Non-vanishing Superpotentials in Heterotic String Theory and Discrete Torsion

We study the non-perturbative superpotential in E_8 x E_8 heterotic string theory on a non-simply connected Calabi-Yau manifold X, as well as on its simply connected covering space \tilde{X}. The superpotential is induced by the string wrapping holomorphic, isolated, genus 0 curves. According to the residue theorem of Beasley and Witten, the non-perturbative superpotential must vanish in a large class of heterotic vacua because the contributions from curves in the same homology class cancel each other. We point out, however, that in certain cases the curves treated in the residue theorem as lying in the same homology class, can actually have different area with respect to the physical Kahler form and can be in different homology classes. In these cases, the residue theorem is not directly applicable and the structure of the superpotential is more subtle. We show, in a specific example, that the superpotential is non-zero both on \tilde{X} and on X. On the non-simply connected manifold X, we explicitly compute the leading contribution to the superpotential from all holomorphic, isolated, genus 0 curves with minimal area. The reason for the non-vanishing of the superpotental on X is that the second homology class contains a finite part called discrete torsion. As a result, the curves with the same area are distributed among different torsion classes and, hence, do not cancel each other.


Introduction
Compactification of E 8 × E 8 heterotic string theory on smooth Calabi-Yau (CY) threefolds can lead to realistic particle physics models. For example, heterotic M-theory vacua consisting of stable, holomorphic SU(4) vector bundles defined by "extension" over a class of Schoen CY threefolds can produce exactly the spectrum of the minimal supersymmetric standard model (MSSM) with gauged B − L symmetry [1,2,3,4]. Similarly, heterotic M-theory compactified on other classes of CY threefolds, such as the tetraquadric, carrying "monad" vector bundles can lead to the MSSM at low energy, with or without gauged B−L symmetry [5,6,7,8,9]. Although these string vacua realize the correct spectrum and interactions of low energy particle physics, there remains a fundamental problem; that is, that the associated threefolds and vector bundles have moduli that generically have no potential energy. Therefore, the vacuum values of these fields can be dynamically unstable and, even if time-independent, cannot be uniquely specifiedthus rendering explicit predictions of the values of supersymmetry breaking and physical parameters impossible. It follows that the stabilization of both geometric and vector bundle moduli is one of the most important problem in heterotic string theory.
A non-vanishing potential energy for the geometric moduli, that is, the complex structure [10] and Kahler moduli, can occur for specific heterotic string vacua due to both perturbative and non-perturbative effects. This leads to partial, and in some toy cases complete, stabilization of these moduli [11]. However, the situation for vector bundle moduli is more difficult. Here, there is no perturbative contribution to their potential energy and one must examine possible non-perturbative effects. A non-perturbative superpotential can, in principal, be generated by string instantons [12,13,14,15,16,17,18,19,20]. It depends (inversely) exponentially on the Kahler moduli, and also contributes to a potential energy for both complex structure and, importantly, the vector bundle moduli through 1-loop determinants. However, it is difficult to compute these 1-loop quantities. So far, this has only been carried out for specific examples of elliptically fibered CY threefolds with spectral cover vector bundles [21,22]. It is important, therefore, to generalize these constructions to more realistic vacua, such as those mentioned above. Even then, to find the complete superpotential one has to sum up the contributions from all holomorphic, isolated, genus 0 curves. Beasley and Witten showed that, in a large class of models, these contributions cancel against each other [23,24]. Hence, in addition to calculating the instanton generated superpotential for specific curves in more realistic vacua, one must then show that these contributions do not cancel each other; that is, that the Beasley-Witten theorem is not applicable to these theories. In this paper, we take a first step in that direction by explicitly calculating the complete leading order instanton superpotential for a heterotic vacuum consisting of a Schoen [25] threefold geometry and a simple "extension" SU(3) vector bundle-similar, but not identical, to the heterotic standard model in [4]. Although our Schoen threefold is a complete intersection CY manifold (CICY) and the vector bundle descends from a vector bundle on the ambient space-two of the three main conditions required by the Beasley-Witten theorem-we find in this theory that the Beasley-Witten theorem is not applicable and that the superpotential indeed does not vanish. Extending this work to exact heterotic standard model vacua will be carried out elsewhere.
We start our analysis with a theory on a Schoen threefoldX which is a CICY in the ambient space A = P 1 × P 2 × P 2 . We will also consider only those vector bundlesṼ onX that descend from a vector bundle V on A. These vacua satisfy two of the three conditions of the residue theorem of Beasley and Witten and, therefore, one might expect the complete non-perturbative superpotential to vanish. However, we point out that the Beasley-Witten residue theorem additionally assumes that the area of all holomorphic curves on the CICY is computed using the restriction of the Kahler form on the ambient space. Usually, this restriction does give the complete Kahler form on the CY manifold-but there are cases when it does not. These more subtle cases arise when the CICY manifold has more (1, 1) classes than does the ambient space. As a result, curves which have the same area with respect to the restriction of the Kahler form of the ambient space, can actually have different area with respect to the true Kahler form on the Calabi-Yau space and, hence, can lie in different homology classes. The Schoen manifold studied in the paper has this property. It has 19 (1, 1) classes whereas the ambient space has only 3. We show that there are holomorphic, isolated, genus 0 curves in this manifold which are unique in their homology classes despite having the same area with respect to the restriction of the Kahler form of A. Thus, for an arbitrary vector bundle the contributions to the non-perturbative superpotential due to these curves cannot cancel each other because they are weighted with different area. This way, one can get around the Beasley-Witten residue theorem.
Furthermore, our CICY Schoen threefold is chosen to have a freely acting Z 3 × Z 3 symmetry group. We then mod our this discrete action, to obtain a non-simply connected Calabi-Yau space with π 1 = Z 3 ×Z 3 . For a toy choice of a vector bundle which descends from the ambient space, the non-perturbative superpotential for all holomorphic, isolated, genus 0 curves with minimal area is computed. For simplicity, we perform our calculations for a fixed complex structure. Hence, the only 1-loop determinant which needs to be computed is the Pfaffian of the Dirac operator on these curves. Since we do not know either the metric or the gauge connection, we use an algebraic method (similar to the one developed in [21,22]) to compute the Pfaffians. They turn out to be homogeneous, degree 2 polynomials on the moduli space of vector bundles. We show that the sum of the contributions from these curves is non-zero. Here, the main reason for the non-vanishing of the superpotential is the discrete part of the second homology group, called discrete torsion. Due to torsion, curves which have the same area actually lie in different classes of the second homology group with integer coefficients. These different classes are labeled by the characters of the torsion subgroup-which in the present case is Z 3 ⊕ Z 3 . Hence, in this case the non-vanishing of the superpotential can also be attributed to existence of holomorphic, isolated, genus 0 curves which are unique in their integral homology classes.
The paper is organized as follows. In Section 2, we start with reviewing the structure of the nonperturbative superpotential in heterotic string theory, mostly following [17]. Then we review the residue theorem of Beasley and Witten, pointing out that it is directly applicable only when the Kahler form on 2 Non-perturbative superpotentials in heterotic string theory 2.1 The general structure of non-perturbative superpotentials We consider E 8 ×E 8 heterotic string theory compactified to four-dimensions on a Calabi-Yau threefold X. As was extensively studied in a variety of contexts and papers [12,13,14,15,16,17,18,19,20,21,22], the effective low-energy field theory may, in principle, develop a non-perturbative superpotential for the moduli fields generated by worldsheet/worldvolume instantons. The structure of the instantons is slightly different in the weakly and strongly coupled heterotic string theories. Be that as it may, the superpotential has the same generic form. For concreteness, we will discuss the weakly coupled case where the superpotential is generated by strings wrapping holomorphic, isolated, genus 0 curves in X. 1 Furthermore, for simplicity, we will restrict our discussion to the "observable" sector; that is, to the superfields associated with the first E 8 factor of the gauge group.The superpotential is then determined by the classical Euclidean worldsheet action S cl evaluated on the instanton solution and by the 1-loop determinants of the fluctuations around this solution. Let C be a holomorphic, isolated, genus 0 curve in X. Then the general form of the superpotential induced by a string wrapping C is [17] .
Let us review various ingredients in this formula. The expression in the exponent is the classical Euclidean action evaluated on C. In the first term, A(C), is the area of the curve given by where ω is the Kahler form on X. In the second term, B is the heterotic string B-field which, in this expression, can be taken to be a closed 2-form, dB = 0. Let ω I be a basis of (1, 1)-forms on X, I = 1, . . . , h 1,1 . Then we can expand Let us define the complexified Kahler moduli Then the exponential prefactor becomes By construction Re(iα I (C)T I ) < 0. Note that the exponential factor in (2.1) can also be understood as a map from the curve C to the non-zero complex numbers C * . That is, Since the value of the integrals depends only on the homology class of the curve, the map is more appropriately expressed as e −S cl : However, here there is an important caveat. In eqs. (2.3), (2.7) we are assuming that the moduli space of the B-field is connected. As we will discuss below, this is not necessarily the case. Hence, the map (2.7) needs to be refined.
Let us now discuss the 1-loop determinants. The first determinant is the Pfaffian of the Dirac operator which comes from integrating over the right moving fermions in the worldsheet theory. In heterotic compactifications, we have to specify the internal gauge field A on X which satisfies the Hermitian Yang-Mills equations where m andm are holomorphic and anti-holomorphic indices on X and g mn is the Ricci flat metric on X.
According to the theorem of Donaldson-Uhlenbeck-Yau, A is a connection on a holomorphic polystable vector bundle V on X whose structure group is a subgroup of E 8 . Then the Pfaffian in (2.1) is the Pfaffian of the Dirac operator depending on the connection A restricted to the curve C. Since the spin bundle on a genus 0 curve is O C (−1), we additionally tensor V with O C (−1) and denote V C (−1) = V| C ⊗ O C (−1). Pfaff(∂ V C (−1) ) depends on the moduli of the vector bundle V. In principle, it can be explicitly expressed as a function of the gauge connection A using the WZW model [22]. However, since no explicit solutions to the Hermitian Yang-Mills equations on X are known, it is unclear how to use this in practice. Since right moving worldsheet fermions are Weyl, the Pfaff(∂ V C (−1) ) is anomalous. However, this anomaly is cancelled by the variation of the B-field [22]. As the result, the Pfaffian of the Dirac operator is not a function on the moduli space of V but, rather, a section of some line bundle. In the denominator in (2.1), det(∂ N C ) comes from integrating over bosonic fluctuations and is the determinant of the∂operator on the normal bundle to the curve C. For an isolated, genus 0 curve, the normal bundle is is the∂-operator on the trivial line bundle which is a constant.
In general, a given homology class of X contains more than 1 holomorphic, isolated, genus 0 curve. The number of these curves is referred to as to Gromov-Witten invariant. All such curves in the same homology class have the same area, the same classical action and the same exponential prefactor in (2.1). However, the 1-loop determinants, in general, are different. Hence, the contribution to the superpotential from all curves C i in the homology class [C] of the curve C is given by (for simplicity, we remove the where n [C] is the number of the holomorphic, isolated, genus 0 curves in the homology class [C]. To find the complete non-perturbative superpotential W , we then have to sum over all homology classes. That is, (2.10)

