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A Construction of Infinitely Many Solutions to the Strominger System

In this paper we construct explicit smooth solutions to the Strominger system on generalized Calabi-Gray manifolds, which are compact non-K\"ahler Calabi-Yau 3-folds with infinitely many distinct topological types and sets of Hodge numbers.


Introduction
The Strominger system [18,30] is a system of partial differential equations characterizing the compactification of heterotic superstrings with torsion. Mathematically speaking, we may think of the Strominger system as a generalization of the Ricci-flat metrics on non-Kähler Calabi-Yau 3-folds, which is simultaneously coupled with the Hermitian Yang-Mills equation on a gauge bundle.
Let X be a complex 3-fold, preferably compact, with holomorphically trivial canonical bundle. We fix a nowhere vanishing holomorphic (3,0)-form Ω on X. Let ω be a Hermitian metric on X, and denote by Ω ω the norm of Ω with respect to the metric ω. In addition, let (E, h) be a holomorphic vector bundle over X equipped with a Hermitian metric. We denote by R and F the endomorphism-valued curvature 2-forms of the holomorphic tangent bundle T 1,0 X and E respectively. Finally, let α ′ ∈ R be a constant. We can write down the Strominger system as follows [19]: In the literature, Equations (1), (2) and (3) are known as the Hermitian Yang-Mills equation, the anomaly cancellation equation and the conformally balanced equation respectively. In this paper, we will only use the Chern connection to compute the curvatures R and F and work with positive α ′ .
It is not hard to see that when ω is a Kähler metric, by Yau's theorem [34] the full Strominger system can be solved if we take ω to be Ricci-flat and simultaneously embed the spin connection into the Yang-Mills connection, meaning to set R = F . Such a solution corresponds to the torsion-free compactification of

The Generalized Calabi-Gray Construction
To obtain interesting solutions to the Strominger system, we need first to construct compact non-Kähler Calabi-Yau 3-folds with balanced metrics. A class of nice such examples is what we call the generalized Calabi-Gray construction [7], which can be phrased as follows.
Let (M, g) be a compact hyperkähler 4-manifold. It is well-known that M is either a flat 4-torus or a K3 surface with a Calabi-Yau metric. We denote by I, J and K the compatible complex structures with IJK = −id and write ω I , ω J and ω K for the corresponding Kähler forms, defined as usual by . In fact, for any real numbers α, β and γ such that we have a compatible complex structure αI + βJ + γK, whose associated Kähler form is given by αω I + βω J + γω K .
Following [17], let us first describe the twistor space Z of M . By stereographic projection, we may parameterize Moreover, we always equip CP 1 ∼ = S 2 the round (Fubini-Study) metric The twistor space Z of M is defined to be the manifold Z = CP 1 × M with the tautological almost complex structure I given by (a). I is an integrable complex structure, making (Z, I) a compact non-Kähler 3-fold whose natural product metric is balanced.
(c). Let Λ 2 Ω Z/CP 1 be the determinant line bundle of the relative cotangent bundle associated to the holomorphic fibration π. Then the line bundle Λ 2 Ω Z/CP 1 ⊗ π * O(2) on Z has a global section which defines a holomorphic symplectic form on each fiber of π.
Now let Σ be a compact Riemann surface of genus g and let ϕ : Σ → CP 1 be a nonconstant holomorphic map. We may treat ϕ either as a nonconstant meromorphic function ζ on Σ or, by identifying CP 1 with the unit sphere in R 3 , as a map ϕ = (α, β, γ) into R 3 . These two viewpoints are related by the stereographic projection formula (4). By pulling back the holomorphic fibration π : Z → CP 1 over ϕ : Σ → CP 1 , we get a holomorphic fibration p : X = ϕ * Z → Σ. As a complex manifold, X is topologically Σ × M with a twisted complex structure J 0 = j Σ ⊕ (αI x + βJ x + γK x ). We have Proposition 2.2. [6,7,8] (a). X has trivial canonical bundle if and only if where K Σ is the canonical bundle of Σ.
From now on, we shall call condition (7) plus that ϕ is not the constant map the "pullback condition". Assuming the pullback condition, it is clear that S = ϕ * O(1) is a square root of K Σ , which is known as a theta characteristic in algebraic geometry, or a spin structure according to Atiyah [1]. Furthermore, the linear system associated to the line bundle S is basepoint-free.
Conversely, if we start with a basepoint-free theta characteristic S on Σ, we may choose s 1 , s 2 ∈ H 0 (Σ, S) such that s 1 and s 2 have no common zeroes, then ζ = s 1 /s 2 is a meromorphic function on Σ. Moreover, ζ defines a holomorphic map ϕ : Σ → CP 1 such that the pullback condition holds.
Therefore in this paper, we shall call a pair (Σ, ϕ) such that the pullback condition holds a vanishing spinorial pair. The generalized Calabi-Gray construction says that from any vanishing spinorial pair, one can construct a compact non-Kähler Calabi-Yau 3-fold X with balanced metrics. In particular, it includes the classical construction of Calabi [3] and Gray [14], where Σ is an immersed minimal surface in a flat 3-torus and ϕ its Gauss map. We also say such an X is a genus g generalized Calabi-Gray manifold, where g is the genus of Σ.

