Calabi-Yau Hypersurfaces Research Articles

Families of Calabi–Yau hypersurfaces in [formula omitted]-Fano toric varieties

We provide a sufficient condition for a general hypersurface in a Q-Fano toric variety to be a Calabi–Yau variety in terms of its Newton polytope. Moreover, we define a generalization of the Berglund–Hübsch–Krawitz construction in case the ambient is a Q-Fano toric variety with torsion free class group and the defining polynomial is not necessarily of Delsarte type. Finally, we introduce a duality between families of Calabi–Yau hypersurfaces which includes both Batyrev and Berglund–Hübsch–Krawitz mirror constructions. This is given in terms of a polar duality between pairs of polytopes Δ1⊆Δ2, where Δ1 and Δ2⁎ are canonical.

Open mirror symmetry for higher dimensional Calabi-Yau hypersurfaces

Compactifications with fluxes and branes motivate us to study various enumerative invariants of Calabi-Yau manifolds. In this paper, we study non-perturbative corrections depending on both open and closed string moduli for a class of compact Calabi-Yau manifolds in general dimensions. Our analysis is based on the methods using relative cohomology and generalized hypergeometric system. For the simplest example of compact Calabi-Yau fivefold, we explicitly derive the associated Picard-Fuchs differential equations and compute the quantum corrections in terms of the open and closed flat coordinates. Implications for a kind of open-closed duality are also discussed.

Chain integral solutions to tautological systems

We give a new geometrical interpretation of the local analytic solutions to a differential system, which we call a tautological system τ, arising from the universal family of Calabi–Yau hypersurfaces Y_a in a G-variety X of dimension n. First, we construct a natural topological correspondence between relative cycles in H_n (X−Y_a, ∪ D−Y_a) bounded by the union of G-invariant divisors ∪D in X to the solution sheaf of τ, in the form of chain integrals. Applying this to a toric variety with torus action, we show that in addition to the period integrals over cycles in Y_a, the new chain integrals generate the full solution sheaf of a GKZ system. This extends an earlier result for hypersurfaces in a projective homogeneous variety, whereby the chains are cycles [3, 7]. In light of this result, the mixed Hodge structure of the solution sheaf is now seen as the MHS of H_n (X−Y_a,∪ D−Y_a). In addition, we generalize the result on chain integral solutions to the case of general type hypersurfaces. This chain integral correspondence can also be seen as the Riemann–Hilbert correspondence in one homological degree. Finally, we consider interesting cases in which the chain integral correspondence possibly fails to be bijective.

Mirror symmetry, D-brane superpotentials and Ooguri–Vafa invariants of Calabi–Yau manifolds**Supported by Y4JT01VJ01 and NSFC(11475178)

The D-brane superpotential is very important in the low energy effective theory. As the generating function of all disk instantons from the worldsheet point of view, it plays a crucial role in deriving some important properties of the compact Calabi–Yau manifolds. By using the generalized GKZ hypergeometric system, we will calculate the D-brane superpotentials of two non-Fermat type compact Calabi–Yau hypersurfaces in toric varieties, respectively. Then according to the mirror symmetry, we obtain the A-model superpotentials and the Ooguri–Vafa invariants for the mirror Calabi–Yau manifolds.

The $$F$$ F -pure threshold of a Calabi–Yau hypersurface

We compute the \(F\)-pure threshold of the affine cone over a Calabi–Yau hypersurface, and relate it to the order of vanishing of the Hasse invariant on the versal deformation space of the hypersurface.

Extremal Transitions from Nested Reflexive Polytopes

Using an inclusion of one reflexive polytope into another is a well-known strategy for connecting the moduli spaces of two Calabi-Yau families. In this paper we look at the question of when an inclusion of reflexive polytopes determines a torically-defined extremal transition between smooth Calabi-Yau hypersurface families. We show this is always possible for reflexive polytopes in dimensions two and three. However, in dimension four and higher, obstructions can occur. This leads to a smooth projective family of Calabi-Yau threefolds that is birational to one of Batyrev's hypersurface families, but topologically distinct from all such families.

Heavy tails in Calabi-Yau moduli spaces

We study the statistics of the metric on K\"ahler moduli space in compactifications of string theory on Calabi-Yau hypersurfaces in toric varieties. We find striking hierarchies in the eigenvalues of the metric and of the Riemann curvature contribution to the Hessian matrix: both spectra display heavy tails. The curvature contribution to the Hessian is non-positive, suggesting a reduced probability of metastability compared to cases in which the derivatives of the K\"ahler potential are uncorrelated. To facilitate our analysis, we have developed a novel triangulation algorithm that allows efficient study of hypersurfaces with $h^{1,1}$ as large as 25, which is difficult using algorithms internal to packages such as Sage. Our results serve as input for statistical studies of the vacuum structure in flux compactifications, and of the distribution of axion decay constants in string theory.

