Calabi-Yau Manifolds and SU(3) Structure

We show that non-trivial SU(3) structures can be constructed on large classes of Calabi-Yau three-folds. Specifically, we focus on Calabi-Yau three-folds constructed as complete intersections in products of projective spaces, although we expect similar methods to apply to other constructions and also to Calabi-Yau four-folds. Among the wide range of possible SU(3) structures we find Strominger-Hull systems, suitable for heterotic or type II string compactifications, on all complete intersection Calabi-Yau manifolds. These SU(3) structures of Strominger-Hull type have a non-vanishing and non-closed three-form flux which needs to be supported by source terms in the associated Bianchi identity. We discuss the possibility of finding such source terms and present first steps towards their explicit construction. Provided suitable sources exist, our methods lead to Calabi-Yau compactifications of string theory with a non Ricci-flat, physical metric which can be written down explicitly and in analytic form.


Introduction
Calabi-Yau (CY) manifolds together with their Ricci-flat metrics lead to a large class of string solutions, which has been the main starting point for string compactifications and string model building to date. The construction of CY manifolds is relatively straightforward: apart from a complex and Kähler structure, only the vanishing of the first Chern class of the tangent bundle is required, a condition which is easily checked. Correspondingly, large classes of CY manifolds have been constructed, notably the traditional set of complete intersection CY three-folds in products of projective spaces (CICYs) [1][2][3] and CY hypersurfaces in toric four-folds [4]. The existence of Ricci-flat metrics on CY manifolds is guaranteed by Yau's theorem [5], although computing this metric is difficult and currently only possible with numerical methods [6][7][8][9]. Fortunately, some of the physics of CY compactifications, in particular the spectrum of massless particles and holomorphic quantities in the low-energy theory, can be extracted with methods of algebraic geometry without any recourse to the metric. However, other crucial pieces of physics, such as the physical Yukawa couplings, do depend on the metric and are quite difficult to compute.
Another problematic aspect of CY compactifications is moduli stabilisation. Successful stabilisation of moduli seems to require flux which leads into the realm of manifolds with SU (3) structure. An SU (3) structure on a six-dimensional manifold X can be defined by a pair (J, Ω) of a two-form J and a three-form Ω, subject to certain constraints. As we will review, such structures are classified by five torsion classes W 1 , . . . , W 5 . Manifolds with SU (3) structure carry a globally defined spinor and, therefore, may preserve some supersymmetry in the context of string compactifications. Whether a given manifold with SU (3) structure does indeed provide a solution to string theory depends on the pattern of torsion classes which have to match constraints arising from the chosen flux (as well as the dilaton solution and the geometry of the un-compactified space). The case of CY manifolds with a Ricci-flat metric corresponds to the special case of SU (3) structure where all torsion classes vanish, W 1 = · · · = W 5 = 0. A closely related case is that of a CY manifold with a metric obtained by conformal re-scaling of the Ricci-flat metric, which leads to an SU (3) structure characterised by W 1 = W 2 = W 3 = 0 and 3W 4 = 2W 5 . In the following, we will refer to all other cases as non-trivial SU (3) structures.
Although manifolds with non-trivial SU (3) structures form a very large class, 1 a limited set of examples which fit into string theory have been constructed. The explicitly known geometries, suitable for heterotic string compactifications, consist of either homogeneous examples [11][12][13][14] or torus fibrations over certain four-dimensional base spaces [15][16][17]. An interesting solution-generating method is also provided in Ref. [18]. For type II compactifications, example geometries have been constructed on twistor spaces [19][20][21], on more generic cosets [22], solvmanifolds [23] and on toric varieties [24][25][26]. While these geometries have led to interesting examples of string vacua, their construction is somewhat tedious. The problem is that there is no analogue of Yau's theorem for the types of non-trivial SU (3) structures required by string theory. This means that SU (3) structures have to be constructed explicitly, for example by constructing the forms (J, Ω). Only after this, often laborious, task is completed is it possibly to decide whether the structure is compatible with string theory. Frequently, the answer turns out to be "no" and one has to start over in this "unguided" search for suitable SU (3) structures. On the upside, once an SU (3) structure relevant to string theory has been found explicitly, the associated metric can be computed from the forms (J, Ω) and is therefore readily available for computing quantities in the associated effective theory, such as physical Yukawa couplings.
The above discussion suggests that string compactification involves a choice between two options, both of which have considerable downsides. Calabi-Yau compactifications have the benefit of relying on large classes of available spaces whose algebraic properties are well explored. However, the differential geometry of CY manifolds, which is ultimately required in a string theory context, is hugely inaccessible due to the difficulties in computing the Ricci-flat metric. On the other hand, compactifications on manifolds with a non-trivial SU (3) structure, offer some hope of a more accessible differential geometry. However, very few such spaces relevant to string theory are known. 2 In this paper, we explore whether the advantages of CY compactifications and those of compactifications on manifolds with SU (3) structure can be combined in a new approach. Specifically, we would like to ask and answer the following two questions: • Can CY manifolds carry non-trivial SU (3) structures with explicit metrics?
• If so, are these non-trivial, explicit SU (3) structures on CY manifolds relevant for string compactification?
We will answer the first of these questions with a resounding "yes" and the second one with a somewhat tentative "yes". The plan of the paper is as follows. In the next section, we start with a brief review of SU (3) structure, in order to prepare the ground and set the notation. Section 3 studies the possibility of non-trivial SU (3) structures on the quintic, as a warm-up exercise. In Section 4, these results will be generalised to all CICY manifolds. Further explicit examples are presented in Section 5 and in Section 6 we discuss the requirement of satisfying the Bianchi identity for the flux. We conclude in Section 7. Some technical results relevant to the discussion in the main text are collected in Appendix A.

SU (3) structure
In this section, we review a number of well-known facts about SU (3) structures (see, for example, Refs. [33][34][35] for more detailed accounts), in order to set the scene and fix our notation. We begin with the mathematical background and then move on to some aspects of SU (3) structures relevant in the context of string theory.