Minimal Surfaces in Flat 3-Spaces
In this section, we construct vanishing spinorial pairs by studying minimal surfaces in flat T 3 . Let T 3 = R 3 /Γ be a 3-torus equipped with the standard flat metric. Let x = (x 1 , x 2 , x 3 ) be the coordinate function on R 3 , which we also use as local flat coordinates on T 3 . Now assume that Σ ⊂ T 3 is a closed (immersed) oriented surface of genus g with induced metric and let (u, v) be local isothermal coordinates on Σ. It is well-known that the (conformal class of) induced metric on Σ determines a complex structure on Σ and z = u + iv is a local holomorphic coordinate. The metric on Σ can be expressed as where ·, · is the ambient metric on T 3 . Moreover we can write down the Kähler formω aŝ Locally, define The isothermal condition implies that φ 2 1 + φ 2 2 + φ 2 3 = 0. It is a well-known result that Σ is a minimal surface if and only if φ is holomorphic, or equivalently, on Σ. Notice that for any smooth function u on Σ, we have 2i∂∂u = ∆u ·ω. Now let us assume that Σ is minimal and denote by Q the Fermat quadric Therefore we get a map ν : Σ → Q given by which is a globally defined holomorphic map and it does not depend on the choice of local isothermal coordinates. This is known as the tangential Gauss map. A simple genus calculation indicates that Q is biholomorphic to CP 1 . Let O(1) be the positive generator of the Picard group of Q and let H be the hyperplane line bundle on CP 2 . It is easy to see that Moreover each φ j can be thought of as a section of H, which corresponds to a globally defined holomorphic 1-form µ j = φ j dz on Σ. From this, we see that in other words, (Σ, ν) is a vanishing spinorial pair.
In fact, an explicit identification g : CP 1 → Q can be constructed as follows Through the stereographic projection (4), it follows that the composition ϕ = g −1 • ν : Σ → CP 1 = S 2 is exactly the classical Gauss map given by the unit normal vector field. Moreover, the Gauss-Bonnet Theorem tells us that ϕ : Σ → CP 1 is of degree g − 1.
According to the work of Meeks [20] and Traizet [32], for every g ≥ 3, there exist minimal surfaces of genus g in T 3 . Conversely, it is an easy exercise in algebraic geometry that the existence of vanishing spinorial pair implies that the genus is at least three.
A pair (Σ, ϕ) with pullback condition is not so far from being minimal. It turns out that (7) provides the Weierstrass data for the universal cover of Σ to be minimally immersed into R 3 such that its Gauss map is given by ϕ itself. In this sense, we may think of (Σ, ϕ) with pullback condition satisfied as "minimal surfaces in T 3 " without solving the period problem.