The moduli space of heterotic line bundle models: a case study for the tetra-quadric

It has recently been realised that polystable, holomorphic sums of line bundles over smooth Calabi-Yau three-folds provide a fertile ground for heterotic model building. Large numbers of phenomenologically promising such models have been constructed for various classes of Calabi-Yau manifolds. In this paper we focus on a case study for the tetra-quadric - a Calabi-Yau hypersurface embedded in a product of four CP1 spaces. We address the question of finiteness of the class of consistent and physically viable line bundle models constructed on this manifold. Further, for a specific semi-realistic example, we explore the embedding of the line bundle sum into the larger moduli space of non-Abelian bundles, both by means of constructing specific polystable non-Abelian bundles and by turning on VEVs in the associated low-energy theory. In this context, we explore the fate of the Higgs doublets as we move in bundle moduli space. The non-Abelian compactifications thus constructed lead to SU(5) GUT models with an additional global B-L symmetry. The non-Abelian compactifications inherit many of the appealing phenomenological features of the Abelian model, such as the absence of dimension four and dimension five operators triggering fast proton decay.

Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space

We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d > 2 (for example, d = 3 is the quintic three-fold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the `d-dimensional pair of pants'; the introduction of the `relative Fukaya category', and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an `orbifold' Fukaya category); a Morse-Bott model for the relative Fukaya category that allows one to make explicit computations; and the introduction of certain graded categories of matrix factorizations mirror to the relative Fukaya category.

Line bundle twisted chiral de Rham complex and bound states of D-branes on toric manifolds

In this note we calculate elliptic genus in various examples of twisted chiral de Rham complex on two-dimensional toric compact manifolds and Calabi–Yau hypersurfaces in toric manifolds. At first the elliptic genus is calculated for the line bundle twisted chiral de Rham complex on a compact smooth toric manifold and K3 hypersurface in P3. Then we twist chiral de Rham complex by sheaves localized on positive codimension submanifolds in P2 and calculate in each case the elliptic genus. In the last example the elliptic genus of chiral de Rham complex on P2 twisted by SL(N) vector bundle with instanton number k is calculated. In all the cases considered we find the infinite tower of open string oscillator contributions and identify directly the open string boundary conditions of the corresponding bound state of D-branes.

Nef and big divisors on toric weak Fano 3-folds

We show that a nef and big line bundle whose adjoint bundle has non-zero global sections on a nonsingular toric weak Fano 3-fold is normally generated. As a consequence, we see that any ample line bundle on a nonsingular toric waek Fano 3-fold is normally generated. As an application, we see that an ample line bundle on a Calabi-Yau hypersurface in a nonsingular toric Fano 4-fold is normally generated. Mathematics Subject Classification: Primary 14M25; Secondary 14M30, 14M45, 52B20

Monodromy of inhomogeneous Picard-Fuchs equations

We study low-degree curves on one-parameter Calabi-Yau hypersurfaces, and their contribution to the space-time superpotential in a superstring compactification with D-branes. We identify all lines that are invariant under at least one permutation of the homogeneous variables, and calculate the inhomogeneous Picard-Fuchs equation. The irrational large volume expansions satisfy the recently discovered algebraic integrality. The bulk of our work is a careful study of the topological integrality of monodromy under navigation around the complex structure moduli space. This is a powerful method to recover the single undetermined integration constant that is itself also of arithmetic significance. The examples feature a variety of residue fields, both abelian and non-abelian extensions of the rationals, thereby providing a glimpse of the arithmetic D-brane landscape.

OPEN VIRTUAL STRUCTURE CONSTANTS AND MIRROR COMPUTATION OF OPEN GROMOV–WITTEN INVARIANTS OF PROJECTIVE HYPERSURFACES

In this paper, we generalize Walcher's computation of the open Gromov–Witten invariants of the quintic hypersurface to Fano and Calabi–Yau projective hypersurfaces. Our main tool is the open virtual structure constants. We also propose the generalized mirror transformation for the open Gromov–Witten invariants, some parts of which are proven explicitly. We also discuss possible modification of the multiple covering formula for the case of higher-dimensional Calabi–Yau manifolds. The generalized disk invariants for some Calabi–Yau and Fano manifolds are shown and they are certainly integers after resummation by the modified multiple covering formula. This paper also contains the direct integration method of the period integrals for higher-dimensional Calabi–Yau hypersurfaces in the Appendix.

