Spectral bundles and the DRY-Conjecture

Supersymmetric heterotic string models, built from a Calabi-Yau threefold X endowed with a stable vector bundle V, usually start from a phenomenologically motivated choice of a bundle V_v in the visible sector, the spectral cover construction on an elliptically fibered X being a prominent example. The ensuing anomaly mismatch between c₂(V_v) and c₂(X), or rather the corresponding differential forms, is often ‘solved’, on the cohomological level, by including a fivebrane. This leads to the question whether the difference can be alternatively realized by a further stable bundle. The ‘DRY’-conjecture of Douglas, Reinbacher and Yau in math.AG/0604597 gives a sufficient condition on cohomology classes on X to be realized as the Chern classes of a stable sheaf. In 1010.1644 [hep-th], we showed that infinitely many classes on X exist for which the conjecture is true. In this note, we give the sufficient condition for the mentioned fivebrane classes to be realized by a further stable bundle in the hidden sector. Using a result obtained in 1011.6246 [hep-th], we show that corresponding bundles exist, thereby confirming this version of the DRY-Conjecture.     

Geometric Model for Complex Non-Kähler Manifolds with <Emphasis Type="Italic">SU</Emphasis> (3) Structure

For a given complex n-fold M we present an explicit construction of all complex (n+1)-folds which are principal holomorphic T²-fibrations over M. For physical applications we consider the case of M being a Calabi-Yau 2-fold. We show that for such M, there is a subclass of the 3-folds that we construct, which has natural families of non-Kähler SU(3)-structures satisfying the conditions for supersymmetry in the heterotic string theory compactified on the 3-folds. We present examples in the aforementioned subclass with M being a K3-surface and a 4-torus.     

Heterotic string from a higher dimensional perspective

The (abelian bosonic) heterotic string effective action, equations of motion and Bianchi identity at order α^′ in ten dimensions, are shown to be equivalent to a higher dimensional action, its derived equations of motion and Bianchi identity. The two actions are the same up to the gauge fields: the latter are absorbed in the higher dimensional fields and geometry. This construction is inspired by heterotic T-duality, which becomes natural in this higher dimensional theory.We also prove the equivalence of the heterotic string supersymmetry conditions with higher dimensional geometric conditions. Finally, some known Kähler and non-Kähler heterotic solutions are shown to be trivially related from this higher dimensional perspective, via a simple exchange of directions. This exchange can be encoded in a heterotic T-duality, and it may also lead to new solutions.     

Heterotic String Compactification and New Vector Bundles

We propose a construction of Kähler and non-Kähler Calabi–Yau manifolds by branched double covers of twistor spaces. In this construction we use the twistor spaces of four-manifolds with self-dual conformal structures, with the examples of connected sum of n\({\mathbb{P}^{2}}\)s. We also construct K3-fibered Calabi–Yau manifolds from the branched double covers of the blow-ups of the twistor spaces. These manifolds can be used in heterotic string compactifications to four dimensions. We also construct stable and polystable vector bundles. Some classes of these vector bundles can give rise to supersymmetric grand unified models with three generations of quarks and leptons in four dimensions.     

Sequestering by global symmetries in Calabi-Yau string models

We study the possibility of realizing an effective sequestering between visible and hidden sectors in generic heterotic string models, generalizing previous work on orbifold constructions to smooth Calabi-Yau compactifications. In these theories, genuine sequestering is spoiled by interactions mixing chiral multiplets of the two sectors in the effective Kähler potential. These effective interactions however have a specific current-current-like structure and can be interpreted from an M-theory viewpoint as coming from the exchange of heavy vector multiplets. One may then attempt to inhibit the emergence of generic soft scalar masses in the visible sector by postulating a suitable global symmetry in the dynamics of the hidden sector. This mechanism is however not straightforward to implement, because the structure of the effective contact terms and the possible global symmetries is a priori model-dependent. To assess whether there is any robust and generic option, we study the full dependence of the Kähler potential on the moduli and the matter fields. This is well known for orbifold models, where it always leads to a symmetric scalar manifold, but much less understood for Calabi-Yau models, where it generically leads to a non-symmetric scalar manifold. We then examine the possibility of an effective sequestering by global symmetries, and argue that whereas for orbifold models this can be put at work rather naturally, for Calabi-Yau models it can only be implemented in rather peculiar circumstances.     

