Yau Threefolds Research Articles

Calabi–Yau Double Coverings of Fano–Enriques Threefolds

AbstractThis note is a report on the observation that the Fano–Enriques threefolds with terminal cyclic quotient singularities admit Calabi–Yau threefolds as their double coverings. We calculate the invariants of those Calabi–Yau threefolds when the Picard number is one. It turns out that all of them are new examples.

Nef Line Bundles on Calabi–Yau Threefolds, I

Abstract We prove that a nef line bundle ${\mathcal{L}}$ with $c_1({\mathcal{L}})^2 \ne 0$ on a Calabi–Yau threefold $X$ with Picard number $2$ and with $c_3(X) \ne 0$ is semiample, that is, some multiple of $\mathcal L$ is generated by global sections.

Genus-One Gromov–Witten Invariants of Quintic Three-folds via MSP Localization

Abstract The moduli stack of Mixed Spin P-fields (MSP) provides an effective algorithm to evaluate all genus Gromov–Witten (GW) invariants of the quintic Calabi–Yau (CY) three-folds. This paper is to apply the algorithm to the genus-one case. We use the localization formula, the proposed algorithm in [ 10, 11], and Zinger’s packaging technique to compute the genus-one GW invariants of the quintic CY three-folds. Our approach to the formula suggests a correspondence between each type of MSP graphs with each physics’ phase: CY, Landau–Ginzburg, or conifold point. In this process, new differential relations among Givental’s I-functions are also discovered.

Machine learning CICY threefolds

The latest techniques from Neural Networks and Support Vector Machines (SVM) are used to investigate geometric properties of Complete Intersection Calabi–Yau (CICY) threefolds, a class of manifolds that facilitate string model building. An advanced neural network classifier and SVM are employed to (1) learn Hodge numbers and report a remarkable improvement over previous efforts, (2) query for favourability, and (3) predict discrete symmetries, a highly imbalanced problem to which both Synthetic Minority Oversampling Technique (SMOTE) and permutations of the CICY matrix are used to decrease the class imbalance and improve performance. In each case study, we employ a genetic algorithm to optimise the hyperparameters of the neural network. We demonstrate that our approach provides quick diagnostic tools capable of shortlisting quasi-realistic string models based on compactification over smooth CICYs and further supports the paradigm that classes of problems in algebraic geometry can be machine learned.

On multi-pointed non-commutative deformations and Calabi–Yau threefolds

We will develop a theory of multi-pointed non-commutative deformations of a simple collection in an abelian category, and construct relative exceptional objects and relative spherical objects in some cases. This is inspired by a work by Donovan and Wemyss.

Dimensional reductions of DFT and mirror symmetry for Calabi–Yau three-folds and K3 × T2

We perform dimensional reductions of type IIA and type IIB double field theory in the flux formulation on Calabi–Yau three-folds and on K3×T2. In addition to geometric and non-geometric three-index fluxes and Ramond–Ramond fluxes, we include generalized dilaton fluxes. We relate our results to the scalar potentials of corresponding four-dimensional gauged supergravity theories, and we verify the expected behavior under mirror symmetry. For Calabi–Yau three-folds we extend this analysis to the full bosonic action including kinetic terms.

On the motive of intersections of two Grassmannians in $$\varvec{\mathbb {P}}^9$$ P 9

Using intersections of two Grassmannians in $$\mathbb {P}^9$$ , Ottem–Rennemo and Borisov–Căldăraru–Perry have exhibited pairs of Calabi–Yau threefolds X and Y that are deformation equivalent, L-equivalent and derived equivalent, but not birational. To complete the picture, we show that X and Y have isomorphic Chow motives.

Intersections of two Grassmannians in ℙ9

Abstract We study the intersection of two copies of Gr ⁢ ( 2 , 5 ) {\mathrm{Gr}(2,5)} embedded in ℙ 9 {{{\mathbb{P}}}^{9}} , and the intersection of the two projectively dual Grassmannians in the dual projective space. These intersections are deformation equivalent, derived equivalent Calabi–Yau threefolds. We prove that generically they are not birational. As a consequence, we obtain a counterexample to the birational Torelli problem for Calabi–Yau threefolds. We also show that these threefolds give a new pair of varieties whose classes in the Grothendieck ring of varieties are not equal, but whose difference is annihilated by a power of the class of the affine line. Our proof of non-birationality involves a detailed study of the moduli stack of Calabi–Yau threefolds of the above type, which may be of independent interest.