The residue theorem of Beasley-Witten
In [23] (also see earlier papers [29,30,31,32]) Beasley and Witten showed that, under some rather general assumptions, the sum (2.9) must vanish for each homology class [C]. Here, we review their assumptions since they will be important later in the paper. LetX be a complete intersection Calabi-Yau threefold in the product of projective spaces 2 A = P n 1 × · · · × P na . That is,X is given by a set of polynomial equations p 1 = 0, . . . , p m = 0 where a i=1 n i − m = 3. Additionally, assume that the Kahler form ωX descends from the ambient space, that is, ωX = ω A |X, and that the vector bundleṼ onX is obtained as a restriction of a vector bundle V on A,Ṽ = V|X. Then, it was shown by Beasley and Witten that if these assumptions are satisfied, the sum (2.9) vanishes for any homology class. This result was proven in [23] and interpreted as a residue theorem.
The proof in [23] is based on standard arguments of topological field theory and localization. First, they constructed a topological worldsheet action with target space A such that there exists a set of supersymmetric vacuum solutions-all with CICY threefoldX, ωX = ω A |X andṼ = V|X. Each such vacuum is associated with a holomorphic curve C ⊂X. By the standard arguments of topological field theory, the correlation functions in this theory do not depend on the coupling. In one limit of the coupling, the correlators are localized on this set of supersymmetric vacua solutions. This leads to eq. (2.10) for the total superpotential, where one uses the fact that non-isolated and/or higher genus curves only contribute to higher F-term interactions,. In another limit, the same correlators vanish because of unsaturated fermionic zero modes. Hence, W = 0. Since the exponential factor is different for each homology class, Beasley and Witten concluded that the sum (2.9) vanishes for any homology class. The assumptions of Beasley and Witten are rather general, which means that in a large class of heterotic string models a non-perturbative superpotential cannot be generated. This raises a question of whether moduli in heterotic compactifications can ever be completely stabilized.
The aim of this paper is to present explicit examples where the non-perturbative superpotential is indeed non-zero.

Applicability of the residue theorem
As we have discussed, in the analysis of Beasley and Witten in [23] there is the assumption that ωX = ω A |X. This assumption is necessary in order for their analysis to be a topological theory on the ambient space withX as a vacuum solution-and, hence, to use their residue theorem. It follows that, in their theorem, the area of all curves in (2.9), (2.10) is measured using the Kahler form ω A on A restricted tõ X. However, there are cases when this restriction is not the same as the physical Kahler form onX. Indeed, it is possible that h 1,1 (X) is not the same as h 1,1 (A) because there can be classes inX which do not come as a restriction of classes from the ambient space. Hence, the residue theorem, strictly speaking, is valid only if h 1,1 (X) = h 1,1 (A). 3 If h 1,1 (X) > h 1,1 (A) the residue theorem, though still valid in the topological theory, is not directly applicable to the physical heterotic string theory. In the former case, the area of holomorphic curves is measured using ω A |X. But in the physical theory, it is measured using the actual Kahler form ωX onX. As a result, the curves which have the same area with respect to ω A |X might have different area with respect to ωX and, hence, might lie in different homology classes. More precisely, if h 1,1 (X) > h 1,1 (A) we have where ∆ωX is the contribution to the Kahler form onX from the (1, 1) classes which do not come as a restriction of classes from the ambient space. Then the actual area of a curve C is given by Two curves C 1 and C 2 which satisfy and appear to lie in the same homology class from the viewpoint of the residue theorem can actually have different area due to different contributions from ∆ωX and can lie in different homology classes.
To say it differently, if h 1,1 (X) > h 1,1 (A) the correlation functions in the topological theory studied in [23] do not coincide with correlation functions in the physical heterotic string theory onX. Hence, the cancellation in the residue theorem does not imply an analogous cancellation in the physical theory. However, we can still apply the residue theorem to the physical theory. If in the physical theory we ignore ∆ωX and measure the area of all curves using ω A |X only, then we should have the same cancellation as in the topological theory. Nevertheless, it is important to emphasize that now the cancellation happens among the 1-loop determinants of the curves in different homology classes but having the same area measured by ω A |X. If we restore the actual area using the Kahler form ωX onX, the contributions of these curves might no longer cancel each other because they might lie in different homology classes and have different area. That is, in the physical theory whether or not curves in a given homology class cancel each other cannot be directly deduced from the residue theorem. Below, we will give an example where the cancellation cannot happen simply because each curve is unique in its homology class.