Constraining the Kähler Moduli in the Heterotic Standard Model

Phenomenological implications of the volume of the Calabi-Yau threefolds on the hidden and observable M-theory boundaries, together with slope stability of their corresponding vector bundles, constrain the set of Kähler moduli which give rise to realistic compactifications of the strongly coupled heterotic string. When vector bundles are constructed using extensions, we provide simple rules to determine lower and upper bounds to the region of the Kähler moduli space where such compactifications can exist. We show how small these regions can be, working out in full detail the case of the recently proposed Heterotic Standard Model. More explicitly, we exhibit Kähler classes in these regions for which the visible vector bundle is stable. On the other hand, there is no polarization for which the hidden bundle is stable.     

Supersymmetric configurations, geometric transitions and new non-Kähler manifolds

We give a detailed derivation of a supersymmetric configuration of wrapped D5 branes on a two-cycle of a warped resolved conifold. Our analysis reveals that the resolved conifold should support a non-Kähler metric with an SU(3) structure. We use this as a starting point of the geometric transition in type IIB theory. A mirror, and a subsequent flop transition using an intermediate M-theory configuration with a G₂ structure, gives rise to the complete IR geometric transition in type IIA theory. A further mirror transformation gives the type IIB gravity dual of the IR gauge theory on the wrapped D5 branes. Expectedly non-Kähler deformations of the resolved and the deformed conifolds appear as the gravity duals of the confining gauge theories in type IIA and type IIB theories respectively, although in more generic cases these manifolds could also be non-geometric. In the local limit we reproduce precisely the scenarios presented in our earlier works. Our present work should therefore be viewed as providing a supergravity proof of geometric transitions in the full global scenarios in type II theories.     

Exactly solvable string compactifications on manifolds of SU(N) holonomy

A variety of heterotic string compactifications on the K3 surface, manifolds of SU(3) holomony, and higher holomony manifolds, are solved exactly. An example of the quintic hypersurface in CP₄ is worked out in detail. It is conjectured, and demonstrated in part, that any supersymmetric compactification of the heterotic string with an N=2 superconformal theory is equivalent to a compactification on a manifold of SU(N) holonomy, and in particular an arbitrary gluing of the discrete models with c=9 gives a solvable heterotic string compactification on some Calabi-Yau manifold. Calabi-Yau compactifications are seen to be exact vacua of string theory, retaining their topological and geometrical characteristics. Previously unknown enhanced gauge symmetries are found to arise for certain backgrounds.     

The tangent bundle of a Calabi-Yau manifold-deformations and restriction to rational curves

The tangent bundle _X of a Calabi-Yau threefoldX is the only known example of a stable bundle with non-trivial restriction to any rational curve onX. By deforming the direct sum of _X and the trivial line bundle one can try to obtain new examples. We use algebro-geometric techniques to derive results in this direction. The relation to the finiteness of rational curves on Calabi-Yau threefolds is discussed.     

Solutions of the Strominger System via Stable Bundles on Calabi-Yau Threefolds

We prove that a given Calabi-Yau threefold with a stable holomorphic vector bundle can be perturbed to a solution of the Strominger system provided that the second Chern class of the vector bundle is equal to the second Chern class of the tangent bundle. If the Calabi-Yau threefold has strict SU(3) holonomy then the equations of motion derived from the heterotic string effective action are also satisfied by the solutions we obtain.     

Non-perturbative superpotentials in string theory

The non-perturbative superpotential can be effectively calculated in M-theory compactification to three dimensions on a Calabi-Yau four-fold X. For certain X, the superpotential is identically zero, while for other X, a non-perturbative superpotential is generated. Using F-theory, these results carry over to certain Type IIB and heterotic string compactifications to four dimensions with N = 1 supersymmetry. In the heterotic string case, the non-perturbative superpotential can be interpreted as coming from space-time and world-sheet instantons; in many simple cases contributions come only from finitely many values of the instanton numbers.     

Cohomological Yang–Mills theories on Kähler 3-folds

We study topological gauge theories with N_c=(2,0) supersymmetry based on stable bundles on general Kähler 3-folds. In order to have a theory that is well defined and well behaved, we consider a model based on an extension of the usual holomorphic bundle by including a holomorphic 3-form. The correlation functions of the model describe complex 3-dimensional generalizations of Donaldson–Witten type invariants. We show that the path integral can be written as a sum of contributions from stable bundles and a complex 3-dimensional version of Seiberg–Witten monopoles. We study certain deformations of the theory, which allow us to consider the situation of reducible connections. We shortly discuss situations of reduced holonomy. After dimensional reduction to a Kähler 2-fold, the theory reduces to Vafa–Witten theory. On a Calabi–Yau 3-fold, the supersymmetry is enhanced to N_c=(2,2). This model may be used to describe classical limits of certain compactifications of (matrix) string theory.     