Definition and properties of SU (3) structure
An SU (3) structure of a six-dimensional manifold X is defined as a sub-bundle of the frame bundle which has structure group SU (3). This means there are local frames A , where A = 1, . . . , 6, of the (co-)tangent bundle which patch together with SU (3) transition functions. More explicitly, if we introduce a frame e a , where a = 1, 2, 3, together with the complex conjugatesē a of the (complexified) co-tangent bundle by then the presence of an SU (3) structure corresponds to the frame transformations e a → U a b e b with U ∈ SU (3). It is then immediately clear that the two forms  In terms of the real six-bein A the above forms can also be written as where 12 is a short-hand notation for 1 ∧ 2 , and similar for the other expressions of this type. Alternatively, an SU (3) structure on the six-dimensional manifold X can be specified by a pair (J, Ω) of a globally-defined real two-form J (which has to be positive everywhere on X) and a globallydefined three-form Ω on X satisfying Note that the (local) expressions (2.2) for J and Ω do indeed satisfy those equations. Conversely, forms (J, Ω) satisfying (2.5) can always be written locally as in (2.2). The torsion classes can be read off from the exterior derivatives of J and Ω, which can be cast into the form where we decompose W 1 = W + 1 + iW − 1 and W 2 = W + 2 + iW − 2 . The torsion classes W 1 , W 4 and W 5 can be computed from dJ and dΩ via where the contraction of forms is normalised such that, for example, 12 1234 = 34 . The remaining torsion classes are constrained by Altogether, this implies the SU (3) representations for the torsion classes as given in Table 1. For illustration and later reference, Table 2 lists a number of mathematical properties of six-dimensional manifolds and the associated vanishing pattern of the torsion classes.

SU (3) structure in string theory
In string theory, one is interested in 10-dimensional spacetimesM of the formM = X × M , where X is a six-dimensional compact manifold and M is the four-dimensional non-compact spacetime. The metric onM has product or warped product structure, with the metric on X induced from an SU (3) structure on X. This SU (3) structure also implies the existence of a globally defined spinor, as is required to preserve a minimal amount of supersymmetry, a feature usually desirable in string theory.

Property
Vanishing torsion class Complex Table 2: Mathematical properties and associated pattern of torsion classes, taken from Ref. [36]. Superscripts ± indicate, respectively, the real or imaginary part of the torsion class.

4D geometry String vacuum
Non-vanishing torsion SU (3) type The required type of SU (3) structure depends on the choice of flux, the dilaton profile, and the noncompact space M and its metric. The latter is frequently chosen to be Minkowski space, AdS or dS with the maximally symmetric metric. To give a flavour of the relevant types of SU (3) structures, we list various supersymmetric string compactifications in Table 3, together with the required pattern of torsion classes on X. The interested reader may find more detailed accounts on these SU (3) structure string vacua, and their various generalisations, in [35,[37][38][39][40][41][42][43][44][45][46][47] and the reviews [36,48,49]. Of particular importance for the rest of the paper will be the Strominger-Hull system [35,37,38] which is characterised by a pattern of torsion classes given by (2.9) From Table 2, this means the manifold X has a complex structure but is, in general, not Kähler (they are, however, conformally balanced). The Strominger-Hull system can provide string solutions both in the case of the heterotic and the type II string. In either case, the dilaton φ is specified by with the string coupling g S = e φ . For the heterotic case, the NS three-form field strength H can be expressed in terms of the torsion classes as where the superscripts denote the component of the torsion class with the indicated number of holomorphic and anti-holomorphic indices. Heterotic compactifications also involve a vector bundle V → X with connection A and associated field strength F . In order for the gauge bundle to preserve supersymmetry, the connection has to satisfy the conditions (2.12) In addition, the field strength associated with this gauge connection is related to the curvature R − of the Hull connection ∇ − on X by the Bianchi identity Here, R − is the curvature of the connection ∇ − , whose Christoffel symbols are given by Γ −m np = Γ m np − 1 2 H m np (where Γ m np are the symbols of the Levi-Civita metric associated to the SU (3) structure), and the last approximation retains only leading order terms in the α expansion.
It should be noted that the α corrections to the Bianchi identity (2.13) are required for solutions with non-vanishing H flux on compact manifolds [50]. Furthermore, while the constraints (2.9)-(2.13) can be shown to be necessary for a heterotic N = 1 Minkowski solution, they do not directly imply the equations of motion. As was shown in Ref. [39] (see also [18,51]) one needs to require in addition that ∇ − is an SU (3) instanton, (2.14) To see that this condition is satisfied, we first note that the vanishing of the gravitino variation δψ m = ∇ + m η implies that ∇ + is an instanton. Second, it is straightforward to show that where the right hand side is O(α ) by the Bianchi identity (2.13). Hence, ∇ − is also an SU (3) instanton, but only up to first order α corrections. However, these corrections appear at the same order in α as other terms that have already been neglected in the equations of motion, and should therefore be discarded as well. We refer the reader to Appendix A of [51] for a recent thorough discussion of this issue. For type II string theory, H can play the role of the NS three-form (as in the heterotic case), or it can be interpreted as the RR three-form. In the former case, the relations (2.10) and (2.11) remain valid, while the Bianchi identity for H reads dH = NS 5-brane sources .
(2. 16) If H is identified with the RR three-form in type IIB string theory, the relation between the torsion classes and the flux is modified to For much of the following discussion, we will be focusing on the pattern of torsion classes (2.9) for the Strominger-Hull system and only return to the task of satisfying the Bianchi identity in Section 6.

A warm-up example: the quintic
In this section, we discuss possible non-trivial SU (3) structures on the quintic CY, defined as the anti-canonical hypersurface in the ambient space A = P 4 . This is a warm-up example for the next section, where this discussion will be generalised to all CICY manifolds. We begin with some general background and notation for the projective space P 4 .