Discrete torsion
Our discussion so far has been missing an important ingredient called discrete torsion. In general, for an arbitrary complex manifold, X, the second homology group with integer coefficients is of the form where Z k is the free part and G tor is a finite group called discrete torsion. For example, a discrete torsion factor of H 2 (X, Z) can arise when X is a quotient of another Calabi-Yau manifold by a freely acting discrete isometry group K-as we will discuss below. The existence of the torsion classes affects the B-field. Indeed, the B-field is an arbitrary closed 2-form dB = 0. However, in general, it implies that the field strength H = dB vanishes in H 3 (X, R) but not necessarily in H 3 (X, Z). In the later case the integral C B is not defined because the moduli space of the B-field is not connected. From the Universal Coefficient Theorem (see e.g. [34]) it follows that This means that there is one-to-one correspondence between the torsion elements of H 2 (X, Z) and the number of the connected components of the moduli space of the B-field. These connected components can be labeled by the characters of the discrete group G tor . Since the B-field is not continuous, we have to replace the exponential prefactor in (2.1) with a more general map from H 2 (X, Z) → C * [35]. While we will continue to denote this map by Then the map (2.6) can be understood as where, in the last step, we view ω C as the Poincare dual 4-cycle and ω C · C is the intersection of this 4-cycle with the curve C. However, ω C defined in (2.17) is Poincare dual only to an element of the free part of H 4 (X, Z). Clearly, ω C should also contain a torsion part. Let G tor have r generators β 1 , . . . , β r . Then the complete expression for ω C is given by where, slightly abusing notation, we continue to use the same symbol for the complexified Kahler form including torsion. Since β α are torsion elements, it follows that for any α there is an integer m α for which m α β α = 0. Hence, we obtain Since m α β α = 0 and C is arbitrary, it follows that χ α = e is α is an m α -th root of unity. Hence, s α can take only discrete values parametrizing the connected components of the moduli space of the B-field. It also follows that χ α is a character of G tor . We conclude that mapping (2.6) now generalizes to The precise values of χ α depends on the choice of the torsion part of B; that is, on the choice of the connected component. Clearly, all curves in the same homology class of H 2 (X, Z) have the same value of β α (C) and, hence, pick up the same character-dependent factor in (2.21).
Let us now refine eq. (2.9) in the presence of discrete torsion. Let [C] be the homology class of the curve C in H 2 (X, R) = R k . As we have just discussed, the curves in [C] do not necessarily lie in the same homology class in H 2 (X, Z) because they might belong to different torsion classes. Curves belonging to different torsion classes pick up different characters under the map (2.21). Hence, equation (2.9) is modified to become [C] ∈ H 2 (X, R) .

(2.22)
To find the complete non-perturbative superpotential, we have to sum over all homology classes [C] ∈ H 2 (X, R). Later in the paper, we will analyze expression (2.22) for a specific example.
3 The Schoen manifold and the prepotential

The Schoen manifold
Having presented the generic discussion above, we now proceed to calculate the non-perturbative superpotential for a specific Calabi-Yau threefold. This manifold, denoted by X, is the quotient of a simply connected, complete intersection Calabi-Yau threefold,X-chosen to be a specific Schoen manifold [25]with respect to its fixed-point free symmetry group K = Z 3 × Z 3 . This Schoen threefold is defined as follows. We constructX as a compete intersection in the ambient space A = P 1 × P 2 × P 2 with homogeneous coordinates X is then given by a common zero locus of two polynomial equations Here P 1 , . . . , P 4 are homogeneous polynomials of degree 1 and Q 1 , . . . , Q 4 are homogeneous polynomials of degree 3. For the purposes of this paper, we will restrictX to be given by the following polynomials This manifold is self-mirror with h 1,1 = h 2,1 = 19 [25,36]. Note that It follows that onX there are 16 (1, 1) classes which do not arise as the restriction of (1, 1) classes from the ambient space. The manifoldX defined by (3.3) is invariant under the action of the K = Z 3 × Z 3 symmetry generated by where ζ = e 2πi/3 . Note that this discrete symmetry does not act on P 1 . This action has fixed points on the ambient space A, but not onX. Having constructedX with a free Z 3 × Z 3 action, we define This manifold is also self-mirror with h 1,1 = h 2,1 = 3 [27]. From these Hodge numbers, it follows that H 2 (X, R) = R 3 . However, H 2 (X, Z) is more involved. It was shown in [27] that it contains the discrete torsion subgroup That is, the complete H 2 (X, Z) is given by Let us point out that from the Universal Coefficient Theorem it follows that Hence, in the present case, the torsion groups of H 2 (X, Z) and of H 2 (X, Z) are the same. Let us present some mathematical details ofX and X following [27]. From eq. (3.3) we see that for fixed [t 0 : t 1 ] we have two elliptic curves, one in each P 2 . Thus, each equation in (3.3) defines a rational elliptic surface dP 9 ∈ P 1 × P 2 and, hence,X is a double elliptic fibration over P 1 . The structure ofX can be illustrated using the diagram Here B 1 and B 2 are the dP 9 surfaces given by the individual equations in (3.3). The Z 3 × Z 3 action descends to B 1 and B 2 . Since its action is trivial on P 1 , on each B k for k = 1, 2 the Z 3 × Z 3 must act by translation along the fiber by two independent sections of order 3. To simplify notation, we denote these sections on either B k by the same symbols µ and ν-unless it is necessary to distinguish them. Additionally, each B k has the zero section σ. This determines the structure of Kodaira fibers and the Mordell-Weil group to be [37] sing(B 1 ) = sing(B 2 ) = 4I 3 , The Mordell-Weil group is generated by the zero section σ and by the sections µ and ν of order 3. Each B k has 4 I 3 singular fibers, each containing 3 exceptional classes intersecting in a triangle. These classes will be denoted by θ ji , where j = 1, . . . , 4 labels the singular fibers and i = 1, 2, 3 labels the exceptional classes in each such fiber. As was shown in [38,39], the basis in H 2 (B k , Z) can be chosen to be where f is the class of the elliptic fiber. The intersection numbers of these classes can be found in [27]. Out of these basis elements, it is possible to construct divisors which are Poincare dual to Z 3 ×Z 3 invariant (1, 1) classes in B k (here we will use the same notation for divisors and Poincare dual (1, 1)-forms). The invariant cohomology group of B k is two-dimensional and generated by [27] where t is a specific linear combination of the classes in (3.12) given by (3.14) The intersection numbers of f and t are [27] Using the invariant cohomology classes in B k , we can now construct divisors inX Poincare dual to the invariant (1, 1) classes onX. The invariant generators can be defined using the diagram (3.10) and the invariant classes in (3.13): Let us now denote the corresponding Poincare dual (1, 1)-forms as ω φ , ω τ 1 , ω τ 2 . They form a basis of the invariant cohomology group H 2 (X, Z) K , and will descend to the quotient manifold X. Their triple intersection numbers can be found using the diagram (3.10) and eq. (3.15) to be The remaining triple intersection numbers are zero. We can now, somewhat abusing notation, define the form ω X onX by which will descend to the non-torsion part of the Kahler form on X. Let us emphasize, however, that ω X is not the same as the Kahler form onX. This follows from the fact that onX there are additional classes of H 2 (X, Z) which are not invariant under Z 3 ×Z 3 . Indeed, as we stated previously, h 1,1 (X) = 19. This means that there are 16 (1, 1) classes onX in addition to ω φ , ω τ 1 and ω τ 2 . That is, the complete Kahler form onX is given by where ∆ωX stands for the contribution from the additional 16 non-invariant classes. Comparing this expression with (2.11), we conclude that We will give an explanation for this relationship in the following subsection.

