remain two coefficients that still need to be adjusted. This can be achieved by introducing the two warp factors of eq. (3.14) as we now show. Let us go back to the expression of the metric in terms of Dz i . In local coordinates we have, -25 -

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It follows that the term 3 i=1 Dz i ⊗ Dz i gives the hermitian metric of CP 2 plus a |ε| 2 term, whereas Dz 4 , Dz 5 only contribute to |ε| 2 . Let us define, We can then adjust F ,G, and ρ so that, These equations can easily be decoupled by first solving for ρ, then for F and finally for G.

D General SCTV base
In the following we give the details of the derivation of eq. (4.8). The first step is writingω in terms of Dz. However this exercice is rather involved, since the Dz are not independent and because of the ambiguity in the decomposition of wedge products. Our starting point is eq. (2.31),Ω In this expression, we notice that the expansion of the contraction with the horizontal vectors amounts to choosing a set S of s + 1 integers between 1 and k + 2, corresponding to the indices of the contracted coordinates. We compute, cf. (2.33), where Q S is the determinant of the submatrix of Q A I whose columns are indexed by S. Notice that if S contains duplicates, or if it does not select independent columns, the determinant vanishes. Thus the sum selects only the sets S for which the matrix Q B A is invertible. The sign (−1) S is the signature of the permutation required to put the s + 1 indices of S in the first position, namely: We would now like to decomposeΩ with respect to the bundle structure. We therefore distinguish four cases: In the first case we get Q S = 0, since rank q a i = d < d + 1. In cases 2 and 3 we can easily see that Q S = qŜ := det(q b a ) a∈Ŝ , while (−1) S = (−1)Ŝ(−1) d for case 2, and (−1) S = (−1)Ŝ(−1) d+1 for case 3. We can now write, with Σ 4 to be determined. In case 4 we get, In the sum, if i ∈Š, the determinant cancels out, leaving only a sum over Š , so that, We are now ready include this sum in the one over theŜ, which appears in cases 2 and 3: we just need to make the change of variableŜ =Š ∪ {β}. However dz β appears in the product, thus we need to shift it to the last position. At the same time we need to move it to its right place inside det(q a , q β ) so as to maintain the increasing order ofŜ. The number of shifts needed to do so is the number of shifts required to bring β from its place to the end in Š plus the number of shifts to bring it from the end to its place inŠ; since Š ∪Š = [|1, k|], this is exactly the number of shifts required to bring β from its place to the end in [|1, k|], i.e. k − β. The last sign we need to compute is,

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To get a more symmetrical expression we can simply complete the sum dz b /z b , since the missing terms can be trivially added thanks to the wedge product. The final expression is thus, The dz were ultimately replaced by Dz becauseΩ is vertical. Now recall that the expression (A.3) decomposes Dz i into base and fiber parts. Since the metric decomposes correctly into (A.6), theDz i are orthogonal to K. Besides, the fiber part can be shown to cancel out in the first factor, so that the first parenthesis is orthogonal to K. Thus we can take the second factor to be proportional to K, and the proportionality factor can be found by computing, h k+2 i .

D.1 Torsion classes
For a generic SCTV base all torsion classes are nonvanishing. We will not write them down explicitly in this case, as they are rather cumbersome and not particularly illuminating. The computation boils down to determining the exterior differentials ofω and K. In the following we give some details of the calculation.

JHEP09(2017)133 F Twistor spaces
There is an alternative description of the total space of the CP 1 fibration over CP 2 in terms of twistor spaces. More generally, for the purposes of the present section we may replace the CP 2 base by any four-dimensional Kähler space B 4 . Consider B 4 equipped with its canonical complex structureÎ and a hermitian metric g. Let us introduce a complex zweibein z 1 , z 2 , so thatÎz k = i z k , for k = 1, 2. These forms are of course only locally defined, since B 4 is not parallelizable in general. We can thus express the metric and the local SU(2) structure on B 4 in terms of the complex zweibein, At any one point x ∈ B 4 ,ĵ x ,ω x form an SU(2) structure on the tangent space T x B 4 . The latter is equipped with a complex structure and a scalar product given byÎ x and g x respectively. Moreover the relation,Î k m = g knĵ mn , allows us to identify the complex structure with a real selfdual form. The latter are parameterized as follows, see appendix B,

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so that If k = if k . In terms of the new zweibein the local SU(2) structure and the metric read, which is precisely of the form of (3.20). Let us also note that the choice of zweibein compatible with I is only determined up to a phase. The latter leaves j and the metric invariant but acts nontrivially on ω, thus changing the SU(2) structure.
We have seen that I x (θ, ψ) defines an almost complex structure on the base. Together with the natural complex structure of the sphere (thought of as a CP 1 ) we can construct an almost complex structure on the total space, so that f 1 , f 2 and K = dθ +i sin θ(dψ +A) are eigenforms of I ± with eigenvalue ±i. We can thus take (f 1 , f 2 , K) as the vielbein on Tw(B 4 ). More generally we could modify (f 1 , f 2 , K) by introducing "warp factors" as in (5.3) below.

G Torsion classes for Kälher base
As mentioned in section 5 we may relax the condition on the base of S 2 (B 4 ), so that B 4 is a generic four-dimensional Kähler manifold. The torsion classes can also be straightforwardly calculated in this case. Note however that this is only a local calculation: without additional constraints, we do not expect there to exist a global extension to a complete space. Let us postulate a globally-defined SU(3) structure as in (5.3) on a CP 1 bundle with metric, g 6 = |h| 2 g 4 + KK * ; K = f dθ + ig(dψ + A) , (G.1) where f, g, h are a priori complex functions; θ and ψ parameterize the S 2 fiber; the oneform A satisfies (B.1), (B.4) for (J, Ω) → (ĵ,ω). We will impose further restrictions on f, g, h; these functions must be regular and non-vanishing, except for g which must vanish at θ = 0 and θ = π. The most general situation we will consider here is that df , dg, dh live on the space spanned by K, K * (this restricts the dependance on the coordinates). Explicitly we expand, and similarly for g, h. It is also possible restrict the dependance on θ alone.