Derivation of the Equation
Starting from a vanishing spinorial pair (Σ, ϕ), we can construct a compact non-Kähler Calabi-Yau 3-fold X as described in Subsection 2.1. Now let us consider solving the Strominger system on X, following the procedure and calculations given in [5,7].
We start with the conformally balanced equation (3). Identifying CP 1 with Q, we obtain a holomorphic map By pulling back φ 1 , φ 2 and φ 3 as sections of H| Q , we get holomorphic 1-forms µ 1 , µ 2 and µ 3 on Σ such that and that the linear system spanned by {µ j } 3 j=1 is basepoint-free. As is known in the literature relating to Theorem 2.1(c), one can check that is a nowhere vanishing holomorphic (3,0)-form on X. Indeed, using local coordinates z on Σ and ζ = z 2 /z 1 on CP 1 , we have the local expression A similar expression arises using ξ = z 1 /z 2 on CP 1 . A computation shows that Ω(∂ z , v, x + iJ 0 x) = 0 for all v, x ∈ T M .
Next, the expression defines a Kähler metric on Σ. Using local coordinates z on Σ and ζ = z 2 /z 1 on CP 1 , we have A direct calculation reveals that the Gauss curvature κ of ω is given by , we obtain that hence ω has non-positive Gauss curvature. If we add in the hyperkähler fiber metrics of the fibration p : X → Σ, we get a natural Hermitian metric ω 0 = ω + αω I + βω J + γω K on X. By definition, We know that Ω ω0 is a constant by its design since Here vol M is the volume form of (M, g). Furthermore, the balanced condition d(ω 2 0 ) = 0 holds as well, therefore ω 0 solves the conformally balanced equation (3). Let f : Σ → R be any smooth function on Σ, and define the Hermitian metric on X. For convenience of notation, we write ω ′ = αω I + βω J + γω K , whose exterior derivative is given by

An elementary calculation indicates
that , and hence ω f = e 2f ω + e f ω ′ solves the conformally balanced equation (3) for arbitrary f . We now consider the anomaly cancellation equation (2). The idea is the following: first we compute the curvature term Tr(R f ∧ R f ) with respect to the ansatz metric ω f and then fix a gauge bundle which solves the Hermitian Yang-Mills equation (1). We then choose f to solve the anomaly cancellation equation. The calculation of the curvature term is essentially the same as we did in [5]. The result is where ∇ϕ 2 = −2κ is with respect to ω and R ′ is the curvature form of the relative cotangent bundle Ω X/Σ with respect to the metric induced from ω 0 . Therefore the Tr(R ′ ∧ R ′ )-term can be cancelled by the curvature term from the gauge bundle, and the anomaly cancelation equation (2) reduces to This choice of gauge bundle makes sense since the relative cotangent bundle solves the Hermitian Yang-Mills equation (1) automatically for arbitrary ω f . Roughly speaking this is because the fiber metrics of p : X → Σ are hyperkähler, and we refer to [5] for more details.

Remark 2.3.
In the case that M is a flat T 4 , the term Tr(R ′ ∧ R ′ ) disappears [7]. In particular we may use any flat vector bundle as our gauge bundle.
Notice that u = e f + α ′ κ 2e f is a function depending only on Σ, and we want to solve Direct calculation shows that The decomposition of dω ′ (9) into its (2, 1) and (1, 2) parts can be seen by acting with the complex structure J 0 and using the following identities Next, a computation shows that α, β and γ all satisfy the equation i.e., they live in the kernel of the operator −∆ + 2κ. Therefore (11) i∂∂ω ′ = −κ ω ∧ ω ′ and the anomaly cancellation equation (2) is further reduced to (12) −∆u + 2κu = 0.
In conclusion, we have demonstrated that to solve the full Strominger system on generalized Calabi-Gray manifolds, we may use the ansatz (8) and the whole system reduces to a quadratic algebraic equation and a linear PDE on Σ: In particular, we get a smooth solution if and only if we can find a function u in the kernel of the operator −∆+2κ such that u is positive at all ramification points of ϕ. Consequently the solvability of the Strominger system on X is closely related to spectral properties of the operator −∆ + 2κ = −∆ − ∇ϕ 2 on Σ, which we shall explore in the next subsection. It is also clear that there are no solutions to the Strominger system using our ansatz if α ′ ≤ 0.
A byproduct of the above calculation is the following statement: A generalized Calabi-Gray manifold X does not admit any pluriclosed metrics.
Proof. Suppose ω 1 is a pluriclosed metric on X, that is, ω 1 is a positive (1,1)-form such that i∂∂ω 1 = 0. Let us consider the integral On one hand, by integration by part, this integral vanishes. On the other hand, i∂∂ω ′ = −κ ω ∧ ω ′ is positive away from a set of measure zero, hence we get a contradiction.
This corollary generalizes a result in [7].