The ambient space description of the invariant (1, 1) classes
The above description of the invariant (1, 1) classes onX is somewhat abstract. Here, we will give a simpler description of ω φ , ω τ 1 , ω τ 2 in terms of the forms on the ambient space. Let J 1 , J 2 , J 3 be the Kahler forms on the three projective spaces P 1 , P 2 , P 2 forming the ambient space, normalized as In the cohomology class of J 2 (and similarly of J 3 ), one can choose a representative to be the Kahler form of the Fubini-Study metric with the Kahler potential This means that the cohomology classes of J 2 and J 3 are also invariant classes.
Let us now define the (1, 1) classes onX by restriction By construction, the cohomology classes of J 1 , J 2 , J 3 are invariant classes in H 2 (X, Z) and, hence, form a basis in H 2 (X, Z) K . The triple intersection numbers of J 1 , J 2 , J 3 can be computed by the standard methods of complete intersection Calabi-Yau manifolds (see e.g. [40]) 5 with the following result with the remaining ones being zero. Comparing eq. (3.24) with (3.17) we conclude that That is, the invariant (1, 1) classes ω φ , ω τ 1 , ω τ 2 , which were constructed in the previous subsection in a rather abstract way, are simply the restriction of the Kahler forms on the projective spaces forming the ambient space-thus explaining expression (3.20). Due to the normalization properties (3.21), the forms J 1 , J 2 , J 3 can be viewed as first Chern classes of the following line bundles on A: This implies the following relations between the line bundles These relations will be useful later. For emphasis, we again note note that the Kahler form (3.19) can be written as We see, therefore, that the Kahler form onX is not simply given by the restriction of the Kahler form from the ambient space but, rather, contains an additional term ∆ωX.

The prepotential and Gromov-Witten invariants
The number of holomorpic, isolated, genus 0 curves in each homology class of X can be read off from the prepotential in type II string theory. The prepotential on X was computed in [27,26,28]. In this subsection, we will review the result. Since the (1, 1) classes {ω φ , ω τ 1 , ω τ 2 } are Z 3 × Z 3 invariant onX, they descend to cohomology classes on X. To simplify our notation, we will label these cohomology classes using the same symbols. Let with the other integrals being zero. Let us define Since H 2 (X, Z) contains torsion classes, we also have to introduce the image under the map e −S cl of the Z 3 × Z 3 torsion generators-which we denote by b 1 and b 2 respectively and satisfy b 3 be a homology class of the form Then, as shown in (2.21), the image of this class under e −S cl is given by The prepotential in type II string theory is defined by the expression Here the sum is over all holomorphic, isolated, genus 0 curves and the polylogarithm Li 3 takes proper care of multiple wrappings. If we know F X , we can expand it in powers of p, q, r, b 1 , b 2 and read off the Gromov-Witten invariants n [C] . The prepotential F X for the quotient Calabi-Yau manifold X in (3.6) was computed in [27,26,28]. Here, we present the result to low orders in p, q, r. It is given by where the polynomial P(q) is of the form Let us discuss some simple consequences of eqs. (3.34), (3.35). It follows that there are no terms ∼ p 0 . In other words, there are no isolated, genus 0 curves in the homology classes (0, n 2 , n 3 , m 1 , m 2 ). However, there are terms in F X that are ∼ p 1 . Hence, there are isolated, genus zero 0 curves in the (1, n 2 , n 3 , m 1 , m 2 ) homology classes. It follows from (3.30) that the contribution of these classes to e −S cl and, hence, the superpotential is proportional to e iT 1 +in 2 T 2 +in 3 T 3 . This means that the leading contribution to the superpotential is ∼ e iT 1 ; all terms with n 2 > 0 and/or n 3 > 0 being exponentially suppressed. Similarly, the contribution to the superpotential of any class with n 1 > 1 is also suppressed relative to e iT 1 . Therefore, since we are interested in computing the superpotential, we will focus on the homology classes of the form (1, 0, 0, m 1 , m 2 ). The number of isolated, genus 0 curves in each such class can be read off from the most leading term in (3.34). This is given by It follows that in each torsion class there is precisely 1 curve. That is, The main aim of the rest of this paper will be to compute the non-perturbative superpotential (2.22) summed over these 9 isolated, genus 0 curves. All of them are in the same homology class in H 2 (X, R) and, hence, have the same area with respect to the Kahler form on X. However, they are distributed in 9 different homology classes once we take discrete torsion into account.

Explicit construction of the isolated, genus 0 curves
It is possible to explicitly visualize these 9 curves as follows. We see from eq. (3.30) that, ignoring torsion, they are in the same homology class [C φ ] which is dual to the class of the (1, 1)-form ω φ . Let us lift these curves from X toX. Since onX we have ω φ = J 1 = J 1 |X, where J 1 is the Kahler form on P 1 ⊂ A, the pre-image of these 9 curves gives 81 holomorphic, genus 0 curves onX which can be parametrized by [t 0 : t 1 ]. Hence, we can visualize these curves by demanding that eqs. (3.3) are solved for arbitrary [t 0 : t 1 ]. This is equivalent to solving the system of equations It is easy to see that this system is solved by 9 × 9 = 81 distinct points on P 2 × P 2 . Since the solutions of (3.38) are distinct points, all the corresponding curves inX are isolated. In Appendix A we compute the normal bundle for each of these curves and check that it is indeed O(−1) ⊕ O(−1). Due to the Z 3 × Z 3 symmetry, the above 81 curves split into 9 orbits under the action of Z 3 × Z 3 -each orbit containing 9 curves. The curves in the same orbit are obtained from each other by the action of the Z 3 × Z 3 group. When we descend to the quotient manifold X, all curves in one orbit yield the same curve in X. Hence, we obtain 9 isolated, genus 0 curves in X which are precisely the curves discussed at the end of the previous subsection. It follows from the prepotential (3.36) they are in the same homology class in H 2 (X, R) but in 9 different homology classes in H 2 (X, Z). We now present 9 curves inX which do not lie in the Z 3 ×Z 3 orbits of each other and which, therefore, descend to 9 distinct curves in X. To accomplish this, let us write the generators g 1 and g 2 in (3.5) in the matrix form Since Z 3 × Z 3 acts simultaneously on both P 2 's, it is convenient to combine [x 0 : x 1 : x 2 ] and [y 0 : y 1 : y 2 ] into a 6-vector (x 0 , x 1 , x 2 , y 0 , y 1 , y 2 ) T . In this basis, the generators of Z 3 × Z 3 are Now choose one arbitrary solution of (3.38). For example, pick where the symbol "T" means think of this as a column vector. It corresponds to the curve Let us now construct the remaining 8 curves C i = P 1 × s i , i = 2, . . . , 9 by acting on s 1 as follows: One can check that these curves solve eqs. (3.38) and cannot be obtained from each other by the action of Z 3 × Z 3 .

