Calabi-Yau Hypersurfaces Research Articles

Toric K3-fibred Calabi-Yau manifolds with del Pezzo divisors for string compactifications

AbstractWe analyse several explicit toric examples of compact K3-fibred Calabi-Yau three-folds. These manifolds can be used for the study of string dualities and are crucial ingredients for the construction of LARGE Volume type IIB vacua with promising applications to cosmology and particle phenomenology. In order to build a phenomenologically viable model, on top of the two moduli corresponding to the base and the K3 fibre, we demand also the existence of two additional rigid divisors: the first supporting the non-perturbative effects needed to achieve moduli stabilisation, and the second allowing the presence of chiral matter on wrapped D-branes. We clarify the topology of these rigid divisors by discussing the interplay between a diagonal structure of the Calabi-Yau volume and D-terms. Del Pezzo divisors appearing in the volume form in a completely diagonal way are natural candidates for supporting non-perturbative effects and for quiver constructions, while ‘non-diagonal’ del Pezzo and rigid but not del Pezzo divisors are particularly interesting for model building in the geometric regime. Searching through the existing list of four dimensional reflexive lattice polytopes, we find 158 examples admitting a Calabi-Yau hypersurface with a K3 fibration and four Kähler moduli where at least one of the toric divisors is a ‘diagonal’ del Pezzo. We work out explicitly the topological details of a few examples showing how, in the case of simplicial polytopes, all the del Pezzo divisors are ‘diagonal’, while ‘non-diagonal’ ones appear only in the case of non-simplicial polytopes. A companion paper will use these results in the study of moduli stabilisation for globally consistent explicit Calabi-Yau compactifications with the local presence of chirality.

Fermionic screenings and line bundle twisted chiral de Rham complex on CY manifolds

We present a generalization of Borisov’s construction of the chiral de Rham complex in the case of the line-bundle-twisted chiral de Rham complex on a Calabi-Yau hypersurface in a projective space. We generalize the differential associated with a polytope Δ of the projective space ℙ d − 1 by allowing nonzero modes for the screening currents forming this differential. It is shown that the numbers of screening current modes define the support function of the toric divisor of a line bundle on ℙ d − 1 that twists the chiral de Rham complex on the Calabi-Yau hypersurface.

Heterotic bundles on Calabi-Yau manifolds with small Picard number

We undertake a systematic scan of vector bundles over spaces from the largest database of known Calabi-Yau three-folds, in the context of heterotic string compactification. Specifically, we construct positive rank five monad bundles over Calabi-Yau hypersurfaces in toric varieties, with the number of Kahler moduli equal to one, two, and three and extract physically interesting models. We select models which can lead to three families of matter after dividing by a freely-acting discrete symmetry and including Wilson lines. About 2000 such models on two manifolds are found.

Orlov spectra: bounds and gaps

The Orlov spectrum is a new invariant of a triangulated category. It was introduced by D. Orlov building on work of A. Bondal-M. van den Bergh and R. Rouquier. The supremum of the Orlov spectrum of a triangulated category is called the ultimate dimension. In this work, we study Orlov spectra of triangulated categories arising in mirror symmetry. We introduce the notion of gaps and outline their geometric significance. We provide the first large class of examples where the ultimate dimension is finite: categories of singularities associated to isolated hypersurface singularities. Similarly, given any nonzero object in the bounded derived category of coherent sheaves on a smooth Calabi-Yau hypersurface, we produce a new generator by closing the object under a certain monodromy action and uniformly bound this new generator's generation time. In addition, we provide new upper bounds on the generation times of exceptional collections and connect generation time to braid group actions to provide a lower bound on the ultimate dimension of the derived Fukaya category of a symplectic surface of genus greater than one.