Basics
On P 4 we introduce homogeneous coordinates x A , where A = 0, . . . , 4, and we define the standard patches U A = {[x 0 : · · · : x 4 ] | x A = 0}. We will frequently be working on the patch U 0 where we denote the affine coordinates by z a = x a /x 0 , where a = 1, . . . , 4. Two useful quantities, which will appear throughout, are where the first is the homogeneous version and the second its affine counterpart. In terms of these quantities, the Fubini-Study Kähler form J can be written as The normalisation is chosen such that P 4 J 4 = 1. A quintic X ⊂ P 4 is defined as the zero locus of a polynomial P = P (x) which is homogeneous of degree five in the coordinates x A . The affine version of this polynomial is denoted by p = p(z) and it is related to its homogeneous counterpart via The above Kähler form J can be restricted to the quintic, which leads to a Kähler form on X. In practice, this restriction can be carried out by solving the defining equation p = 0 for, say, z 4 in terms of the remaining three coordinates z α , α = 1, 2, 3, and by replacing the differential dz 4 with Besides J 0 , another standard differential form on X is the (3, 0)-form Ω 0 which, on the patch U 0 , can be explicitly written as [52,53] We can ask if the above forms (J 0 , Ω 0 ) already define an SU (3) structure on X. Given the index structure of both forms we clearly have so that the second condition (2.5) for an SU (3) structure is satisfied. In order to check the first condition, we carry out an explicit calculation, using Eqs. (3.2), (3.4) and (3.6), which leads to The function F on X reads in homogeneous and affine form respectively, where ∇P denotes the gradient of P in terms of homogeneous coordinates. Since this function is non-trivial, the first conditions (2.5) is not satisfied and the pair (J 0 , Ω 0 ) does not define an SU (3) structure. We note from Eq. (3.9) that F is well-defined on P 4 since it is homogeneous of degree zero in x A andx A and it is non-singular since σ does not vanish on P 4 . Moreover, the quintic X (defined by P = 0) is smooth precisely if ∇P = 0 everywhere on X, so F is a strictly positive function.

SU (3) structures on the quintic
As we have seen, the above forms (J 0 , Ω 0 ) do not define an SU (3) structure due to the appearance of the non-trivial function F in Eq. (3.8). However, this problem can be easily fixed by a conformal re-scaling. Indeed, by virtue of Eqs. (3.7) and (3.8), the re-scaled forms satisfy the relations (2.5) and, hence, define an SU (3) structure for any real number k. Note, in particular, that F is strictly positive for a smooth quintic and, hence, J is a positive form, as required.
What are the torsion classes associated to this SU (3) structure? Using dJ 0 = dΩ 0 = 0, the exterior derivatives are easily computed as A comparison with the general equations (2.6) for these derivates shows that the torsion classes are given by Since W 1 and W 2 vanish, we know from Table 3 that we have an associated complex structure on X.
If we further specialise to k = 1, we find which defines a Strominger-Hull system with W 3 = 0, as can be seen by comparing the expression with Eq. (2.9). The dilaton is fixed by dφ = d(ln F) , (3.14) so that the string coupling g S = const × F can be kept perturbative everywhere on X, for a suitable choice of the integration constant. If H is interpreted as an NS flux, it is explicitly given by In conclusion, starting from the Fubini-Study metric on P 4 and the standard (3, 0) form on the quintic, we can construct a family of SU (3) structures, parametrised by a real number k, on every smooth quintic. For a special choice, k = 1, this SU (3) structure is of the Strominger-Hull form with W 3 = 0. The dilaton varies non-trivially, but can be kept in the perturbative range and we have non-zero NS flux.
We note that these SU (3) structures do not corresponds to the conformally Ricci-flat case and are, hence, non-trivial in the sense defined earlier. Indeed, the last row in Table 2 shows that a conformally Ricci-flat SU (3) structure is characterised by 3W 4 − 2W 5 = 0, while, from Eq. (3.12), our SU (3) structures satisfy Our next step will be to generalise this discussion to all CICY manifolds.

CICYs and SU (3) structure
We will now discuss complete intersection CY manifolds (CICYs) in the ambient space A = P n 1 × · · · × P nm . We first review some standard results and notation and then construct SU (3)-structures on these manifolds, generalising the approach we have taken for the quintic.