Non-vanishing of the superpotential onX
Let us now compactify the E 8 ×E 8 heterotic string on the manifoldX, and consider the non-perturbative superpotential generated by the 81 isolated, genus 0 curves discussed above. To describe the complete string vacuum, one must also introduce a specific holomorphic vector bundle-which we will do in the next section. Although we will compute the superpotential for this specific bundle, the results of the present section are valid for any vector bundle. Since the curves specified by the solutions of (3.38) are all parameterized by [t 0 : t 1 ], it is tempting to conclude that they are in the same homology class dual to the (1, 1) class ω φ = J 1 |X. However, we will see that this is not the case. In fact, we will show that these curves lie in 81 different homology classes. First consider the twofold B 1 ≃ dP 9 ⊂ P × P 2 defined by Let us now examine the genus 0 curves parametrized by [t 0 : t 1 ]. They are specified by the 9 solutions of Each solution is a distinct section of the elliptically fibered surface B 1 ≃ dP 9 . We denote these sections by σ i , i = 1, . . . , 9. Since the order of the Mordell-Weil group MW (B 1 ) = Z 3 ⊕ Z 3 is 9, these sections are in one-to-one correspondence with the elements of MW (B 1 ) [27]. On the other hand, it was shown in [38] that on a dP 9 twofold distinct elements of the Mordell-Weil group are all non-homologous to each other. It follows that the 9 solutions to (4.2) fall into 9 different homology classes. The same is true on B 2 ≃ dP 9 defined by F 2 = (λ 1 t 0 + t 1 )(y 3 0 + y 3 1 + y 3 2 ) + (λ 2 t 0 + λ 3 t 1 )(y 0 y 1 y 2 ) = 0 (4.3) for the 9 genus 0 curves specified by the solutions of For reasons of simplicity, we also denote these sections by σ i , i = 1, . . . , 9.
We now extend this result toX. To start with, note that there is a natural map Let us define This provides a more abstract way to visualize the 81 curves solving eqs. (3.38). Then the map (4.5) acts on the above described solutions as We note that the 81 elements σ i ×σ j , all being distinct, are in one-to-one correspondence with elements of the Mordell-Weil group MW ( and is of order 81 as well. Therefore, we have a map between the set {σ i ×σ j } and the elements of MW (B 1 ) ⊕ MW (B 2 ). By construction, it is a surjective linear map between two finite sets consisting of 81 elements each.
Hence, it is one-to-one. Since all distinct elements of MW (B 1 ) ⊕ MW (B 2 ) are non-homologous to each other, it follows that all 81 curves obtained in (3.38) lie in 81 different homology classes. In particular, it follows that each of these 81 homology classes has precisely 1 isolated, genus 0 curve. Hence, as long as the Pfaffian of the Dirac operator of at least one of these curves is not identically zero-which is expected to be true for a generic vector bundle-the non-perturbative superpotential in this theory is non-zero. We will show this explicitly for a specific holomorphic vector bundle in the remainder of this paper.
To finish this section, let us point out that any of the 81 homology classes discussed above are dual to a (1, 1) class of the form J 1 |X + ∆ωX. 6 Here, it suffices to recall that ∆ωX stands for the classes onX which cannot be obtained as a restriction from the ambient space. All of these 81 curves have equal area with respect to J 1 |X and ω A |X, but have different areas with the respect to the actual Kahler form ωX. Hence, we have an explicit realization of the situation described in Subsection 2.3. As was discussed, this violates one of the assumptions of the Beasley-Witten residue theorem and, hence, one can expect a non-vanishing instanton superpotential. Finally, note that since ∆ωX contains only non-invariant classes under Z 3 × Z 3 which vanish on the quotient manifold X, the area of the images of the 9 curves in eqs. (3.41), (3.42), (3.43) in X is the same with respect to the Kahler form ω X on X. That is, although the original 81 curves were in different homology classes on the covering manifold, their images are in the same homology class on the quotient manifold modulo torsion.

The vector bundle
The rest of the paper will be devoted to an explicit computation of the superpotential in a concrete example of a theory on X. Specifically, we will consider a toy model where the holomorphic vector bundle V is taken to have structure group SU(3). A more phenomenologically realistic class of bundles will be studied elsewhere. Since X is a quotient manifold, it is easiest to first construct a vector bundlẽ V on the covering spaceX which is equivariant under the action of K = Z 3 × Z 3 . The moduli space of an equivariant vector bundleṼ consists of connected components labeled by the characters of K. The vector bundle V on the quotient space X is then defined as First, we will discuss a construction ofṼ in terms of the data onX. We then will express the same vector bundle as the restriction of a vector bundleṼ on the ambient space. The second description is more explicit and will be used in computing the superpotential in the next section.

Construction of the vector bundleṼ onX
We will constructṼ by specifying the line bundles L 1 , L 2 , L 3 onX satisfying the property Then we defineṼ as a sequence of extensions Eq. (5.2) assures that the structure group ofṼ is SU(3) rather than U(3). The structure group of the rank 2 vector bundleW is U(2). ForṼ to descend to the quotient manifold X, it has to be equivariant. To achieve that, it is sufficient to require that the line bundles L 1 , L 2 , L 3 are equivariant. A discussion of equivariant line bundles on the Schoen manifold can be found in [3]. The action of the discrete group in [3] was chosen to be different from ours in (3.5). However, the conclusions on equivariance are the same. Here we will simply state the conclusions, referring to [3] for additional details. First, any equivariant line bundle L onX has to be constructed out of the invariant divisors in (3.16). That is, it has to be of the form where c 1 , c 2 , c 3 are integers. In addition, the sum c 2 + c 3 has to be divisible by 3. In our toy model, we will choose L 1 , L 2 , L 3 to be Note that eq. (5.2) is satisfied. We will take the trivial choice of the equivariant structure; that is, we will assume that the moduli space of V given by (5.1) consists of the component of the moduli space of V which is invariant under Z 3 × Z 3 .
For V to have structure group SU(3) rather than its subgroup, we have to make sure that there exist non-trivial extensions in (5.3). The spaces of non-trivial extensions are given by respectively. For simplicity, we will often denote them by [W ] and [ṼW ]. Note that H 1 (X,W ⊗ L * 3 ) is the space of extensions for a fixed extensionW in [W ]. That is why we denote it by [ṼW ]. Each element in the extension class defines a vector bundle. However, it is important to take into account that different elements in the extension class can define isomorphic vector bundles. LetW 1 andW 2 be two vector bundles from the same extension class [W ]. That is, they both satisfy For any line bundle L there is an isomorphism L → λL, where we multiply all elements of the fiber of L by a non-zero complex number λ. Let us consider the following isomorphisms of L 1 and L 2 : L 1 → L 1 , L 2 → λL 2 . Then from the "five" lemma (see e.g. [34]), it follows that W 1 and W 2 are isomorphic. This means that elements in H 1 (X, L 1 ⊗ L * 2 ) related by a multiplication by λ ∈ C * correspond to the same vector bundle. Therefore, the moduli space M(W ) of vector bundles corresponding to the extension class [W ] is the projectivization of H 1 (X, L 1 ⊗ L * 2 ): Similarly, the moduli space M(ṼW ) of vector bundles corresponding to the extension class [ṼW ] is given by The full moduli space M(Ṽ ) ofṼ can be then understood as a fibration over M(W ) = PH 1 (X, L 1 ⊗ L * 2 ), where the fiber at a fixed extensionW is given by M(ṼW ) = PH 1 (X,W ⊗ L * 3 ). In Appendix B, we compute the dimensions of the spaces in (5.6). We find Note that if we introduce coordinates in the vector space of extensions, it is straightforward to introduce coordinates on its projectivization; that is, we simply treat these coordinates as homogeneous ones. Since the line bundles L 1 , L 2 , L 3 are equivariant, they descend to the quotient manifold X. To simplify notation, we will denote the corresponding line bundles on X by the same letters L 1 , L 2 , L 3 . Hence, the vector bundles W and V on X, obtained by modding outW andṼ by the action of Z 3 × Z 3 , can be defined by the similar extension sequences on X (5.14) As we mentioned before, we will take the trivial choice of the equivariant structure. This means that the the extension classes [W ] and [V W ] can be taken to be the invariant components of (5.6). That is, To show thatṼ and, hence, V admits an Hermitian connection satisfying eq. (2.8), we need to prove that the extensions described above correspond to stable vector bundles. This is discussed in Appendix C.