Solutions
In the previous section we manifested that the solvability of the Strominger system on generalized Calabi-Gray manifolds is closely related to the spectral property of the operator −∆ + 2κ = −∆ − ∇ϕ 2 . In fact, this operator falls into the larger class of Schödinger operators associated with holomorphic maps from Riemann surfaces to complex manifolds as below. Let ϕ : Σ → N be a holomorphic map from a compact Riemann surfaces to a complex manifold. By fixing a Hermitian metric on N and choosing a metric ω on Σ, it makes sense to talk about the Schödinger operator L ϕ = −∆ − ∇ϕ 2 on Σ. Here, ∆ is the Laplace-Beltrami operator associated to ω, and the norm ∇ϕ is measured with respect to the chosen metrics on both domain and target. Clearly, L ϕ is a self-adjoint operator on the Sobolev space W 1 (Σ) with respect to the L 2 -norm. The associated quadratic form is Since the domain has real dimension two, this quadratic form is conformally invariant. Therefore, the kernel of L ϕ and the number Ind L ϕ of negative eigenvalues of L ϕ , known as the index, do not depend on the choice of the metric ω. For example, if ϕ is the (extended) Gauss map of a minimal surface Σ in flat 3-space, then L ϕ = −∆ + 2κ is the Jacobi operator, or the stability operator of the minimal surface, which comes from the second variation of the area functional. As mentioned in [22], this operator also shows up naturally in the study of Willmore surfaces and Polyakov quantum string theory. Throughout this paper, the target N is always the projective line CP 1 equipped with the Fubini-Study metric (5). Now let ϕ : Σ → CP 1 be a nonconstant holomorphic map. We would like to understand the space ker L ϕ . By the stereographic projection (4), we may think of ϕ = (α, β, γ) as a map into R 3 . It is easy to check that α, β and γ live in the kernel of L ϕ , therefore dim ker L ϕ ≥ 3.
Let V ϕ denote the 3-dimensional space spanned by α, β and γ. Montiel and Ros [22] showed that the space ker L ϕ /V ϕ is in 1-1 correspondence to complete branched minimal immersions into R 3 with finite total curvature and planar ends (up to translation) such that its extended Gauss map is ϕ. However, it seems that in general we do not know how to compute ker L ϕ and there are examples that dim L ϕ is strictly greater than 3. For this reason, to solve the reduced the Strominger system (13) on generalized Calabi-Gray manifolds, let us consider only the case u ∈ V ϕ . In this scenario, the condition that u is positive at all ramification points of ϕ is equivalent to that the branched points of ϕ on CP 1 all lie in an open hemisphere, which we abbreviate as the "hemisphere condition". Clearly if (Σ, ϕ) satisfies the hemisphere condition and τ : Σ → Σ is an unramified covering of Riemann surfaces, then ( Σ, ϕ • τ ) also satisfies the hemisphere condition.
Given any vanishing spinorial pair (Σ, ϕ), we may make the hemisphere condition holds by composing ϕ with a suitable automorphism of CP 1 , since there exist Möbius transformations on CP 1 pushing all points on the south hemisphere to the north hemisphere. As a consequence, there exist vanishing spinorial pairs satisfying the hemisphere condition for every genus g ≥ 3 and the moduli of curves with vanishing spinorial pairs satisfying the hemisphere condition is exactly the moduli of curves with basepoint-free theta characteristics, which roughly forms a divisor in the moduli space of curves [31]. However it seems unknown that whether there exists or not a minimal surface in T 3 with its Gauss map satisfying the hemisphere condition. The hemisphere condition fails for the 5-dimensional family of triply periodic minimal surfaces constructed by Meeks [20].
Summarizing our previous results, we have proved that Theorem 2.5. Let (Σ, ϕ) be a vanishing spinorial pair with the hemisphere condition satisfied, then we may construct explicit solutions to the Strominger system on the associated generalized Calabi-Gray manifold X. In fact, we get a family of such solutions of real dimension of dim ker L ϕ ≥ 3. As a consequence, for every genus g ≥ 3, there exist smooth solutions to the Strominger system on genus g generalized Calabi-Gray manifolds. They have infinitely many distinct topological types and sets of Hodge numbers.
The genus g = 3 case is of particular interest. Assuming that Σ is of genus 3 and (Σ, ϕ) a vanishing spinorial pair, then clearly ϕ : Σ → CP 1 is of degree 2, therefore Σ is hyperelliptic. In this case, the map ϕ : Σ → CP 1 is a double covering branched over 8 points on CP 1 . Conversely, from any 8 distinct points on CP 1 , one can construct a double covering ϕ : Σ → CP 1 branched over them. By Riemann-Hurwitz, such Σ's are hyperelliptic genus 3 curves. In addition, the pullback condition ϕ * O(2) = K Σ is automatically satisfied. Therefore we can construct vanishing spinorial pairs satisfying the hemisphere condition on any hyperelliptic genus 3 curve, hence solving the Strominger system.
It is also worth pointing out that Theorem 2.5 can be strengthen to an "if and only if" statement for the genus 3 case. This is because that when genus is 3 and ϕ : Σ → CP 1 the hyperelliptic covering, there exists the hyperelliptic involution ι on Σ commuting with ϕ. Since ι is an isometry of ω, it also acts on the space ker L ϕ . Consequently ker L ϕ breaks up into eigenspaces of ι with eigenvalue 1 and -1: The eigenspace V 1 of eigenvalue 1 consists of functions pulling back from CP 1 , hence we have since functions in V 1 have to be the first eigenfunctions of the spherical Laplacian on CP 1 . On the other hand ι acts as -1 on V −1 , therefore for any function in V −1 , it must vanish on the fixed points of ι, i.e., the ramification points of ϕ, hence we cannot use functions in V −1 to produce smooth solutions of the Strominger system. So we have proved Corollary 2.6. Suppose Σ has genus 3 and let (Σ, ϕ) be a vanishing spinorial pair. Using our ansatz, the Strominger system has a smooth solution if and only if (Σ, ϕ) satisfies the hemisphere condition.