Basics
As mentioned above, the ambient space is given by a product A = P n 1 × · · · × P nm of m projective factors with dimensions n i , where i = 1, . . . , m, and total dimension d = m i=1 n i . Homogeneous coordinates for each projective factor are denoted by x i = (x iA ), where A = 0, 1 . . . , n i . Their affine counterparts in the patch x i0 = 0 are called z i = (z ia ) with z ia = x ia /x i0 and a = 1, . . . , n i . We will frequently work in the patch U 0 of A where all x i0 = 0, using the coordinates (z 1 , . . . , z m ). In analogy with the quintic case, we define the quantities which can be used to write down the Fubini-Study Kähler forms J i for each projective factor. In affine coordinates z i they are explicitly given by The CICY three-fold X is defined as the common zero locus of K polynomials P u = P u (x 1 ,. . ., x m ), u = 1, . . . , K. They are homogeneous with multi-degree q u = (q i u ), where q i u is the degree of homogeneity of P u in the coordinates x i of the i th projective factor. The polynomials are related to their affine counterparts p u = p u (z 1 , . . . , z m ) on the patch U 0 by The information about the multi-degrees of the defining polynomials, together with the dimensions of the projective ambient space factors, is often summarised by the configuration matrix where the two non-trivial Hodge numbers h 1,1 and h 2,1 are attached as superscripts and the Euler number η = 2(h 1,1 − h 2,1 ) as a subscript. The Calabi-Yau condition, c 1 (T X) = 0, simply translates into the conditions K u=1 q i u = n i + 1, for i = 1, . . . , m, on the degrees. Using this notation, the quintic in P 4 discussed in the previous section is described by the configuration [P 4 | 5] 1, 101 −200 . There is an infinite number of CICY configuration matrices of the above type but it turns out that different configurations can correspond to the same topological class of Calabi-Yau manifolds. Taking this identification into account, the number of topological types of CICY three-folds becomes finite and the classification of Ref. [1,2] leads to 7890 topological types. 3 This list provides one representative configuration matrix for each topological type and we will use some examples from this list in the next section. For now the discussion will be carried out in terms of a general configuration matrix and, hence, applies to all CICY manifolds and all configuration matrices.
There are a number of obvious differential forms which can be defined on X. First of all we can restrict the Fubini-Study Kähler forms (4.2) to obtain the Kähler forms for i = 1, . . . , m on X. In particular, note that these forms are closed, that is There are configurations for which the forms J i provide a basis for the second cohomology of X and these are sometimes referred to as favourable configurations. In fact, out of the 7890 configurations provided in the standard list of Ref. [1,2], some 60% turn out to be favourable in this sense. Furthermore, it has recently been shown [54] that almost all of the other entries in the list have equivalent, favourable configurations. In other words, for almost all of the 7890 different topological types can a configuration matrix be found such that the forms (4.5) span the entire second cohomology of X. This means that the subsequent construction, which will be based on the forms J i , can be thought of as exhausting the entire available space of Kähler classes. We recall that the triple intersection numbers λ ijk of X can be expressed in terms of the forms J i by As on every Calabi-Yau manifold, we have of course also the holomorphic (3, 0) form Ω 0 . For CICYs this form can be explicitly constructed [1,3] by first defining the ambient space (3, 0) formΩ viâ and then restricting this form to X, that is We note that Ω 0 has the properties the latter as a trivial consequence of the index structure. The forms J i ∧ J j ∧ J k as well as Ω 0 ∧Ω 0 are top forms on X and must, therefore, be related by certain functions Λ ijk on X such that We note that integrating this equation over X and using Eq. (4.7) leads to an alternative expression for the intersection numbers, Eq. (4.11) is key in our construction of SU (3) structures on CICY manifolds and generalises the relation (3.8) for the quintic. The computation of the single function Λ 111 = F in Eq. (3.9) for the quintic can be generalised, and a general result for Λ ijk for an arbitrary configuration matrix can be found. Since the details are somewhat involved and the general formula turns out to be complicated this calculation has been relegated to Appendix A. One general rule, which is easily stated, is that Λ ijk = 0 whenever the corresponding triple intersection number λ ijk vanishes. For co-dimension one configuration matrices, that is, for K = 1 and a single defining polynomial P , the expression for Λ ijk is relatively manageable. In this case, we have for all i, j, k with λ ijk = 0 that where c ijk are combinatorial constants and ∇ i P is the gradient of P with respect to the homogeneous coordinates x iA of the i th projective factor. The combinatorial factors c ijk can be explicitly computed, for example by using Eq. (4.12) or from the general expression in Appendix A. We find that all c ijk ≥ 0, with equality only if λ ijk = 0. We note that the RHS of Eq. (4.13) is homogeneous of degree zero in all coordinates x i and, hence, the Λ ijk are indeed well-defined on A and on X. Further, it follows from Eq. (4.13) that the Λ ijk are non-singular (since all σ i are non-zero on P n i ) and Λ ijk ≥ 0 everywhere on X. These properties of Λ ijk in the co-dimension one case are indeed general and also hold for higher co-dimension, as can be seen from the results in Appendix A.

SU (3) structures on CICY manifolds
We will now construct SU (3) structures on arbitrary CICY manifolds by specifying a pair (J, Ω) of a two-and three-form, starting with the Ansatz where the (1, 1)-forms J i and the (3, 0)-form Ω 0 have been defined in Eqs. (4.5) and (4.9), respectively. Further, the a i are real, smooth functions on X which are constrained to be strictly positive (so that J is a positive form) but are otherwise arbitrary. The function A on X is real or complex, smooth and should be everywhere non-vanishing. By virtue of the second relation (4.10) we have J ∧ Ω = 0, so that the second requirement for an SU (3) structure in Eq. (2.5) is satisfied independently of the choice of functions a i and A. Using Eq. (4.11), we find the first condition (2.5) for an SU (3) structure is satisfied iff (4.15) Using the explicit expressions for the structure functions Λ ijk , it can be shown that where g αβ = −2iJ αβ is the metric associated to J and det(B), defined in (A.3), is the generalization of the factor |p ,4 | 2 that appears in the denominator of Ω in (3.6).
In conclusion, this means that the forms (J, Ω) in Eq. (4.14) define an SU (3) structure on X iff the functions a i and A satisfy the constraint (4.15). Note that this leaves considerable freedom in the construction. Basically, we can start by choosing any set of real, smooth and strictly positive functions a i , and then use Eq. (4.15) to define A. Since all Λ ijk ≥ 0 and all a i ≥ 0, the RHS of Eq. (4.15) is positive, so A defined in this way can be taken to be real and positive as well. For the quintic, where m = 1, this leads to a form J given by the conformal re-scaling of a Kähler form.
In general, for m > 1, this is no longer necessarily the case, since the functions a i can be chosen independently.
Since dJ i = dΩ = 0, it is easy to compute the exterior derivatives of J and Ω and we find Comparison with the general expressions (2.6) for dJ and dΩ then leads to the torsion classes Since W 1 and W 2 vanish, we know from Table 2 that there is always a complex structure. The other generic feature of this class of SU (3) structures is that W 5 is always an exact one-form. Further details depend on the choice of functions a i . This leaves considerably scope for constructing SU (3) structures based on the Ansatz (4.14), which we only begin to explore in the present paper.
As an example consider the expressions which are generalisations of σ i = σ 1,i . Since these quantities are nowhere vanishing on A and homogeneous of bi-degree (s, s) in (x i ,x i ) they are well-suited to construct smooth, strictly positive functions a i . For example, we can set where a is a smooth, strictly positive function on X and the t i > 0 are real constants. In this case, the forms (J, Ω) can be written as Note that the above J 0 is a Kähler form and the constants t i can thus be interpreted as Kähler parameters, while J is obtained from J 0 by a conformal re-scaling with a. Inserting this into Eq. (4.15), we find the forms (J, Ω) in Eq. (4.22) define an SU (3) structure iff We will refer to this sub-class as "universal" SU (3) structures. The structure functions Λ ijk enter the construction of these universal SU (3) structures only through the function F, defined above, which can be explicitly computed using the expression (4.13) for Λ ijk in the case of co-dimension one configurations, or the expression (A.8) in the general case. Specialising Eq. (4.16) to the universal case we have in particular that where g 0,αβ = −2iJ 0,αβ is the metric associated to the Kähler form J 0 and B is defined in (A.2). This result shows that F is always a strictly positive function (provided all the Kähler parameters t i > 0). In fact, F generalises the function of the same name we have defined for the quintic, see Eq. (3.9). For universal SU (3) structures, we have the exterior derivatives 25) and comparison with Eq. (2.6) shows that the corresponding torsion classes are given by Of course, W 1 and W 2 are still zero, so that we have a complex structure and W 5 remains exact. In addition to the properties of the generic case (4.18), universal SU (3) structures also have a vanishing W 3 torsion class. Also, the class W 4 is exact and related to W 5 by The function a is still at our disposal. If it is chosen such that |d ln a| |d ln F| everywhere on X, we have 3W 4 2W 5 from Eq. (4.27). Table 2 shows that this corresponds to an (approximate) conformally Ricci-flat situation, where the conformal factor a dominates over the effect of non Ricciflatness of the underlying Fubini-Study metric, which causes the appearance of the d ln F term in Eq. (4.27). In contrast to the exact conformal Calabi-Yau structure normally used in the construction of N = 1 type IIB vacua [40], the present approximate structure comes equipped with an explicit, albeit approximate, metric.
Another interesting and obvious choice for a is for any real number k. This leads to a pattern of torsion classes with W 4 and W 5 proportional to one another. For this choice, the limit in which the SU (3) structure becomes approximately conformally Ricci-flat, that is, 3W 4 2W 5 , can be made more explicit and it corresponds to k → ∞.
If we set k = 1 so that a = F , then the torsion classes specialise further to This means that W 5 = 2W 4 and, hence, we have a Strominger-Hull system with W 3 = 0 and a dilaton φ specified by This shows that, for a suitable choice of integration constant, the string coupling can be kept perturbative. The torsion classes in Eq. (4.31) represent the direct generalisation of the quintic results to all CICY manifolds. In particular, we find that a Strominger-Hull system can be realised on all CICY manifolds. Let us finally remark that the SU (3) structure constructed here does not relate in any obvious way to the unique integrable SU (3) structure that exist on a CY manifold. Naturally, the (3,0)forms of these two SU (3) structure are necessarily proportional. However, we cannot determine how the Hermitian form J or the Kähler forms J i relates to the Kähler form of the integrable SU (3) structure, since the latter is not known explicitly. As mentioned in the Introduction, this ignorance of the integrable Kähler form is one of our motivations to construct explicit SU (3) structures on CY manifolds.