The ambient space description ofṼ
As was shown in the previous section, the line bundles OX (φ), OX(τ 1 ), OX(τ 2 ) can be obtained as restrictions of line bundles on the ambient space. Using (3.27), we find that L 1 , L 2 , L 3 are also restrictions of line bundles on A. Let us define Then L 1 = L 1 |X , L 2 = L 2 |X, L 3 = L 3 |X. This implies that the extensionsW andṼ are also restrictions of extensions on A, which we denote byW andṼ respectively. They satisfy Similarly to our discussion in the previous subsection, we can introduce the moduli spaces of the corresponding vector bundles Let us now study how the spaces (5.19), (5.20) are related to the similar spaces in (5.6), (5.8), (5.10). The dimension of the first cohomology group in (5.19) can be computed using the Kunneth formula and the Bott's formula We then find that h 1 (A, L 1 ⊗ L * 2 ) = 18 . The relation between cohomology groups on A and onX ⊂ A can be obtained using the Koszul sequence-as we now explain. The Calabi-Yau threefoldX is defined as a submanifold in A using eqs. (3.3).
SinceX is of co-dimension 2, its normal bundle is a rank 2 vector bundle. From eqs. (3.3) we find that it is a restriction of the following vector bundle on A : Let L be a vector bundle on A and L = L|X. They are related to each other by the Koszul sequence The map r is the restriction map, the map F is multiplication (from the left) by the row vector (F 1 , F 2 ) of the defining polynomials in (3.3) and the map F ′ is determined by the composition rule F • F ′ = 0, which follows from the exactness of (5.26). This implies that F ′ is a column vector (F 2 , −F 1 ) T . Note that the sequence (5.26) is not short and, hence, we cannot write the long exact cohomology sequence directly. However, one can split (5.26) into two short exact sequences by introducing an auxiliary sheaf S: where the maps H 1 , H 2 satisfy H 2 • H 1 = F . Writing the long exact cohomology sequences for (5.27) allows us to compute the cohomology groups of L in terms of the cohomology groups of L, N * ⊗ L and ∧ 2 N * ⊗ L. These, in turn, can be calculated using the Kunneth and Bott formulas. We will not present the details of these laborious calculations and only give the results.
To compute the space of extensions [W ] we apply the Koszul sequence (5.25) to L = L 1 ⊗ L * 2 . Then we find that To compute the space of extensions [ṼW ], we apply the Koszul sequence (5.26) to L =W ⊗ L * 3 . But first we have to find the cohomology groups ofW ⊗ L * 3 ,W ⊗ L * 3 ⊗ N * andW ⊗ L * 3 ⊗ ∧ 2 N * using (5.18). We obtain Now, from the Koszul sequence (5.26) applied to L =W ⊗ L * 3 we find Here F 1 is the first defining polynomial in (3.3). It can be viewed as an element of H 0 (A, O A (1, 3, 0)). When we multiply F 1 by a differential in H 1 (A, O A (−5, 2, 1)), we naturally obtain a differential in where the first factor is the projectivization of the vector space in (5.28) and the second factor is the projectivization of the vector space in (5.32).

Parametrization of the moduli space
The aim of this subsection is to derive a parameterization of the moduli spaces M(ṼW ) and M(V W ). Parametrization of the moduli spaces of extensions M(W ) and M(W ) can be derived in a similar way, but is not required in this paper. First, we consider the numerator in (5.32). According to the Kunneth and Bott formulas, Let us consider the vector space where we have used Serre duality. This vector space is 2-dimensional and we denote its basis as {r 0 , r 1 }. This basis is chosen to be dual to the basis {t 0 , t 1 } of homogeneous degree 1 polynomials on P 1 . The vector space of interest, is 3-dimensional with a natural basis {r 2 0 , r 0 r 1 , r 2 1 } dual to the basis {t 2 0 , t 0 t 1 , t 2 1 } of degree 2 polynomials on P 1 . It follows that an arbitrary element v ∈ H 1 (A, O A (−4, 5, 1)) can be written as v = r 2 0 f 1 (x, y) + r 0 r 1 f 2 (x, y) + r 2 1 f 3 (x, y) , (5.37) where f 1 , f 2 , f 3 are homogeneous polynomials on P 2 ×P 2 of degree (5,1). Here, to simplify our notation, we let x denote the coordinates on the first P 2 , x ≡ [x 0 : x 1 : x 2 ] and, similarly, y denotes the coordinates on the second P 2 , y ≡ [y 0 : y 1 : y 2 ]. The coefficients in the polynomials f 1 , f 2 , f 3 can be viewed as coordinates on H 1 (A, O A (−4, 5, 1)). As we computed in (5.29), there are 189 such coefficients. Since eventually we are interested in the moduli space of the vector bundle V on X, we restrict H 1 (A, O A (−4, 5, 1)) to its subspace consisting of elements v inv which are invariant under Z 3 × Z 3 . Since the discrete group does not act on P 1 (see eq. (3.5)), the elements r 0 and r 1 are automatically invariant. Hence, v inv is of the form (5.37) where the polynomials f 1 , f 2 , f 3 are restricted to be the Z 3 × Z 3 invariant polynomials of degree (5, 1). Let us introduce a basis for these invariant polynomials: The invariant polynomials f 1 , f 2 , f 3 are then given by where (a α , b α , c α ) are coordinates on the 21(=189/9)-dimensional invariant subspace of H 1 (A, O A (−4, 5, 1)). However, to obtain the invariant part of the extension class [ṼW ] we have to mod out by those elements which can be obtained by multiplying the defining polynomial F 1 by elements of H 1 (A, O A (−5, 2, 1)). We discuss this in detail in Appendix D. Here we simply state the result. Dividing by the the denominator in (5.32) is equivalent to imposing the following constraints on the coordinates (a α , b α , c α ): a 1 + a 2 + a 3 = 0 , a 4 + a 5 + a 6 = 0 , 6 The superpotential on X In this section, we will compute the leading non-perturbative superpotential, that is, ∼ p = e iT 1 (see eq. (3.30)), in a heterotic string vacuum specified by (X, V ). To simplify our analysis, we will perform the calculations for fixed complex structure. Then (det∂ O C i (−1) ) in (2.22) become numerical constants which will not play any role and will be ignored. Our aim in this section will be to compute the Pfaffians. First, we will calculate them on the covering spaceX and then on the quotient space X. Since we would like to compare these two calculations, in the theory onX we will restrict ourselves to the invariant component of the moduli space which will descend to X. The method of computing the Pfaffians will be similar to the one introduced in [21,22]. Since we do not know either the metric or the connection, we will rely on an algebraic approach whose essence is to understand under which conditions each Pfaffian vanishes. The conditions will be derived as a homogeneous polynomial equation on the moduli space.
Since the Pfaffian is a section of a line bundle on the moduli space and the moduli space is a projective space, this polynomial will be the Pfaffian up to a numerical coefficient which cannot be determined by our algebraic method. Let us now review the general condition for the vanishing of a Pfaffian on a holomorphic, isolated, genus 0 curve C [15,21,22]. The Pfaffian vanishes if and only if the operator∂ V C (−1) has a zero mode. The zero modes of∂ are elements of the Dolbeault cohomology group. In the present case, the cohomology group of interest is H 0 (C, V | C ⊗ O C (−1)). Hence, the Pfaffian vanishes if and only if h 0 (C, V | C ⊗ O C (−1)) = 0. Since h 0 (C, V | C ⊗ O C (−1)) is not a topological invariant, it depends on where we are in the moduli space of V . For generic values of the moduli, h 0 (C, V | C ⊗ O C (−1)) will be zero and ∂ V C (−1) will not have zero modes. However, at a specific co-dimension 1 subspace of the moduli space h 0 (C, V | C ⊗ O C (−1)) will jump-thus producing a zero mode. The Pfaffian of∂ V C (−1) will be determined by the equation defining this co-dimension 1 subspace.