Geometric Consequences of the Hemisphere Condition
Though the hemisphere condition is trivial for algebraic geometry, it has nontrivial consequences in differential geometry. In this subsection, we prove a simple result regarding the index of the operator −∆ − ∇ϕ 2 mentioned in Subsection 2.4.
Let ϕ : Σ → CP 1 be a holomorphic map from a compact Riemann surface to the projective line. Fix the Fubini-Study metric on CP 1 . As we have seen, the index Ind L ϕ , i.e., the number of negative eigenvalues of the operator −∆ − ∇ϕ 2 , does not depend on the choice of the metric on Σ. Moreover, we have the following estimates Theorem 2.7. (Tysk [33]) Ind L ϕ ≤ 7.68183 · deg ϕ. Ind L ϕ ≥ C · deg ϕ for some absolute constant C.
Assuming the hemisphere condition, we can derive Proposition 2.9.
If the branched points of ϕ in CP 1 all lie in an open hemisphere, then we have the estimate Ind L ϕ ≥ deg ϕ.
Proof. As mentioned in Subsection 2.4, the index Ind L ϕ does not depend on the choice of metrics on Σ. In particular, we may use the singular metric ω ϕ := ϕ * ω F S to calculate Ind L ϕ , because the eigenvalues of the corresponding operator −∆ ϕ + ∇ϕ 2 ωϕ = −∆ ϕ + 2 can still be defined by variational characterization [33], where ∆ ϕ is the singular Laplace-Beltrami operator associated to ω ϕ . Consequently we only need to estimate the number of eigenvalues of −∆ ϕ less than 2.
Without loss of generality, we may assume all branched points of ϕ lie in the south hemisphere S. Let N be the north hemisphere, then we know that ϕ −1 (N ) consists of deg ϕ copies of disks and ϕ −1 (S) is a connected compact Riemann surface with deg ϕ copies of S 1 as its boundary. Applying [22,Lemma 12] to this decomposition of Σ, and using the counting result from [22,Lemma 11(b)], we prove the proposition.

Remark 2.10.
The hemisphere condition is necessary for the above proposition, otherwise we have the counterexample due to Ross [29], where he showed that the Schwarz' P and D surfaces (genus 3 minimal surfaces in flat T 3 ) have Ind L ϕ = 1.