Further examples
While the discussion in the previous section was general and applies to all CICY manifolds it is still useful to work out a few cases other than the quintic more explicitly. We begin with the two CICY manifolds which are arguably the simplest after the quintic, the bi-cubic hypersurface in P 2 × P 2 and the tetra-quadric hypersurface in P 1 × P 1 × P 1 × P 1 . Finally, we analyse a more complicated example of a co-dimension two CICY.

The bi-cubic
The bi-cubic hypersurface in the ambient space A = P 2 × P 2 (number 7884 in the list of Ref. [1]) is characterised by the configuration matrix where we have also listed the homogeneous coordinates and the affine coordinates on the patch U 0 defined by x 0 = 0, y 0 = 0. The bi-cubic is defined as the zero locus of a bi-cubic polynomial P = P (x, y), which is related to its affine counterpart p = p(z 1 , . . . , z 4 ) on U 0 by the two Fubini-Study Kähler forms on U 0 can be written as To restrict these forms to X we can, for example, solve the equation p = 0 for z 4 = f (z), where z = (z α ) = (z 1 , z 2 , z 3 ) and use In this way, we can obtain explicit equations for the restricted Kähler forms J 1 = J 1 | X and J 2 = J 2 | X as well as for the holomorphic (3, 0) form The only non-vanishing triple intersection numbers of the bi-cubic are λ 112 = λ 122 = 3 (along with the ones obtained by index permutation) and, hence, we have only two non-vanishing structure functions Λ 112 and Λ 122 . They can be computed from the above expressions for J i and Ω 0 . A somewhat tedious calculation shows that where ∇ 1 P and ∇ 2 P are the gradients of P with respect to the x and y coordinates, respectively. Note that the above structure functions have the properties mentioned in the general discussion. They are homogeneous of degree zero in both the x and y coordinates and their complex conjugates and are, hence, well-defined functions on the ambient space A and on the CY X. Furthermore, they are non-singular (as σ 1 and σ 2 do not vanish) and they are clearly positive everywhere. With these ingredients, the construction of SU (3) structures on the bi-cubic proceeds following the general logic explained in the previous section. A pair (J, Ω) given by the Ansatz provides an SU (3) structure iff the constraint is satisfied. Inserting the bi-cubic structure functions (5.7), this constraint becomes explicitly Any choice of two smooth and strictly positive functions a 1 and a 2 on the bi-cubic now leads to an SU (3) structure. We simply insert these two functions into the RHS of Eq. (5.10), which is always strictly positive since ∇ 1 P and ∇ 2 P cannot vanish simultaneously for a smooth bi-cubic. Then, demanding that A > 0 everywhere fixes A and, hence, an SU (3) structure whose torsion classes are of the form (4.18) and can be explicitly computed from these equations. In this way, choices for a i such as the ones proposed in Eqs. (4.19) and (4.20), give rise to a large class of SU (3) structures. For the universal case a i = a t i with |A| 2 = a 3 F, the function F reads explicitly The bi-cubic Strominger-Hull system is then characterised by the general equations (4.30)-(4.32) with the above functions F inserted.