Non-perturbative gravitational corrections in a class of N = 2 string duals

We investigate the non-perturbative equivalence of some heterotic/type II dual pairs with N = 2 supersymmetry. The perturbative heterotic scalar manifolds are respectively SU(1, 1)/U(1) × SO(2,2+N_V)/SO(2) × SO(2+N_V) and SO(4,4+N_H)/SO(4) × SO(4+N_H) for moduli in the vector multiplets and hypermultiplets. The models under consideration correspond, on the type II side, to self-mirror Calabi-Yau threefolds with Hodge numbers h^1,1 = N_V + 3 = h^2,1 = N_H + 3, which are K3 fibrations. We consider three classes of dual pairs, with N_V = N_H = 8, 4 and 2. The models with h^1,1 = 7 and 5 provide new constructions, while the h^1,1 = 11, already studied in the literature, is reconsidered here. Perturbative R²-like corrections are computed on the heterotic side by using a universal operator whose amplitude has no singularities in the (T, U) space, and can therefore be compared with the type II side result. We point out several properties connecting K3 fibrations and spontaneous breaking of the N = 4 supersymmetry to N = 2. As a consequence of the reduced S- and T- duality symmetries, the instanton numbers in these three classes are restricted to integers, which are multiples of 2, 2 and 4, for N_V = 8, 4 and 2, respectively.     

CNM models,holomorphic functions and projective superspace c-maps

Continuing the investigation of CNM (chiral-non-minimal) hypermultiplet non-linear σ-models, we propose extensions of the concept of the c-map which relate holomorphic functions to hyper-Kähler geometrics. In particular, we show that a whole series of hyper-Kähler potentials can be derived by replacing the role of the 4D, N = 1 tensor multiplet in the original c-map by 4D, N = 1 non-minimal multiplets and auxiliary superfields. The resulting N = 2 models appear to have interesting connections to Calabi-Yau manifolds and algebraic varieties. These models also emphasize the fact that special hyper-Kähler manifolds (the analogs of special Kähler manifolds) without isometries exist.     

Gauge Theoretical Construction of Non-compact Calabi-Yau Manifolds

We construct the non-compact Calabi-Yau manifolds interpreted as the complex line bundles over the Hermitian symmetric spaces. These manifolds are the various generalizations of the complex line bundle over CP^N−1. Imposing an F-term constraint on the line bundle over CP^N−1, we obtain the line bundle over the complex quadric surface Q^N−2. On the other hand, when we promote the U(1) gauge symmetry in CP^N−1 to the non-abelian gauge group U(M), the line bundle over the Grassmann manifold is obtained. We construct the non-compact Calabi-Yau manifolds with isometries of exceptional groups, which we have not discussed in the previous papers. Each of these manifolds contains the resolution parameter which controls the size of the base manifold, and the conical singularity appears when the parameter vanishes.     

Vector Bundle Moduli

We give the general presciption for calculating the number of moduli of irreducible, stable U(n) holomorphic vector bundles with positive spectral covers over elliptically fibered Calabi–Yau threefolds. Explicit results are presented for Hirzebruch base surfaces B = F _r. Vector bundle moduli appear as gauge singlet scalar fields in the effective low-energy actions of heterotic superstrings and heterotic M-theory.     

Toric <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> and <Emphasis Type="Italic">Spin</Emphasis>(7) Holonomy Spaces from Gravitational Instantons and Other Examples

Non-compact G ₂ holonomy metrics that arise from a T ² bundle over a hyper-Kähler space are constructed. These are one parameter deformations of certain metrics studied by Gibbons, Lü, Pope and Stelle in [1]. Seven-dimensional spaces with G ₂ holonomy fibered over the Taub-Nut and the Eguchi-Hanson gravitational instantons are found, together with other examples. By using the Apostolov-Salamon theorem [2], we construct a new example that, still being a T ² bundle over hyper-Kähler, represents a non-trivial two parameter deformation of the metrics studied in [1]. We then review the Spin(7) metrics arising from a T ³ bundle over a hyper-Kähler and we find a two parameter deformation of such spaces as well. We show that if the hyper-Kähler base satisfies certain properties, a non-trivial three parameter deformation is also possible. The relation between these spaces with half-flat and almost G ₂ holonomy structures is briefly discussed.