Hori-Vafa mirror periods, Picard-Fuchs equations, and Berglund-Hübsch-Krawitz duality

This paper discusses the overlap of the Hori-Vafa formulation of mirror symmetry with some other constructions. We focus on compact Calabi-Yau hypersurfaces \( {\mathcal{M}_{\,G}} = \left\{ {G = 0} \right\} \) in weighted complex projective spaces. The Hori-Vafa formalism relates a family \( \left\{ {{M_G} \in WCP_{{Q_1}, \ldots, {Q_m}}^{m - 1}\left[ s \right]\left| {\sum\nolimits_{i = 1}^m {{Q_i} = s} } \right.} \right\} \) Ginzburg mirror theory. A technique suggested by Hori and Vafa allows the Picard-Fuchs equations satisfied by the corresponding mirror periods to be determined. Some examples in which the variety \( {\mathcal{M}_G} \) is crepantly resolved are considered. The resulting Picard-Fuchs equations agree with those found elsewhere working in the Batyrev-Borisov framework. When G is an invertible nondegenerate quasihomogeneous polynomial, the Chiodo-Ruan geometrical interpretation of Berglund-Hübsch-Krawitz duality can be used to associate a particular complex structure for \( {\mathcal{M}_G} \) with a particular Kähler structure for the mirror \( {\tilde{\mathcal{M}}_G} \). We make this association for such G when the ambient space of \( {\mathcal{M}_G} \) is CP 2, CP 3, and CP 4. Finally, we probe some of the resulting mirror Kähler structures by determining corresponding Picard-Fuchs equations.

(0,2) deformations of linear sigma models

We study (0,2) deformations of a (2,2) supersymmetric gauged linear sigma model for a Calabi-Yau hypersurface in a Fano toric variety. In the non-linear sigma model these correspond to some of the holomorphic deformations of the tangent bundle on the hypersurface. Combinatorial formulas are given for the number of these deformations, and we show that these numbers are exchanged by mirror symmetry in a subclass of the models.

Moduli spaces of stable bundles on Calabi–Yau varieties and Donaldson–Thomas invariants

Let Y be a smooth Calabi–Yau hypersurface of P 1 × P where P stands for a P d -bundle over P 1 . We will prove that for many ample line bundles L and certain Chern characters c , the moduli space M ¯ L ( c ) (resp. M L ( c ) ) of L -Gieseker semistable (resp. L -stable ) rank two torsion free sheaves (resp. vector bundles) on Y with Chern character c are smooth and irreducible and we will compute its dimension. Moreover, we will prove that both moduli spaces coincide. As a byproduct of the geometrical description of these moduli spaces, we will compute the Donaldson–Thomas invariants of some Calabi–Yau 3-folds.

LG/CY correspondence: The state space isomorphism

We prove the mirror duality conjecture for the mirror pairs constructed by Berglund, Hübsch, and Krawitz. Our main tool is a cohomological LG/CY correspondence which provides a degree-preserving isomorphism between the cohomology of finite quotients of Calabi–Yau hypersurfaces inside a weighted projective space and the Fan–Jarvis–Ruan–Witten state space of the associated Landau–Ginzburg singularity theory.

Five-brane superpotentials, blow-up geometries and SU(3) structure manifolds

We investigate the dynamics of space-time filling five-branes wrapped on curves in heterotic and orientifold Calabi-Yau compactifications. We first study the leading N=1 scalar potential on the infinite deformation space of the brane-curve around a supersymmetric configuration. The higher order potential is also determined by a brane superpotential which we compute for a subset of light deformations. We argue that these deformations map to new complex structure deformations of a non-Calabi-Yau manifold which is obtained by blowing up the brane-curve into a four-cycle and by replacing the brane by background fluxes. This translates the original brane-bulk system into a unifying geometrical formulation. Using this blow-up geometry we compute the complete set of open-closed Picard-Fuchs differential equations and identify the brane superpotential at special points in the field space for five-branes in toric Calabi-Yau hypersurfaces. This has an interpretation in open mirror symmetry and enables us to list compact disk instanton invariants. As a first step towards promoting the blow-up geometry to a supersymmetric heterotic background we propose a non-Kaehler SU(3) structure and an identification of the three-form flux.