The tetra-quadric
The tetra-quadric hypersurface in the ambient space A = P 1 × P 1 × P 1 × P 1 (number 7862 in the list of Ref. [1]) is described by the configuration −128 where the homogeneous coordinates for each P 1 factor and their affine counterparts on the patch U 0 = {x i,0 = 0} are listed on the right. The tetra-quadric is defined as the zero locus of a polynomial P = P (x 1 , . . . , x 4 ) of multi-degree (2, 2, 2, 2), related to its affine counterpart p = p(z 1 , . . . , z 4 ) by As usual we define which leads to the Fubini-Study Kähler forms We can restrict these forms to the tetra-quadric, J i = J i | X , by solving, for example, for z 4 = f (z), where z = (z α ) = (z 1 , z 2 , z 3 ) and use On the patch U 0 , this leads to while the holomorphic (3, 0) form on X can be written as The only non-vanishing triple intersection numbers of the tetra-quadric are λ 123 = λ 124 = λ 134 = λ 234 = 2 and, by inserting the above expressions for J i and Ω 0 into Eq. (4.11), we find for the corresponding structure functions 19) and permutations thereof. All other Λ ijk vanish. We note that, in line with our general statements, all Λ ijk are well-defined, smooth and positive. The Ansatz then leads to an SU (3) structure (J, Ω) on the tetra-quadric iff |A| 2 = 4 i,j,k=1 Λ ijk a i a j a k and, with the above structure functions (5.19), this condition turns into For the universal case a i = a t i , the function A is determined by |A| 2 = a 3 F, where F is explicitly given by For a smooth tetra-quadric not all Λ i vanish simultaneously, so F > 0 everywhere on X. The Strominger-Hull system on the tetra-quadric is described by the general equations (4.30)-(4.32) with the above function F inserted.

A co-dimension two CICY
The purpose of this example is two-fold. First, we show how to generalise the methods from the co-dimension one case to higher co-dimensions. Second, we illustrate how this leads to the general formula for the structure functions Λ ijk given in Appendix A. The example we are working with is a co-dimension two CICY (number 7888 in the list of Ref. [1]) in the projective ambient space P 1 × P 4 , with configuration matrix X ∼ P 1 0 2 P 4 4 1

2,86
−168 We denote the two defining equations by P u and their affine counterparts in the patch U 0 = {x 10 = 0, x 20 = 0} by p u , where u = 1, 2. The explicit computation proceeds similarly to the co-dimension one case. First, we use the equations p u = 0 to solve for, say, z 4 and z 5 in terms of z = (z 1 , z 2 , z 3 ) = (z α ), and replace their differentials dz 4 and dz 5 using to arrive at In affine coordinates, the two Fubini-Study Kähler forms are given by As a next step we need to compute the holomorphic three-form Ω 0 . Applying its general definition (4.8), (4.9) to the present case leads to The only non-vanishing intersection numbers of this CICY are λ 122 = 4 and λ 222 = 8 and the corresponding non-vanishing structure functions can be written in the form cp1,d p 2,ap2,b − p 1,ap1,d p 2,cp2,b − p 1,cp1,b p 2,ap2,d + p 1,ap1,b p 2 2. For Λ 122 , the sums are all symmetric and hence every term occurs twice. That means that upon factoring out the common factor of 2 from the numerator, we get a factor of 1/3 in the denominator, matching the symmetry factors c ijk of (A.6) that appear in (A.5).
3. We always have to skip three columns in the Jacobian. In Λ 122 the index structure shows that we skip the column which contains derivatives w. r. t. the P 1 coordinate z 1 , which is why the sums run from 2 to 5. Likewise, in Λ 222 we delete columns such that a P 1 coordinate and a P 4 coordinate are left, so one sum runs over the P 1 block, i.e. from 1 to 1, and the other over the P 4 block, that is, again from 2 to 5.
4. The indices on p u,s show anti-symmetry structures which arise from a determinant of the Jacobian with three columns deleted.
5. The terms z azb z czd actually sum to zero due to the symmetry/antisymmetry structure of the indices.
6. The expression has the structure one would expect from the terms (∇ i 1 p u 1 ·∇ī 1pū1 )(∇ i 2 p u 2 · ∇ī 2pū2 ) in (A.5), where we have carried out the anti-symmetrisation from the u 1 u 2 factors.
For the universal case a i = a t i , the function A is determined by |A| 2 = a 3 F, where F is obtained by inserting the structure functions (5.30) into the second Eq. (4.23). For the Strominger-Hull system on this co-dimension two CICY, the function F is used in the general equations (4.30)-(4.32).

The Bianchi identity
We have seen that large classes of non-trivial SU (3) structures can be constructed on CICY manifolds. In particular, we have demonstrated the existence of a Strominger-Hull system on every CICY manifold. These SU (3) structures are potentially relevant for string theory, both in the context of heterotic and type II compactifications. While we leave a comprehensive study of the relevant applications in string theory for future work, we present here a brief initial discussion of the Bianchi identities for the anti-symmetric tensor fields. These identities need to be satisfied in order to obtain a full string solution and this proves a somewhat difficult task. Consequently, we limit our discussion to some general remarks and then re-visit the tetra-quadric example.

