Calabi-Yau Hypersurfaces Research Articles

On the hyperplane conjecture for periods of Calabi–Yau hypersurfaces in $\mathbf{P}^n$

In [HLY1], Hosono, Lian, and Yau gave a conjectural characterization of the set of solutions to certain Gelfand–Kapranov–Zelevinsky hypergeometric equations which are realized as periods of Calabi–Yau hypersurfaces in a Gorenstein Fano toric variety $X$. We prove this conjecture in the case where $X$ is a complex projective space.

Mirror symmetry for quasi-smooth Calabi–Yau hypersurfaces in weighted projective spaces

We consider a d-dimensional well-formed weighted projective space P(w¯) as a toric variety associated with a fan Σ(w¯) in Nw¯⊗R whose 1-dimensional cones are spanned by primitive vectors v0,v1,…,vd∈Nw¯ generating a lattice Nw¯ and satisfying the linear relation ∑iwivi=0. For any fixed dimension d, there exist only finitely many weight vectors w¯=(w0,…,wd) such that P(w¯) contains a quasi-smooth Calabi–Yau hypersurface Xw defined by a transverse weighted homogeneous polynomial W of degree w=∑i=0dwi. Using a formula of Vafa for the orbifold Euler number χorb(Xw), we show that for any quasi-smooth Calabi–Yau hypersurface Xw the number (−1)d−1χorb(Xw) equals the stringy Euler number χstr(Xw¯∗) of Calabi–Yau compactifications Xw¯∗ of affine toric hypersurfaces Zw¯ defined by non-degenerate Laurent polynomials fw¯∈ℂ[Nw¯] with Newton polytope conv({v0,…,vd}). In the moduli space of Laurent polynomials fw¯ there always exists a special point fw¯0 defining a mirror Xw¯∗ with a Z∕wZ-symmetry group such that Xw¯∗ is birational to a quotient of a Fermat hypersurface via a Shioda map.

The Gamma and Strominger–Yau–Zaslow conjectures : a tropical approach to periods

We propose a new method to compute asymptotics of periods using tropical geometry, in which the Riemann zeta values appear naturally as error terms in tropicalization. Our method suggests how the Gamma class should arise from the Strominger-Yau-Zaslow conjecture. We use it to give a new proof of (a version of) the Gamma Conjecture for Batyrev pairs of mirror Calabi-Yau hypersurfaces.

Vacua with Small Flux Superpotential.

We describe a method for finding flux vacua of type IIB string theory in which the Gukov-Vafa-Witten superpotential is exponentially small. We present an example with W_{0}≈2×10^{-8} on an orientifold of a Calabi-Yau hypersurface with (h^{1,1},h^{2,1})=(2,272), at large complex structure and weak string coupling.

Searching the landscape of flux vacua with genetic algorithms

In this paper, we employ genetic algorithms to explore the landscape of type IIB flux vacua. We show that genetic algorithms can efficiently scan the landscape for viable solutions satisfying various criteria. More specifically, we consider a symmetric T6 as well as the conifold region of a Calabi-Yau hypersurface. We argue that in both cases genetic algorithms are powerful tools for finding flux vacua with interesting phenomenological properties. We also compare genetic algorithms to algorithms based on different breeding mechanisms as well as random walk approaches.

Gromov–Witten Invariants of Calabi–Yau Manifolds With Two Kähler Parameters

Abstract We study the Gromov–Witten theory of $K_{{{\mathbb{P}}}^1\times{{\mathbb{P}}}^1}$ and some Calabi–Yau hypersurfaces in toric varieties. We give a direct geometric proof of the holomorphic anomaly equation for $K_{{{\mathbb{P}}}^1\times{{\mathbb{P}}}^1}$ in the form predicted by B-model physics. We also calculate the closed formula of genus one quasimap invariants of Calabi–Yau hypersurfaces in ${{\mathbb{P}}}^{m-1}\times{{\mathbb{P}}}^{n-1}$ after restricting the 2nd Kähler parameter to zero. By the wall-crossing theorem between Gromov–Witten and quasimap invariants, we thus obtain their genus one Gromov–Witten invariants.