Calabi–Yau threefolds with small Hodge numbers associated with a one-parameter family of polynomials

We construct several quintic Calabi–Yau threefolds over the rationals with small Hodge numbers, by using certain members of a family of polynomial solutions of a second order linear partial differential equation.

Tonoli's Calabi–Yau threefolds revisited

We find a simple construction of Tonoli's examples of Calabi–Yau threefolds in complex P6. We prove that the rank of the Picard group of elements of one of these families is at least 2.

Rigid realizations of modular forms in Calabi–Yau threefolds

We construct examples of modular rigid Calabi–Yau threefolds, which give a realization of some new weight 4 cusp forms.

Physics of superheavy dark matter in supergravity

New trends in inflationary model building and dark matter production in supergravity are considered. Starobinsky inflation is embedded into [Formula: see text] supergravity, avoiding instability problems, when the inflaton belongs to a vector superfield associated with a [Formula: see text] gauge symmetry, instead of a chiral superfield. This gauge symmetry can be spontaneously broken by the super-Higgs mechanism resulting in a massive vector supermultiplet including the (real scalar) inflaton field. Both supersymmetry (SUSY) and the R-symmetry can also be spontaneously broken by the Polonyi mechanism at high scales close to the inflationary scale. In this case, Polonyi particles and gravitinos become superheavy, and can be copiously produced during inflation by the Schwinger mechanism sourced by the universe expansion. The Polonyi mass slightly exceeds twice the gravitino mass, so that Polonyi particles are unstable and decay into gravitinos. Considering the mechanisms of superheavy gravitino production, we find that the right amount of cold dark matter composed of gravitinos can be achieved. In our scenario, the parameter space of the inflaton potential is directly related to the dark matter one, providing a new unifying framework of inflation and dark matter genesis. A multi-superfield extension of the supergravity framework with a single (inflaton) superfield can result in a formation of primordial nonlinear structures like mini- and stellar-mass black holes, primordial nongaussianity, and the running spectral index of density fluctuations. This framework can be embedded into the SUSY GUTs inspired by heterotic string compactifications on Calabi–Yau three-folds, thus unifying particle physics with quantum gravity.

Singular and Calabi–Yau varieties linked with billiard trajectories and diffusion operators

The images under a certain map of periodic billiard trajectories inside the fundamental region of the affine Weyl group of the root system $$A_{2}$$ are closed curves and configurations of lines described by a one-parameter family of polynomials. The polynomials are related with eigenvectors of symmetric diffusion operators connected with the deltoid. Several associated singular varieties and Calabi–Yau threefolds defined over the rationals are constructed.

Holomorphic anomaly equations and the Igusa cusp form conjecture

Let S be a K3 surface and let E be an elliptic curve. We solve the reduced Gromov–Witten theory of the Calabi–Yau threefold $$S \times E$$ for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture. The proof relies on new results in the Gromov–Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov–Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphic anomaly equation proves a conjecture of Milanov, Ruan and Shen. We further conjecture quasimodularity and holomorphic anomaly equations for the cycle-valued Gromov–Witten theory of every elliptic fibration with section. The conjecture generalizes the holomorphic anomaly equations for elliptic Calabi–Yau threefolds predicted by Bershadsky, Cecotti, Ooguri, and Vafa. We show a modified conjecture holds numerically for the reduced Gromov–Witten theory of K3 surfaces in primitive classes.

Double coverings with h2,0=0 over compact Kähler manifolds

Abstract We give a formula for Hodge numbers of double coverings with h 2 , 0 = 0 over compact Kahler manifolds. As an application, we consider Calabi–Yau double coverings and calculate their Hodge numbers. In this way, we find several pairs ( h 1 , 1 , h 1 , 2 ) of Hodge numbers of Calabi–Yau threefolds that do not come from toric setting.

Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems

We consider the Gopakumar–Ooguri–Vafa correspondence, relating U(N) Chern–Simons theory at large N to topological strings, in the context of spherical Seifert 3-manifolds. These are quotients S^Γ=Γ ∖ S^3 of the three-sphere by the free action of a finite isometry group. Guided by string theory dualities, we propose a large N dual description in terms of both A- and B-twisted topological strings on (in general non-toric) local Calabi–Yau threefolds. The target space of the B-model theory is obtained from the spectral curve of Toda-type integrable systems constructed on the double Bruhat cells of the simply-laced group identified by the ADE label of Γ. Its mirror A-model theory is realized as the local Gromov–Witten theory of suitable ALE fibrations on P^1, generalizing the results known for lens spaces.We propose an explicit construction of the family of target manifolds relevant for the correspondence, which we verify through a large N analysis of the matrix model that expresses the contribution of the trivial flat connection to the Chern–Simons partition function. Mathematically, our results put forward an identification between the 1/N expansion of the slN+1 LMO invariant of S^Γ and a suitably restricted Gromov–Witten/Donaldson–Thomas partition function on the A-model dual Calabi–Yau. This 1/N expansion, as well as that of suitable generating series of perturbative quantum invariants of fiber knots in S^Γ, is computed by the Eynard–Orantin topological recursion.

On homological mirror symmetry of toric Calabi–Yau threefolds

We use Lagrangian torus fibrations on the mirror $X$ of a toric Calabi-Yau threefold $\check X$ to construct Lagrangian sections and various Lagrangian spheres on $X$. We then propose an explicit correspondence between the sections and line bundles on $\check X$ and between spheres and sheaves supported on the toric divisors of $\check X$. We conjecture that these correspondences induce an embedding of the relevant derived Fukaya category of $X$ inside the derived category of coherent sheaves on $\check X$.

The Thom–Sebastiani theorem for the Euler characteristic of cyclic L∞-algebras

Let L be a cyclic L∞-algebra of dimension 3 with finite dimensional cohomology only in dimension one and two. By transfer theorem there exists a cyclic L∞-algebra structure on the cohomology H⁎(L). The inner product plus the higher products of the cyclic L∞-algebra defines a superpotential function f on H1(L). We associate with an analytic Milnor fiber for the formal function f and define the Euler characteristic of L is to be the Euler characteristic of the étale cohomology of the analytic Milnor fiber.In this paper we prove a Thom–Sebastiani type formula for the Euler characteristic of cyclic L∞-algebras. As applications we prove the Joyce–Song formulas about the Behrend function identities for semi-Schur objects in the derived category of coherent sheaves over Calabi–Yau threefolds. A motivic Thom–Sebastiani type formula and a conjectural motivic Joyce–Song formula for the motivic Milnor fiber of cyclic L∞-algebras are also discussed.

Motivic Donaldson–Thomas invariants of some quantized threefolds

This paper is motivated by the question of howmotivic Donaldson–Thomas invariants behave in families. We compute the invariants for some simple families of noncommutative Calabi–Yau threefolds, defined by quivers with homogeneous potentials. These families give deformation quantizations of affine three-space, the resolved conifold, and the resolution of the transversal A_n -singularity. It turns out that their invariants are generically constant, but jump at special values of the deformation parameter, such as roots of unity. The corresponding generating series are written in closed form, as plethystic exponentials of simple rational functions. While our results are limited by the standard dimensional reduction techniques that we employ, they nevertheless allow us to conjecture formulae for more interesting cases, such as the elliptic Sklyanin algebras.

New realizations of modular forms in Calabi–Yau threefolds arising from ϕ4 theory

Brown and Schnetz found that the number of points over Fp of a graph hypersurface is often related to the coefficients of a modular form. We set some of the reduction techniques used to discover such relations in a general geometric context. We also prove the relation for two examples of modular forms of weight 3 and two of weight 4, refine the statement and suggest a method of proving it for three more of weight 4, and use one of the proved examples to construct two new rigid Calabi–Yau threefolds that realize Hecke eigenforms of weight 4 (one provably and one conjecturally).