Generalities
We recall that a Strominger-Hull SU (3) structure (J, Ω) on a CICY manifold X was obtained by setting where J i = J i | X are the ambient space Kähler forms restricted to X and Ω 0 is the holomorphic (3, 0) form on X. The constants t i > 0 can be interpreted as the analogue of Kähler parameters and the function F can be written as where the structure functions have been computed for all CICY manifolds in Eq. (4.13) and Appendix A. It turns out that for a smooth manifolds, F > 0 everywhere on X as must be the case in order for J to be a positive form.
Since dJ i = 0 and dΩ 0 = 0, the exterior derivatives of J and Ω are easily computed, and lead to the torsion classes 3) The dilaton φ, both in the context of type II and heterotic string theory, satisfies so that the string coupling g s = e φ = const × F can be kept perturbative, for a suitable choice of the integration constant. The final ingredient required for this to become a string solution is that the torsion (6.3) is supported by a suitably matching flux. Let us focus on the case where this flux is the NS three-form field strength H, either in the heterotic or type II context. In either case, H is given by Eq. (2.11) and, inserting J from Eq. (6.1) leads to This is the NS flux we need to add to the compactification in order to obtain a string solution. It is important to note that H is not closed, and, hence, this flux needs to be supported by a non-zero source term in the Bianchi identity. In type II string theory this requires an NS5 brane source equal to the RHS of Eq. (6.6) (or a D5 brane source in the case where H is interpreted as a RR flux). We recall that by equation (6.1), F must be a smooth function, and so the RHS of (6.6) cannot be proportional to a δ function. In other words, the source required to satisfy the equation requires some smearing. 4 We leave the subtleties of such constructions for future work, but remark that sources of this type have been discussed for vacua on solvmanifolds in Ref. [23]. Related discussions can also be found for example in Refs. [42,[55][56][57][58][59]. It would be interesting to explore if the methods used in these references can be useful to find the required source term.
In the heterotic case, the Bianchi identity reads (to leading order in the α expansion) 5 dH = α 4 (tr(F ∧ F ) − tr(R ∧ R)) , (6.7) and the task is to match this to the RHS of Eq. (6.6). Since we have specified a metric, the tr(R ∧ R) term can be computed explicitly. Hence, solving the Bianchi identities reduces to the task of finding a suitable vector bundle V → X with a connection A and associated field strength F such that 2i∂∂F ∧ J 0 = α 4 (tr(F ∧ F ) − tr(R ∧ R)) . (6.8) Note that the gauge connection also has to satisfy the supersymmetry conditions (2.12) at the same time. This appears to be a difficult task as there is no obvious guidance for the choice of a suitable bundle V and its connection. In the present paper, we will not attempt to construct such bundles and connections. Instead, we compute the term tr(R ∧ R) in order to gain a clearer picture which contribution from tr(F ∧ F ) is required. Of considerable help for this task is the observation is that tr(R ∧ R) is invariant under a conformal re-scaling of the metric [60]. This means that, instead of using the metric g associated to the SU (3) structure (6.1), we can use the Kähler metric g 0 associated to the Kähler form J 0 . Despite this simplification, it seems difficult to express tr(R ∧ R) in a manageable and suggestive form in complete generality for any CICY manifold. For this reason, we will compute tr(R ∧ R) for an example, namely the tetra-quadric CICY discussed in Section 5.2.

The tetra-quadric re-visited
Our task is to compute the tr(R ∧ R) term in the Bianchi identity for the case of the tetra-quadric with the metric associated to the Strominger-Hull SU (3) structure (6.1). As discussed, this can be done in terms of the Kähler metric g 0 , associated to the Kähler form J 0 = 4 i=1 t i J i . Summing up the explicit forms J i for the tetra-quadric in Eq. (5.17) gives where v α = p ,α /p ,4 and the Kähler potential is given by K = i 2π 4 i=1 t i ln κ i , restricted to X. The associated metric g 0,αβ = −2iJ 0,αβ and its inverse g αβ 0 can then be written as where we have introduced the short-hand notation A useful relation between those quantities is In order to compute the curvature two-form, we use the standard Kähler geometry relations and inserting the above metric and its inverse leads to the connection one-form 14) and the curvature two-form As a useful crosscheck of our calculation, it is straightforward to check that this expression leads to a Ricci-form R = 3 α=1 R α α which satisfies R = ∂∂ ln det g 0 , (6.16) as required for a Kähler metric. The desired quantity tr(R ∧ R) can then be written as It is interesting that this result can be expressed in terms of the tetra-quadric structure functions Λ i given in Eq. (5.19) and the function F in Eq. (5.22) by writing Note that when inserted into Eq. (6.17), the pre-factor v α /v β drops out so that tr(R∧R) only depends onΩ β α and the Kähler forms J α . It is encouraging that tr(R ∧ R) depends on the same quantities as dH. We hope this result will ultimately provide some guidance as to which vector bundle and connection to choose for the tetra-quadric, in order to satisfy the Bianchi identity.