Calculation of the Pfaffians
As was discussed in Section 3.4, onX there are 81 isolated curves of interest. These curves split into 9 orbits under the action of Z 3 × Z 3 with 9 curves in each orbit. If we restrict ourselves to the invariant part of the moduli space, all curves in the same orbit will give an identical contribution. Hence, in this case we need to compute the Pfaffians of the Dirac operator on any 9 curves which do not lie in the orbits of each other. An example of such curves was given in eqs. (3.41), (3.42), (3.10). Let us recall that these curves lie in different homology classes and, hence, have different areas measured by the Kahler form oñ X. However, they have the same area when measured using the invariant part of the Kahler form ω A |X. Therefore, the images of these curves in X have the same area with respect to the Kahler form ω X .
Let us now study under which conditions h 0 (C,Ṽ | C ⊗ O C (−1)) = 0 for the curves C of the type P 1 × x × y ⊂X ⊂ P 1 × P 2 × P 2 . We denote B = P 2 × P 2 and define p B to be the projection p B : A → B with fibers being P 1 . Now consider a particular extension elementW on A, At each point on B and for any line bundle L, p B * L is generated by the cohomology group of the fiber at this point; that is by H 0 (P 1 , L| P 1 ). Similarly, R 1 p B * L is generated by H 1 (P 1 , L| P 1 ). Clearly, for L of the form L = O A (m 1 , m 2 , m 3 ), we have L| P 1 = O P 1 (m 1 ). Then, using the Bott's formula, we compute that 1) . Note that the right hand side in eqs. (6.5) is independent of the choice of the extension representativẽ W.
Recall from eq. (5.32) that modulo the denominator-which will be taken into account later-the space of extensions is given by the elements where v can explicitly be written as (see (5.37)) v = r 2 0 f 1 (x, y) + r 0 r 1 f 2 (x, y) + r 2 1 f 3 (x, y) . (6.11) Comparing eq. (6.9) and eq. (6.10), we conclude that δ(Ṽ ) is given by multiplication by v. As was discussed around eq. (5.35), we can introduce the basis {t 0 , t 1 } for H 0 (P 1 , O P 1 (1)) and the dual basis ). To construct the matrix δ(Ṽ ), we simply multiply v by the basis elements and present the answer in the matrix form This gives If C is a curve corresponding to a specific point x × y ∈ P 2 × P 2 then we get After we evaluate f 1 f 3 − f 2 2 at points of P 2 × P 2 , the right hand side of (6.15) becomes a degree 2 homogeneous polynomial of the parameters of f 1 , f 2 , f 3 . For the purposes of our paper, we can restrict f 1 , f 2 , f 3 to be the invariant polynomials under Z 3 × Z 3 . They are explicitly given by eqs. (5.38), (5.39). Furthermore, we recall that to describe the extensions ofṼ rather than those ofṼ, we have to impose the relations (5.40). We also recall that to describe the moduli space ofṼ, we projectivize the corresponding space of extensions. This simply means that we view the parameters (a α , b α , c α ) of the polynomials f 1 , f 2 , f 3 as homogeneous coordinates. Since the moduli space is a projective space and (f 1 f 3 − f 2 2 )(x, y) is a homogeneous polynomial of degree 2, we conclude that (6.15) is a section of a line bundle of degree 2 on the moduli space. Finally, we notice that eq. (6.15) depends only on the coordinates of M(ṼW ). The coordinates of M(W ) drop out from our calculations.
From eq. (6.8) and the fact the Pfaffian is a section of a line bundle on the moduli space, we conclude that up to a numerical coefficient-which we are not able to compute by our method. Let us now apply this result to the curves (3.41), (3.43). Denote 9 . (6.17) Substituting the points (3.41), (3.43) into (6.17), we obtain the following expressions for R i (ζ = e 2πi/3 ): Note that every AX ,i is non-zero because the Pfaffians vanish only along the zero locus of the polynomials RX ,i and do not vanish identically. We are not able to compute the coefficients AX ,i by our algebraic method. However, it is possible to constrain them using the Beasley-Witten residue theorem, which we will now discuss.

The residue theorem onX
Our theory onX formally satisfies the assumptions of Beasley and Witten in [23], which we reviewed in Subsection 2.2. Indeed, the Calabi-Yau threefoldX is, by construction, a projective complete intersection manifold and the vector bundleṼ is the restriction of a vector bundleṼ. Nevertheless, as we discussed in Subsection 2.3, the residue theorem of Beasley-Witten is not directly applicable here since h 1,1 (X) > h 1,1 (A). However, indirectly we can still apply it. If we measure the area of curves inX using the (1,1) form ω A |X then, according to the residue theorem, the sum of the Pfaffians of all curves with the same area has to vanish. The 81 curves found in Subsection 3.4 have the same area with respect to ω A |X and, hence, we can apply the residue theorem to them. Since we are restricting ourselves to the invariant part of the moduli space, all curves in the same Z 3 × Z 3 orbit will have an identical Pfaffian. This means that it is enough to sum the Pfaffians of the curves which do not lie in the orbit of each other. These Pfaffians are given in eqs. (6.18) and (6.19). Hence, the residue theorem implies that These results will play a role in constructing the superpotential on the quotient Calabi-Yau manifold X.

The explicit formula for the superpotential on X
Since the curves C i corresponding to solutions (3.41), (3.43) lie in different Z 3 × Z 3 orbits, they descend to 9 different curves on X. To simplify our notation, we will still denote these curves by C i . The non-perturbative superpotential for these curves is then given by As was discussed in section 3, these 9 curves lie in the same homology class in H 2 (X, R) but in 9 different homology classes in H 2 (X, Z). This means that the contribution for each curve will pick up a distinct Z 3 × Z 3 character χ i . As we discussed in Subsection 2.4, as long as the curves in the same homology class in H 2 (X, Z) receive the same character and curves in different homology classes receive different characters, the distribution of the characters among the curves is arbitrary and depends on the choice of the connected component of the moduli space of the B-field.
To compute W X ([C]) in (6.22), we notice that since the Pfaffians in the previous subsection were computed for the invariant part of the moduli space. It follows that where the right hand side is given by eqs. (6.19), (6.18), (6.21). In particular, it follows that The reason is that, by construction, the gauge connection on V is the equivariant connection onṼ . For the trivial choice of the equivariant structure, it is just the connection onṼ restricted to the invariant part of the moduli space. Since in the previous subsection we restricted our calculations to the invariant part of the moduli space, the Dirac operators on both sides in (6.23) depend on the same connection and, hence, their Pfaffians are equal. Thus, the superpotential in (6.22) becomes where R X,i are also given by (6.18) and A X,i also satisfy the constraints (6.21). Now, the key observation is that, since the linear combination 9 i=1 A X,i R X,i = 0 due to the residue theorem, the linear combination in (6.25) 9 i=1 χ i A X,i R X,i is non-zero because it is twisted by the characters χ i , most of which are not unity.
Thus, we have explicitly demonstrated that in our model on X a non-vanishing, non-perturbative superpotential can be generated in the low-energy field theory.

Conclusion and future directions
In this paper, we presented examples of heterotic string compactifications with non-vanishing nonperturbative superpotentials. In our examples, the superpotential does not vanish on both the simply connected covering space and the non-simply connected manifold obtained as a quotient by the action of the discrete isometry group. In both cases, the reason for the non-vanishing of the superpotential can be attributed to the existence of holomorphic, isolated, genus 0 curves which are unique in their integer homology classes.
It would be interesting to generalize the ideas developed in this paper for realistic heterotic models and to compute non-perturbative superpotentials in a heterotic MSSM. The heterotic Standard Model constructed in [1,2,3,4] used a different Schoen manifold with a different action of Z 3 × Z 3 . Hence, it would be interesting to see if one can build a heterotic MSSM on the Schoen manifold used in this paper. Then one can extend the results of this paper to compute the non-perturbative superpotential in an MSSM, rather than in a toy model. The result is expected to be non-zero, as in our present examples.
Another possible direction is to apply our methods to realistic heterotic models obtained using the monad construction [5,6,7,8,9,33]. The crucial difference is that such models are built on projective Calabi-Yau manifolds satisfying h 1,1 (X) = h 1,1 (A). Then, according to the Beasley-Witten residue theorem, the non-perturbative superpotential vanishes on the covering manifoldX. However, on the quotient manifold X it might be non-zero because the second homology group of X is expected to contain discrete torsion. It would be interesting to see if one indeed can generate a non-perturbative superpotential in such models.