Hyperconifold transitions, mirror symmetry, and string theory

Multiply-connected Calabi–Yau threefolds are of particular interest for both string theorists and mathematicians. Recently it was pointed out that one of the generic degenerations of these spaces (occurring at codimension one in moduli space) is an isolated singularity which is a finite cyclic quotient of the conifold; these were called hyperconifolds. It was also shown that if the order of the quotient group is even, such singular varieties have projective crepant resolutions, which are therefore smooth Calabi–Yau manifolds. The resulting topological transitions were called hyperconifold transitions, and change the fundamental group as well as the Hodge numbers. Here Batyrevʼs construction of Calabi–Yau hypersurfaces in toric fourfolds is used to demonstrate that certain compact examples containing the remaining hyperconifolds — the Z 3 and Z 5 cases — also have Calabi–Yau resolutions. The mirrors of the resulting transitions are studied and it is found, surprisingly, that they are ordinary conifold transitions. These are the first examples of conifold transitions with mirrors which are more exotic extremal transitions. The new hyperconifold transitions are also used to construct a small number of new Calabi–Yau manifolds, with small Hodge numbers and fundamental group Z 3 or Z 5 . Finally, it is demonstrated that a hyperconifold is a physically sensible background in Type IIB string theory. In analogy to the conifold case, non-perturbative dynamics smooth the physical moduli space, such that hyperconifold transitions correspond to non-singular processes in the full theory.

A (0,2) mirror map

We study the linear sigma model subspace of the moduli space of (0,2) superconformal world-sheet theories obtained by deforming (2,2) theories based on Calabi-Yau hypersurfaces in reflexively plain toric varieties. We describe a set of algebraic coordinates on this subspace, formulate a (0,2) generalization of the monomial-divisor mirror map, and show that the map exchanges principal components of singular loci of the mirror half-twisted theories. In non-reflexively plain examples the proposed map yields a mirror isomorphism between subfamilies of linear sigma models.

Representations on the cohomology of hypersurfaces and mirror symmetry

We study the representation of a finite group acting on the cohomology of a non-degenerate, invariant hypersurface of a projective toric variety. We deduce an explicit description of the representation when the toric variety has at worst quotient singularities. As an application, we conjecture a representation-theoretic version of Batyrev and Borisov's mirror symmetry between pairs of Calabi–Yau hypersurfaces, and prove it when the hypersurfaces have dimension at most 3. An interesting consequence is the existence of pairs of Calabi–Yau orbifolds whose Hodge diamonds are mirror, with respect to the usual Hodge structure on singular cohomology.

Algebraic $K$-theory of toric hypersurfaces

We construct classes in the motivic cohomology of certain 1-parameter families of Calabi–Yau hypersurfaces in toric Fano n-folds, with applications to local mirror symmetry (growth of genus 0 instanton numbers) and inhomogeneous Picard–Fuchs equations. In the case where the family is classically modular the classes are related to Beilinson’s Eisenstein symbol; the Abel–Jacobi map (or rational regulator) is computed in this paper for both kinds of cycles. For the “modular toric” families where the cycles essentially coincide, we obtain a motivic (and computationally effective) explanation of a phenomenon observed by Villegas, Stienstra, and Bertin.

A global mirror symmetry framework for the Landau–Ginzburg/Calabi–Yau correspondence

We show how the Landau–Ginzburg/Calabi–Yau correspondence for the quintic three-fold can be cast into a global mirror symmetry framework. Then we draw inspiration from Berglund–Hubsch mirror duality construction to provide an analogue picture featuring all Calabi–Yau hypersurfaces and certain quotients by finite abelian group actions.