Conclusion
Compactification on Calabi-Yau three-folds and on manifolds with SU (3) structure are usually seen as complementary within string theory. In this paper, we have explored the relation between these two approaches by constructing non-trivial SU (3) structures on Calabi-Yau three-folds and by analysing their possible role in string compactifications.
Focusing on the relatively easily accessible complete intersection Calabi-Yau manifolds in product of projective spaces (CICY manifolds) we have obtained a number of interesting results. We have seen that, using the Kähler forms provided by the projective ambient spaces and the holomorphic (3, 0) form available on a CY manifold as basic building blocks, we can obtain large classes of SU (3) structures on all CICY manifolds. For our construction, all these SU (3) structures have vanishing torsion classes W 1 and W 2 and, hence, have an associated complex structure. The other three torsion classes W 3 , W 4 and W 5 are generically non-zero and are given in terms of a set of smooth, strictly positive but otherwise arbitrary functions a i on the CICY manifold. In a "universal" case, where all functions a i are proportional, we obtain a Strominger-Hull system with W 3 = 0 on every CICY manifold.
Such SU (3) structures can lead to string backgrounds in both heterotic and type II string theory with a non-trivial dilaton profile and non-vanishing NS flux. We have computed the dilaton profile required for the Strominger-Hull system on CICY manifolds and find that the string coupling can be kept in the perturbative range. Further, we have determined the NS flux H supporting this solution and it turns out that it is not closed and, hence, needs to be supported by sources in the Bianchi identity. In the type II case, this requires (smeared) NS five-brane solutions and in the heterotic case a suitable vector bundle.
We have explicitly discussed a number of examples, namely the quintic in P 4 , the bi-cubic hypersurface in P 2 × P 2 , the tetra-quadric hypersurface in P 1 × P 1 × P 1 × P 1 and a co-dimension two CICY manifold. For the tetra-quadric, we have taken preliminary steps towards solving the Bianchi identity by computing the tr(R ∧ R) term.
A number of follow-up questions and further directions are suggested by the results in this paper. First and foremost, it would be desirable to clarify, in the case of the universal Strominger-Hull system, whether the Bianchi identity in the type II or heterotic case can be solved. Only if this can be accomplished have we found a proper string solution. Our construction also leads to a large class of non-universal SU (3) structures, where the functions a i are not proportional to one another, and these should be explored in more detail. It is possible that other SU (3) structures of interest to string theory can be found among those non-universal cases.
Moreover, while we have primarily manipulated the real two-form J of the SU (3) structure in this paper, there is scope to construct interesting non-trivial SU (3) structures by a less restricted Ansatz for the complex three-form Ω. Such generalisations are necessary in order to produce SU (3) structures with non-integrable almost complex structure. It would, for example, be interesting to explore if type IIA SU (3) vacua can be constructed in this way. 6 While our construction was carried out for CICY manifolds, it likely generalises to other classes of CY three-folds, notably to CY three-folds constructed as hypersurfaces in toric four-fold ambient spaces [4]. There is indeed overlap between CICY three-folds and CY hypersurfaces in toric fourfolds and some of the examples considered in this paper, specifically the quintic, the bi-cubic and the tetra-quadric appear in both lists.
Finally, our construction should readily generalise to CY four-folds, in this case leading to SU (4) structures. For example, a suitable modification of the quintic example could be applied to the sextic in P 5 or adding another P 1 factor to the tetra-quadric example would lead to SU (4) structures on the degree (2, 2, 2, 2, 2) CY hypersurface in (P 1 ) 5 . It would be interesting to study this generalisation and its possible applications to F-theory compactifications.
introduce the K × d Jacobi matrix where we define p u,s = ∂p u /∂z s . By deleting three columns r, s, t from A we obtain K × K matrices denoted by B K [r, s, t] and, for ease of notation, we also introduce associated index sets I[r, s, t] = {1, . . . , d}\{r, s, t}. With this notation, we can write Using B K [r, s, t], we can generalise the expression (3.6) for the holomorphic (3, 0)-form Ω 0 to Lastly, in order to describe the general expression, we define for each index s = (ia) the (n i + 1)dimensional gradients where p u,ia = ∂p u /∂z ia . It should be noted that the∇ ia do in fact not depend explicitly on the index a, and hence they are the same for all coordinates z ia from the same ambient space factor i, that is, It turns out that this somewhat redundant definition is helpful in order to incorporate certain combinatorial factors in the final expression for the Λ ijk . Given this notation, a straightforward but tedious computation based on Eq. where the scalar product (∇ i p u · ∇īpū) is the standard Euclidean scalar product of the two vectors. Note that, since the ∇ ia in fact only depend on i, we have only attached a label i to Λ ijk . All Λ ijk that cannot be constructed in this way (for example, since it is impossible to delete three affine coordinates a 1 , a 2 and a 3 from a P 1 or P 2 ) are zero. Furthermore, there are two symmetry factors c ijk andĉ ijk in the expression (A.5). The c ijk arise from symmetries in the indices i, j, k and are given by The factorsĉ ijk arise from an over-counting of different factors in the sums: since∇ ia p u is the same for all a, we get some terms several times. To be more precise, we generically get each factor K times. However, it sometimes happens that by leaving out the indices i, j, k in the sums, some P n i do not enter at all 7 and hence do not give rise to a symmetry factor. We find that Looking at Eq. (4.11), we note the following: Ω 0 on the right-hand side is given by Eq. (A.3) involve derivatives of the defining polynomials in the denominator. In contrast, the Kähler forms J on the left-hand side do not explicitly involve p u . A CICY given by the zero locus of a set of polynomials, p u (z) = 0, is left invariant by a scaling of p u → λ u p u . However, from Eq. (4.11) it would then seem as if the left-hand side does not scale with λ u while the right-hand side does. This is resolved by the observation that expression (A.5) scales as Λ ijk → K u=1 |λ u | 2 Λ ijk . At the same time, |det(B K [r, s, t])| 2 → K u=1 |λ u | 2 |det(B K [r, s, t])| 2 . Since |det(B K [r, s, t])| 2 appears in the denominator in Eq. (A.3), Eq. (4.11) is homogeneous of degree 0.
(A.8) where we use the notation explained in Eqs. (4.1) and (4.3). Here, like in the analogous result for co-dimension one CICYs, Eq. (4.13), ∇ i P u denotes the standard gradient of the u th polynomial with respect to the coordinates x iA of the i th projective ambient space factor. It is instructive to compare Eq. (A.8) with the co-dimension one result (4.13) in more detail. We note that the division by |∇ i P | 2 |∇ j P | 2 |∇ k P | 2 effectively leads to the omission of these terms, making it the analog of the omissions encoded by the index sets I[i, j, k]. For the co-dimension one case, the combinatorial factors c ijk in Eq. (A.8) do indeed specialise to the factors of the same name in Eq. (4.13). When comparing the affine expression for the Λ ijk in Eqn. (A.5) with the expression of the tetra-quadric (5.19), it seems as if there is an additional factor of κ i in the denominator of Eq. (A.5). However, the scalar products (∇ i p u ·∇īpū) lead to an extra κ i , such that the two expressions match exactly.
We still need to demonstrate that the affine and homogeneous formulae for Λ ijk in Eqs. (A.5) and (A.8) are indeed equivalent. To do this we note that and (A.10) The full expression (A.8) involves a product of K scalar products of the type (A.10). Due to the factors u 1 ...u K , ū 1 ...ū K and δ i j ,ī j , the structure functions Λ ijk pick up a pre-factor This can be further simplified by using the Calabi-Yau condition K u=1 q i u = n i + 1 , (A.11) such that we get (∇ i 1 P u 1 · ∇ī 1Pū1 ). . .(∇ i K P u K · ∇ī KPūK ) = |x i 1 0 | 2 . . . |x i K 0 | 2 m l=1 |x n l +1 l0 | 2 (∇ i 1 p u 1 ·∇ī 1pū1 ). . .(∇ i K p u K ·∇ī KpūK ) . (A.12) By writing the numerator |x i 1 0 | 2 . . . |x i K 0 | 2 as a product over all d = K + 3 indices (ia) and dividing by the three that are left out, we find (A.13) Finally, we can use this to rewrite the pre-factor in (A.12) as (A.14) According to Eq. (A.9), this is precisely the factor that converts the σ i appearing in the denominator of Eq. (A.8) into the κ i appearing in the denominator of Eq. (A.5).