A The normal bundle to the curves inX
Here we will compute the normal bundle to the curves in Subsection 3.4. Specifically, we present our calculations for the curve specified by s 1 in eq. (3.41). The other curves can be treated similarly and give the same result.
The curve s 1 is of the form Let us first consider the short exact sequence relating the tangent bundle TX and the normal bundle NX ofX; that is where T A is the tangent bundle of the ambient space given by and we have used the fact that the tangent bundle of P 1 is O P 1 (2). Using eqs.
We now want to restrict the sequence (A.2) to the curve C 1 . For the curve of the form (A.1), we obtain The sequence (A.2) then becomes Let us now analyze the maps h 1 and h 2 . The map h 1 is defined as a map from tangent directions ∂ along A to the column vector (∂F 1 , ∂F 2 ) T . Since T A is of rank 5 and NX is of rank 2, h 1 is a 2 × 5 matrix. Evaluating the derivatives of F 1 and F 2 and restricting the results to the curve (A.1) gives Since the sequence (A.6) is exact, it follows that h 1 and h 2 satisfy the composition rule h 1 • h 2 = 0. This determines h 2 | C 1 to be up to an arbitrary holomorphic section h 0 on C 1 which is a homogeneous polynomials of degree k ≥ 0 in [t 0 : t 1 ]. Since any vector bundle on C 1 ≃ P 1 is a sum of line bundles, TX| C 1 must be of the form where, from (A.6), it follows that m 1 + m 2 + m 3 = 0. Examining the sequence (A.6), it is easy to see that these conditions have only one consistent possibility, namely m 1 = 2, m 2 = m 3 = −1. This means that Finally, let us consider the short exact sequence relating the tangent bundle T C 1 and the normal bundle Using eq. (A.9), we obtain This implies that the only possible form for NC 1 is

B Extension ofW andṼ
In this appendix, we calculate the number of extensions ofW andṼ and prove eq. (5.11). Our calculations will be similar to the ones performed in [3], where additional details can be found.

B.1 Extensions ofW
The extensions ofW are given by the dimension of the cohomology group Let us consider the direct image π 1 * L 1 ⊗ L * 2 under the projection π 1 in the diagram (3.10). Using the definitions of φ, τ 1 , τ 2 in (3.16), we can give L 1 ⊗ L * 2 in the form From the diagram (3.10), it follows that the projections satisfy Then we obtain Computing R 1 π 1 * L 1 ⊗ L * 2 , we find that To show this, note that at each point p on B 1 , R 1 π 1 * L 1 ⊗ L * 2 is generated by the first cohomology group on the elliptic fiber F p of the projection π 1 at p. Using eqs. (3.15), (3.16) we find that the line bundle OX (−2φ + τ 1 + 2τ 2 )| Fp has degree 3 and by the Kodaira vanishing theorem (see e.g [44]) the cohomology group in (B.6) vanishes. This proves (B.5). As the next step we similarly project (B.4) to the base of B 1 . We obtain we find that Since the higher direct images in eqs. (B.5), (B.8) vanish from a Leray spectral sequence, it follows that

B.2 Extensions ofṼ
The number of extensions ofṼ (for a fixed extensionW in [W ]) is given by H 1 (X,W ⊗ L * 3 ). To compute this cohomology group, we consider the short exact sequence The sequence (B.12) implies the following long exact sequence of cohomology groups 14) The cohomology of L 1 ⊗ L * 3 and L 2 ⊗ L * 3 can be computed using direct images, just as in the previous subsection. Using the identities [3] and following the same steps as in the previous subsection, we obtain Then from (B.14) we see that

C Stability ofW andṼ
Since we are only considering a toy model, we will not give a comprehensive proof thatW andṼ are stable. Instead, we examine the most important necessary condition for this to be the case. Let us recall that a vector bundleṼ onX is called stable if for any subsheaf S of lower rank we have µ(S) < µ(Ṽ ) . (C.1) Here, the slope µ(S) is defined by where ωX is the Kahler form onX. From eqs. (5.3), we observe that the line bundle L 1 injects intoW andW injects intoṼ . We now discuss whether L 1 andW destabilizeW andṼ respectively. Using the definition ofW in (5.3), we see thatW has rank 2 and its first Chern class is given by c 1 (L 1 )+c 1 (L 2 ). Then the condition µ(L 1 ) < µ(W ) can be stated as X (c 1 (L 1 ) − c 1 (L 2 )) ∧ ωX ∧ ωX < 0 ⇔ µ(L 1 ⊗ L * 2 ) < 0 .

(C.3)
Since L 1 and L 2 are equivariant and constructed out of the invariant classes, we can replace ωX in (C.3) with its invariant part ω X in (3.18). Using the expression for the invariant part of the Kahler form in (3.18), we can rewrite (C.3) in the form Using the triple intersection numbers (3.17), we then obtain the following inequality for the Kahler parameters: (t 3 ) 2 + 4(t 1 ) 2 + 6t 1 t 3 + 24t 1 t 2 − 6t 2 t 3 < 0 . (C.5) Let us now study the condition that µ(W ) < µ(Ṽ ). Note that c 1 (W ) = c 1 (L 1 ) + c 1 (L 2 ) and, since L 1 ⊗ L 2 ⊗ L 3 is trivial, it follows that c 1 (W ) = c 1 (L * 3 ). Also note that since c 1 (Ṽ ) = 0, it follows that µ(Ṽ ) = 0. Then the condition µ(W ) < µ(Ṽ ) can be stated as Using the triple intersection numbers (3.17), we then obtain the inequality The bundlesW andṼ are not destabilized if there exists a region in the Kahler moduli space where both inequalities (C.5) and (C.7) are simultaneously satisfied. It is easy to see that it is indeed the case. For example, if we take t 2 ≈ t 3 and t 1 ≪ t 2 , t 3 both inequalities are satisfied. where f 1 , f 2 , f 3 are invariant polynomials of degree (5, 1) on P 2 × P 2 and {r 0 , r 1 } is a basis in the vector space H 1 (P 1 , O P 1 (−3)) dual to the basis {t 0 , t 1 } in H 0 (P 1 , O P 1 (1)). The polynomials f 1 , f 2 , f 3 can be expanded in the basis (5.38) 3) The aim of this appendix is to describe the process of factoring out F 1 · H 1 (A, O A (−5, 2, 1)). This will give a parameterization of the invariant part of the moduli space ofṼ and of the moduli space of V . Consider an element u in H 1 (A, O A (−5, 2, 1)). Let us write it in the form similar to (D.2). Using the Kunneth and Bott formulas, we can express H 1 (A, O A (−5, 2, 1)) as H 1 (A, O A (−5, 2, 1)) = H 1 (P 1 , O P 1 (−5)) ⊗ H 0 (P 2 × P 2 , O P 2 ×P 2 (2, 1)) .
Performing the quotient action in (D.1) is now equivalent to finding the cokernel of the matrix F 1 in (D.11), (D.13). This is, in turn, equivalent to finding the kernel of the matrix (F 1 ) T which acts on the parameters of the polynomials f 1 , f 2 , f 3 ; that is, on the column vector (a 1 , . . . , a 7 , b 1 , . . . , b 7 , c 1 , . . . , c 7 ) T . Finding the kernel of (F 1 ) T means solving the linear system of equations