Complete Intersection Calabi-Yau Research Articles

Heterotic instanton superpotentials from complete intersection Calabi-Yau manifolds

We study Pfaffians that appear in non-perturbative superpotential terms arising from worldsheet instantons in heterotic theories. A result by Beasley and Witten shows that these instanton contributions cancel among curves within a given homology class for Calabi-Yau manifolds that can be described as hypersurfaces or complete intersections in projective or toric ambient spaces. We provide a prescription that identifies all ℙ1 curves in certain homology classes of complete intersection Calabi-Yau manifolds in products of projective spaces (CICYs) and cross-check our results by a comparison with the genus zero Gromov-Witten invariants. We then use this construction to study instanton superpotentials on those manifolds and their quotients. We identify a non-toric quotient of a non-favorable CICY with a single genus zero curve in a certain homology class, so that a cancellation à la Beasley-Witten is not possible. In another example, we study a non-toric quotient of a favorable CICY and check that the superpotential still vanishes. From this and related examples, we conjecture that the Beasley-Witten cancellation result can be extended to toric and non-toric quotients of CICYs, but can be avoided if the CICY is non-favorable.

Equivariant A-twisted GLSM and Gromov-Witten invariants of CY 3-folds in Grassmannians

We compute genus-zero Gromov-Witten invariants of Calabi-Yau complete intersection 3-folds in Grassmannians using supersymmetric localization in A-twisted nonAbelian gauged linear sigma models. We also discuss a Seiberg-like duality interchanging Gr(n, m) and Gr(m − n, m).

Minuscule Schubert Varieties and Mirror Symmetry

We consider smooth complete intersection Calabi-Yau 3-folds in minuscule Schubert varieties, and study their mirror symmetry by degenerating the ambient Schubert varieties to Hibi toric varieties. We list all possible Calabi-Yau 3-folds of this type up to deformation equivalences, and find a new example of smooth Calabi-Yau 3-folds of Picard number one; a complete intersection in a locally factorial Schubert variety ${\boldsymbol{\Sigma}}$ of the Cayley plane ${\mathbb{OP}}^2$. We calculate topological invariants and BPS numbers of this Calabi-Yau 3-fold and conjecture that it has a non-trivial Fourier-Mukai partner.

Stringy Chern classes of singular toric varieties and their applications

Let X be a normal projective Q-Gorenstein variety with at worst log-terminal singularities. We prove a formula expressing the total stringy Chern class of a generic complete intersection in X via the total stringy Chern class of X. This formula is motivated by its applications to mirror symmetry for Calabi-Yau complete intersections in toric varieties. We compute stringy Chern classes and give a combinatorial interpretation of the stringy Libgober-Wood identity for arbitrary projective Q-Gorenstein toric varieties. As an application we derive a new combinatorial identity relating d-dimensional reflexive polytopes to the number 12 in dimension d>3.

Holomorphic Yukawa couplings for complete intersection Calabi-Yau manifolds

We develop methods to compute holomorphic Yukawa couplings for heterotic compactifications on complete intersection Calabi-Yau manifolds, generalising results of an earlier paper for Calabi-Yau hypersurfaces. Our methods are based on constructing the required bundle-valued forms explicitly and evaluating the relevant integrals over the projective ambient space. We also show how our approach relates to an earlier, algebraic one to calculate the holomorphic Yukawa couplings. A vanishing theorem, which we prove, implies that certain Yukawa couplings allowed by low-energy symmetries are zero due to topological reasons. To illustrate our methods, we calculate Yukawa couplings for SU(5)-based standard models on a co-dimension two complete intersection manifold.

Landau–Ginzburg/Calabi–Yau correspondence for the complete intersections X3,3 and X2,2,2,2

We define a generalization of Fan–Jarvis–Ruan–Witten theory, a “hybrid” model associated to a collection of quasihomogeneous polynomials of the same weights and degree, which is expected to match the Gromov–Witten theory of the Calabi–Yau complete intersection cut out by the polynomials. In genus zero, we prove that the correspondence holds for any such complete intersection of dimension three in ordinary, rather than weighted, projective space. These results generalize those of Chiodo–Ruan for the quintic threefold, and as in that setting, Givental's quantization can be used to yield a conjectural relation between the full higher-genus theories.

Hodge numbers for CICYs with symmetries of order divisible by 4

We compute the Hodge numbers for the quotients of complete intersection Calabi-Yau three-folds by groups of orders divisible by 4. We make use of the polynomial deformation method and the counting of invariant K\"ahler classes. The quotients studied here have been obtained in the automated classification of V. Braun. Although the computer search found the freely acting groups, the Hodge numbers of the quotients were not calculated. The freely acting groups, $G$, that arise in the classification are either $Z_2$ or contain $Z_4$, $Z_2 \times Z_2$, $Z_3$ or $Z_5$ as a subgroup. The Hodge numbers for the quotients for which the group $G$ contains $Z_3$ or $Z_5$ have been computed previously. This paper deals with the remaining cases, for which $G \supseteq Z_4$ or $G\supseteq Z_2 \times Z_2$. We also compute the Hodge numbers for 99 of the 166 CICY's which have $Z_2$ quotients.