D-brane superpotentials: Geometric and worldsheet approaches

From the worldsheet perspective, the superpotential on a D-brane wrapping internal cycles of a Calabi–Yau manifold is given as a generating functional for disk correlation functions. On the other hand, from the geometric point of view, D-brane superpotentials are captured by certain chain integrals. In this work, we explicitly show for branes wrapping internal two-cycles how these two different approaches are related. More specifically, from the worldsheet point of view, D-branes at the Landau–Ginzburg point have a convenient description in terms of matrix factorizations. We use a formula derived by Kapustin and Li to explicitly evaluate disk correlators for families of D2-branes. On the geometry side, we then construct a three-chain whose period gives rise to the effective superpotential and show that the two expressions coincide. Finally, as an explicit example, we choose a particular compact Calabi–Yau hypersurface and compute the effective D2-brane superpotential in different branches of the open moduli space, in both geometric and worldsheet approaches.

Automorphic realization of residual Galois representations

We show that it is possible in rather general situations to obtain a finite-dimensional modular representation ρ of the Galois group of a number field F as a constituent of one of the modular Galois representations attached to automorphic representations of a general linear group over F , provided one works “potentially.” The proof is based on a close study of the monodromy of the Dwork family of Calabi–Yau hypersurfaces; this in turn makes use of properties of rigid local systems and the classification of irreducible subgroups of finite classical groups with certain sorts of generators.

On equivalences of derived and singular categories

Abstract Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → $$ \mathbb{A}^1 $$, g:Y → $$ \mathbb{A}^1 $$. Assuming that there exists a complex of sheaves on X × $$ \mathbb{A}^1 $$ Y which induces an equivalence of D b(X) and D b(Y), we show that there is also an equivalence of the singular derived categories of the fibers f −1(0) and g −1(0). We apply this statement in the setting of McKay correspondence, and generalize a theorem of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective space, to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef anticanonical class.

Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions

We construct a surprisingly large class of new Calabi–Yau 3-folds $X$ with small Picard numbers and propose a construction of their mirrors $X^∗$ using smoothings of toric hypersurfaces with conifold singularities. These new examples are related to the previously known ones via conifold transitions. Our results generalize the mirror construction for Calabi– Yau complete intersections in Grassmannians and flag manifolds via toric degenerations. There exist exactly 198849 reflexive four-polytopes whose two-faces are only triangles or parallelograms of minimal volume. Every such polytope gives rise to a family of Calabi–Yau hypersurfaces with at worst conifold singularities. Using a criterion of Namikawa we found 30241 reflexive four-polytopes such that the corresponding Calabi–Yau hypersurfaces are smoothable by a flat deformation. In particular, we found 210 reflexive four-polytopes defining 68 topologically different Calabi–Yau 3-folds with $h_11 = 1$. We explain the mirror construction and compute several new Picard–Fuchs operators for the respective oneparameter families of mirror Calabi–Yau 3-folds.

Matrix Factorizations, Massey Products and F-Terms for Two-Parameter Calabi-Yau Hypersurfaces

We discuss B–type tensor product branes in mirrors of two–parameter Calabi–Yau hypersurfaces, using the language of matrix factorizations. We determine the open string moduli of the branes at the Gepner point. By turning on both bulk and boundary moduli we then deform the brane away from the Gepner point. Using the deformation theory of matrix factorizations we compute Massey products. These contain the information about higher order deformations and obstructions. The obstructions are encoded in the F–term equations, which we obtain from the Massey product algorithm. We show that the F– terms can be integrated to an effective superpotential. Our results provide an ingredient for open/closed mirror symmetry for these hypersurfaces.

On Katz's bound for the number of elements with given trace and norm

In this note an improvement of the Katz's bound on the number of elements in a finite field with given trace and norm is given. The improve- ment is obtained by reducing the problem to estimating the number of rational points on certain toric Calabi-Yau hypersurface, and then to use detailed co- homological calculations by Rojas-Leon and the second author for such toric hypersurfaces.