Counting string theory standard models

We derive an approximate analytic relation between the number of consistent heterotic Calabi-Yau compactifications of string theory with the exact charged matter content of the standard model of particle physics and the topological data of the internal manifold: the former scaling exponentially with the number of Kähler parameters. This is done by an estimate of the number of solutions to a set of Diophantine equations representing constraints satisfied by any consistent heterotic string vacuum with three chiral massless families, and has been computationally checked to hold for complete intersection Calabi-Yau threefolds (CICYs) with up to seven Kähler parameters. When extrapolated to the entire CICY list, the relation gives ∼1023 string theory standard models; for the class of Calabi-Yau hypersurfaces in toric varieties, it gives ∼10723 standard models.

Estimating Calabi-Yau hypersurface and triangulation counts with equation learners

We provide the first estimate of the number of fine, regular, star triangulations of the four-dimensional reflexive polytopes, as classified by Kreuzer and Skarke (KS). This provides an upper bound on the number of Calabi-Yau threefold hypersurfaces in toric varieties. The estimate is performed with deep learning, specifically the novel equation learner (EQL) architecture. We demonstrate that EQL networks accurately predict numbers of triangulations far beyond the h1,1 training region, allowing for reliable extrapolation. We estimate that number of triangulations in the KS dataset is 1010,505, dominated by the polytope with the highest h1,1 value.

Exact Computation of the Special Geometry for Calabi–Yau Hypersurfaces of Fermat Type

We continue to develop our method for effectively computating the special K\"ahler geometry on the moduli space of Calabi-Yau manifolds. We generalize it to all polynomial deformations of Fermat hypersurfaces.

Equivalences of families of stacky toric Calabi-Yau hypersurfaces

Given the same anti-canonical linear system on two distinct toric varieties, we provide a derived equivalence between partial crepant resolutions of the corresponding stacky hypersurfaces. The applications include: a derived unification of toric mirror constructions, calculations of Picard lattices for linear systems of quartics in $\mathbf {P}^3$, and a birational reduction of Reid’s list to 81 families.

Calabi Yau Hypersurfaces and SU-Bordism

V. V. Batyrev constructed a family of Calabi–Yau hypersurfaces dual to the first Chern class in toric Fano varieties. Using this construction, we introduce a family of Calabi–Yau manifolds whose SU-bordism classes generate the special unitary bordism ring $${\Omega ^{SU}}[\frac{1}{2}] \cong Z[\frac{1}{2}][{y_i}:i \geqslant 2]$$ . We also describe explicit Calabi–Yau representatives for multiplicative generators of the SU-bordism ring in low dimensions.

A Generalized Construction of Calabi-Yau Models and Mirror Symmetry

We extend the construction of Calabi-Yau manifolds to hypersurfaces in non-Fano toric varieties, requiring the use of certain Laurent defining polynomials, and explore the phases of the corresponding gauged linear sigma models. The associated non-reflexive and non-convex polytopes provide a generalization of Batyrev’s original work, allowing us to construct novel pairs of mirror models. We showcase our proposal for this generalization by examining Calabi-Yau hypersurfaces in Hirzebruch n-folds, focusing on n=3,4 sequences, and outline the more general class of so-defined geometries.

Variations of geometric invariant quotients for pairs, a computational approach

We study the GIT compactifications of pairs formed by a hypersurface and a hyperplane. We provide a general setting to characterize all polarizations which give rise to different GIT quotients. Furthermore, we describe a finite set of one-parameter subgroups sufficient to determine the stability of any GIT quotient. We characterize all maximal orbits of non stable and strictly semistable pairs, as well as minimal closed orbits of strictly semistable pairs. Our construction gives natural compactifications of the space of log smooth pairs for Fano and Calabi-Yau hypersurfaces.