Gromov-Witten/Pairs correspondence for the quintic 3-fold

We use the Gromov-Witten/Pairs (GW/P) descendent correspondence for toric 3-folds and degeneration arguments to establish the GW/P correspondence for several compact Calabi-Yau (CY) 3-folds (including all CY complete intersections in products of projective spaces). A crucial aspect of the proof is the study of the GW/P correspondence for descendents in relative geometries. Projective bundles over surfaces relative to a section play a special role. The GW/P correspondence for Calabi-Yau complete intersections provides a structure result for the Gromov-Witten invariants in a fixed curve class. After a change of variables, the Gromov-Witten series is a rational function in the variable − q = e i u -q=e^{iu} invariant under q ↔ q − 1 q \leftrightarrow q^{-1} .

Globally generated vector bundles on complete intersection Calabi–Yau threefolds

We investigate the globally generated vector bundles on complete intersection Calabi-Yau threefolds with the first Chern class at most 2. We classify all the globally generated vector bundles of an arbitrary rank on quintic in $\mathbb{P}^4$ and investigate the globally generated vector bundles of rank 2 on complete intersection Calabi-Yau threefolds of codimension 2.

On smooth Gorenstein polytopes

A Gorenstein polytope of index $r$ is a lattice polytope whose $r$th dilate is a reflexive polytope. These objects are of interest in combinatorial commutative algebra and enumerative combinatorics, and play a crucial role in Batyrev's and Borisov's computation of Hodge numbers of mirror-symmetric generic Calabi-Yau complete intersections. In this paper we report on what is known about smooth Gorenstein polytopes, i.e., Gorenstein polytopes whose normal fan is unimodular. We classify $d$-dimensional smooth Gorenstein polytopes with index larger than $(d+3)/3$. Moreover, we use a modification of Obro's algorithm to achieve classification results for smooth Gorenstein polytopes in low dimensions. The first application of these results is a database of all toric Fano $d$-folds whose anticanonical divisor is divisible by an integer $r$ satisfying $r \ge d-7$. As a second application we verify that there are only finitely many families of Calabi-Yau complete intersections of fixed dimension that are associated to a smooth Gorenstein polytope via the Batyrev-Borisov construction.

Prioritary omalous bundles on Hirzebruch surfaces

An irreducible algebraic stack is called unirational if there exists a surjective morphism, representable by algebraic spaces, from a rational variety to an open substack. We prove unirationality of the stack of prioritary omalous bundles on Hirzebruch surfaces, which implies also the unirationality of the moduli space of omalous H -stable bundles for any ample line bundle H on a Hirzebruch surface (compare with Costa and Miro-Ŕoig, 2002). To this end, we find an explicit description of the duals of omalous rank-two bundles with a vanishing condition in terms of monads. Since these bundles are prioritary, we conclude that the stack of prioritary omalous bundles on a Hirzebruch surface different from P 1 × P 1 is dominated by an irreducible section of a Segre variety, and this linear section is rational (Ionescu, 2015). In the case of the space quadric, the stack has been explicitly described by N. Buchdahl. As a main tool we use Buchdahl’s Beilinson-type spectral sequence. Monad descriptions of omalous bundles on hypersurfaces in P 4 , Calabi–Yau complete intersection, blowups of the projective plane and Segre varieties have been recently obtained by A.A. Henni and M. Jardim (Henni and Jardim, 2013), and monads on Hirzebruch surfaces have been applied in a different context in Bartocci et al. (2015).

Moduli spaces of Calabi–Yau complete intersections

In this short note, based on the work [7] in 1994, we describe compactifications of moduli spaces of Calabi–Yau complete intersections in Gorenstein toric Fano varieties.

Mirror transitions and the Batyrev-Borisov construction

We consider examples of extremal transitions between families of Calabi-Yau complete intersection threefolds in toric varieties, which are induced by toric embeddings of one toric variety into the other. We show that the toric map induced by the linear dual of the embedding induces a birational morphism between the mirror Calabi-Yau families, and in one case show that it can be extended to the full mirror transition between mirror families.

Wilson lines and Chern-Simons flux in explicit heterotic Calabi-Yau compactifications

We study to what extent Wilson lines in heterotic Calabi-Yau compactifications lead to non-trivial H-flux via Chern-Simons terms. Wilson lines are basic ingredients for Standard Model constructions but their induced H-flux may affect the consistency of the leading order background geometry and of the two-dimensional worldsheet theory. Moreover H-flux in heterotic compactifications would play an important role for moduli stabilization and could strongly constrain the supersymmetry breaking scale. We show how to compute H-flux and the corresponding superpotential, given an explicit complete intersection Calabi-Yau compactification and choice of Wilson lines. We do so by classifying special Lagrangian submanifolds in the Calabi-Yau, understanding how the Wilson lines project onto these submanifolds, and computing their Chern-Simons invariants. We illustrate our procedure with the quintic hypersurface as well as the split-bicubic, which can provide a potentially realistic three generation model.