Another representation of the β form of the inhomogeneous Picard-Fuchs equation

In this letter we give another representation of the β form in the inhomogeneous Picard-Fuchs equation for open topological string for some one-parameter Calabi-Yau hypersurfaces in weighted projective spaces. Furthermore, the corresponding domain wall tensions calculated by using these β forms are consistent with the results that appear in literature. The β form is essential for the calculation of the D-brane domain wall tension, and a convenient choice of β forms should simplify the calculation. The freedom of the choice of β forms shows some symmetries in Calabi-Yau space.

Mirror Map as Generating Function of Intersection Numbers: Toric Manifolds with Two Kähler Forms

In this paper, we extend our geometrical derivation of expansion coefficients of mirror maps by localization computation to the case of toric manifolds with two K\"ahler forms. Especially, we take Hirzebruch surfaces F_{0}, F_{3} and Calabi-Yau hypersurface in weighted projective space P(1,1,2,2,2) as examples. We expect that our results can be easily generalized to arbitrary toric manifold.

Toric elliptic fibrations and F-theory compactifications

The 102581 flat toric elliptic fibrations over P^2 are identified among the Calabi-Yau hypersurfaces that arise from the 473800776 reflexive 4-dimensional polytopes. In order to analyze their elliptic fibration structure, we describe the precise relation between the lattice polytope and the elliptic fibration. The fiber-divisor-graph is introduced as a way to visualize the embedding of the Kodaira fibers in the ambient toric fiber. In particular in the case of non-split discriminant components, this description is far more accurate than previous studies. The discriminant locus and Kodaria fibers groups of all 102581 elliptic fibrations are computed. The maximal gauge group is SU(27), which would naively be in contradiction with 6-dimensional anomaly cancellation.

Picard-Fuchs equations for relative periods and Abel-Jacobi map for Calabi-Yau hypersurfaces

We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a Calabi-Yau hypersurface in toric variety, and deriving a general formula for the $d$-exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for the Abel-Jacobi map, which has played an important role in the recent study of D-branes (by Morrison and Walcher). Using the variation formalism, we prove that the relative periods of toric B-branes on Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system proposed in physics literature (by Alim, Hecht, Mayr, and Mertens), and discuss the relations between several works in the recent study of open string mirror symmetry. We also give the general solutions to the enhanced hypergeometric system.

Homological mirror symmetry for the quintic 3–fold

We prove homological mirror symmetry for the quintic Calabi-Yau 3-fold. The proof follows that for the quartic surface by Seidel (arXiv:math/0310414) closely, and uses a result of Sheridan (arXiv:1012.3238). In contrast to Sheridan's approach (arXiv:1111.0632), our proof gives the compatibility of homological mirror symmetry for the projective space and its Calabi-Yau hypersurface.

Hypersurfaces in Mori dream spaces

Let X be a hypersurface of a Mori dream space Z. We provide necessary and sufficient conditions for the Cox ring R(X) of X to be isomorphic to R(Z)/(f), where R(Z) is the Cox ring of Z and f is a defining section for X. We apply our results to Calabi–Yau hypersurfaces of toric Fano fourfolds. Our second application is to general degree d hypersurfaces in Pn containing a linear subspace of dimension n−2.

Chow rings and decomposition theorems for K3 surfaces and Calabi–Yau hypersurfaces

The decomposition theorem for smooth projective morphisms $\pi:\mathcal{X}\rightarrow B$ says that $R\pi_*\mathbb{Q}$ decomposes as $\oplus R^i\pi_*\mathbb{Q}[-i]$. We describe simple examples where it is not possible to have such a decomposition compatible with cup-product, even after restriction to Zariski dense open sets of $B$. We prove however that this is always possible for families of $K3$ surfaces (after shrinking the base), and show how this result relates to a result by Beauville and the author on the Chow ring of $K3$ surfaces $S$. We give two proofs of this result, the second one involving a certain decomposition of the small diagonal in $S^3$ also proved by Beauville and the author}. We prove an analogue of such a decomposition of the small diagonal in $X^3$ for Calabi-Yau hypersurfaces $X$ in $\mathbb{P}^n$, which in turn provides strong restrictions on their Chow ring.