Distinguished-root formulas for generalized Calabi–Yau hypersurfaces

By a "generalized Calabi-Yau hypersurface" we mean a hypersurface in ${\mathbb P}^n$ of degree $d$ dividing $n+1$. The zeta function of a generic such hypersurface has a reciprocal root distinguished by minimal $p$-divisibility. We study the $p$-adic variation of that distinguished root in a family and show that it equals the product of an appropriate power of $p$ times a product of special values of a certain $p$-adic analytic function ${\mathcal F}$. That function ${\mathcal F}$ is the $p$-adic analytic continuation of the ratio $F(\Lambda)/F(\Lambda^p)$, where $F(\Lambda)$ is a solution of the $A$-hypergeometric system of differential equations corresponding to the Picard-Fuchs equation of the family.

Hemisphere partition function and monodromy

We discuss D-brane monodromies from the point of view of the gauged linear sigma model. We give a prescription on how to extract monodromy matrices directly from the hemisphere partition function. We illustrate this procedure by recomputing the monodromy matrices associated to one-parameter Calabi-Yau hypersurfaces in weighted projected space.

The F-pure threshold of quasi-homogeneous polynomials

Inspired by the work of Bhatt and Singh [3] we compute the F-pure threshold of quasi-homogeneous polynomials. We first consider the case of a curve given by a quasi-homogeneous polynomial f in three variables x,y,z of degree equal to the degree of xyz and then we proceed with the general case of a Calabi–Yau hypersurface, i.e. a hypersurface given by a quasi-homogeneous polynomial f in n+1 variables x0,…,xn of degree equal to the degree of x0⋯xn.

Systematics of axion inflation in Calabi-Yau hypersurfaces

We initiate a comprehensive survey of axion inflation in compactifications of type IIB string theory on Calabi-Yau hypersurfaces in toric varieties. For every threefold with h1,1 ≤ 4 in the Kreuzer-Skarke database, we compute the metric on Kähler moduli space, as well as the matrix of four-form axion charges of Euclidean D3-branes on rigid divisors. These charges encode the possibility of enlarging the field range via alignment. We then determine an upper bound on the inflationary field range Δϕ that results from the leading instanton potential, in the absence of monodromy. The bound on the field range in this ensemble is Δϕ ≲ 0.3Mpl, in a compactification where the smallest curve volume is (2π)2α′, and we argue that the sigma model expansion is adequately controlled. The largest increase resulting from alignment is a factor ≈ 2.6. We also examine a set of threefolds with h1,1 up to 100 and characterize their axion charge matrices. While we find modest alignment in this ensemble, the maximum field range is ultimately suppressed by the volume of the internal space, which typically grows quickly with h1,1. Furthermore, we find that many toric divisors are rigid — and the corresponding charge matrices are relatively trivial — at large h1,1. It is therefore challenging to realize alignment via superpotentials generated only by Euclidean D3-branes, without taking into account the effects of flux, D7-branes, and orientifolding.

The stringy Euler number of Calabi–Yau hypersurfaces in toric varieties and the Mavlyutov duality

We show that minimal models of nondegenerated hypersufaces defined by Laurent polynomials with a $d$-dimensional Newton polytope $\Delta$ are Calabi-Yau varieties $X$ if and only if the Fine interior of $\Delta$ consists of a single lattice point. We give a combinatorial formula for computing the stringy Euler number of $X$. This formula allows to test mirror symmetry in cases when $\Delta$ is not a reflexive polytope. In particular we apply this formula to pairs of lattice polytopes $(\Delta, \Delta^{\vee})$ that appear in the Mavlyutov's generalization of the polar duality for reflexive polytopes. Some examples of Mavlyutov's dual pairs $(\Delta, \Delta^{\vee})$ show that the stringy Euler numbers of the corresponding Calabi-Yau varieties $X$ and $X^{\vee}$ may not satisfy the expected topological mirror symmetry test: $e_{\rm st}(X) = (-1)^{d-1} e_{\rm st}(X^{\vee})$. This shows the necessity of an additional condition on Mavlyutov's pairs $(\Delta, \Delta^\vee)$.

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.

On the Chow group of zero-cycles of Calabi–Yau hypersurfaces

We prove the existence of a canonical zero-cycle \(c_X\) on a Calabi–Yau hypersurface X in a complex projective homogeneous variety. More precisely, we show that the intersection of any n divisors on X, \(n=\mathrm {dim}\,X\) is proportional to the class of a point on a rational curve in X.