Complete intersection Calabi-Yau six-folds

Complete intersection Calabi-Yau six-folds

Topological invariants and fibration structure of complete intersection Calabi-Yau four-folds

We investigate the mathematical properties of the class of Calabi-Yau four-folds recently found in [arXiv:1303.1832]. This class consists of 921,497 configuration matrices which correspond to manifolds that are described as complete intersections in products of projective spaces. For each manifold in the list, we compute the full Hodge diamond as well as additional topological invariants such as Chern classes and intersection numbers. Using this data, we conclude that there are at least 36,779 topologically distinct manifolds in our list. We also study the fibration structure of these manifolds and find that 99.95 percent can be described as elliptic fibrations. In total, we find 50,114,908 elliptic fibrations, demonstrating the multitude of ways in which many manifolds are fibered. A sub-class of 26,088,498 fibrations satisfy necessary conditions for admitting sections. The complete data set can be downloaded at http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/Cicy4folds/index.html .

Localization Computation of One-Point Disk Invariants of Projective Calabi–Yau Complete Intersections

We define one-point disk invariants of a smooth projective Calabi-Yau (CY) complete intersection (CI) in the presence of an anti-holomorphic involution via localization. We show that these invariants are rational numbers and obtain a formula for them which confirms, in particular, a conjecture by Jinzenji-Shimizu.

Mirror symmetry for closed, open, and unoriented Gromov–Witten invariants

In the first part of this paper, we obtain mirror formulas for twisted genus 0 two-point Gromov–Witten (GW) invariants of projective spaces and for the genus 0 two-point GW-invariants of Fano and Calabi–Yau complete intersections. This extends previous results for projective hypersurfaces, following the same approach, but we also completely describe the structure coefficients in both cases and obtain relations between these coefficients that are vital to the applications to mirror symmetry in the rest of this paper. In the second and third parts of this paper, we confirm Walcher's mirror symmetry conjectures for the annulus and Klein bottle GW-invariants of Calabi–Yau complete intersection threefolds; these applications are the main results of this paper. In a separate paper, the genus 0 two-point formulas are used to obtain mirror formulas for the genus 1 GW-invariants of all Calabi–Yau complete intersections.

Determinantal Quintics and Mirror Symmetry of Reye Congruences

We study a certain family of determinantal quintic hypersurfaces in $\mathbb{P}^{4}$ whose singularities are similar to the well-studied Barth-Nieto quintic. Smooth Calabi-Yau threefolds with Hodge numbers $(h^{1,1},h^{2,1})=(52,2)$ are obtained by taking crepant resolutions of the singularities. It turns out that these smooth Calabi-Yau threefolds are in a two dimensional mirror family to the complete intersection Calabi-Yau threefolds in $\mathbb{P}^{4}\times\mathbb{P}^{4}$ which have appeared in our previous study of Reye congruences in dimension three. We compactify the two dimensional family over $\mathbb{P}^{2}$ and reproduce the mirror family to the Reye congruences. We also determine the monodromy of the family over $\mathbb{P}^{2}$ completely. Our calculation shows an example of the orbifold mirror construction with a trivial orbifold group.

Elliptic fibrations with rank three Mordell-Weil group: F-theory with U(1)×U(1)×U(1) gauge symmetry

We analyze general F-theory compactifications with U(1) x U(1) x U(1) Abelian gauge symmetry by constructing the general elliptically fibered Calabi-Yau manifolds with a rank three Mordell-Weil group of rational sections. The general elliptic fiber is shown to be a complete intersection of two non-generic quadrics in P^3 and resolved elliptic fibrations are obtained by embedding the fiber as the generic Calabi-Yau complete intersection into Bl_3 P^3, the blow-up of P^3 at three points. For a fixed base B, there are finitely many Calabi-Yau elliptic fibrations. Thus, F-theory compactifications on these Calabi-Yau manifolds are shown to be labeled by integral points in reflexive polytopes constructed from the nef-partition of Bl_3 P^3. We determine all 14 massless matter representations to six and four dimensions by an explicit study of the codimension two singularities of the elliptic fibration. We obtain three matter representations charged under all three U(1)-factors, most notably a tri-fundamental representation. The existence of these representations, which are not present in generic perturbative Type II compactifications, signifies an intriguing universal structure of codimension two singularities of the elliptic fibrations with higher rank Mordell-Weil groups. We also compute explicitly the corresponding 14 multiplicities of massless hypermultiplets of a six-dimensional F-theory compactification for a general base B.