Calabi-Yau Research Articles

Coefficients of the monodromy matrices of one-parameter families of double octic Calabi-Yau threefolds at a half-conifold point

Doran and Morgan introduced in [12] a rational basis for the monodromy group of the Picard-Fuchs operator of a hypergeometric family of Calabi-Yau threefolds. In this paper we compute numerically the transition matrix between a generalization of the Doran-Morgan basis and the Frobenius basis at a half-conifold point of a one-parameter family of double octic Calabi-Yau threefolds. We identify the entries of this matrix as rational functions in the special values L(f,1) and L(f,2) of the corresponding modular form f and one constant. We also present related results concerning the rank of the group of period integrals generated by the action of the monodromy group on the conifold period.

Modular differential equations of the elliptic genus of Calabi–Yau fourfolds

We study modular differential equations (MDEs) of the elliptic genus of four-dimensional complex varieties with trivial first Chern class. We construct modular differential equations of orders 3, 4, 5 and 6 with respect to the heat operator for every weak Jacobi form of weight 0 and index 2. We prove that the elliptic genus of a Calabi–Yau fourfold satisfies a MDE of the minimal possible order 3 if and only if its Euler number is equal to 48 or −18. We construct MDEs of order 5 of the elliptic genus of hyperkähler fourfolds of types Hilb2(K3) and Kum2(A).

Exponential Networks, WKB and Topological String

We propose a connection between 3d-5d exponential networks and exact WKB for difference equations associated to five dimensional Seiberg-Witten curves, or equivalently, to quantum mirror curves to toric Calabi-Yau threefolds $X$: the singularities in the Borel planes of local solutions to such difference equations correspond to central charges of 3d-5d BPS KK-modes. It follows that there should be distinguished local solutions of the difference equation in each domain of the complement of the exponential network, and these solutions jump at the walls of the network. We verify and explore this picture in two simple examples of 3d-5d systems, corresponding to taking the toric Calabi-Yau $X$ to be either $\mathbb{C}^3$ or the resolved conifold. We provide the full list of local solutions in each sector of the Borel plane and in each domain of the complement of the exponential network, and find that local solutions in disconnected domains correspond to non-perturbative open topological string amplitudes on $X$ with insertions of branes at different positions of the toric diagram. We also study the Borel summation of the closed refined topological string free energy on $X$ and the corresponding non-perturbative effects, finding that central charges of 5d BPS KK-modes are related to the singularities in the Borel plane.

Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties

Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties

The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus

The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus

Symmetric fluxes and small tadpoles

The analysis of type IIB flux vacua on warped Calabi-Yau orientifolds becomes considerably involved for a large number of complex structure fields. We however show that, for a quadratic flux superpotential, one can devise simplifying schemes which effectively reduce the large number of equations down to a few. This can be achieved by imposing the vanishing of certain flux quanta in the large complex structure regime, and then choosing the remaining quanta to respect the symmetries of the underlying prepotential. One can then implement an algorithm to find large families of flux vacua with a fixed flux tadpole, independently of the number of fields. We illustrate this approach in a Calabi-Yau manifold with 51 complex structure moduli, where several reduction schemes can be implemented in order to explicitly solve the vacuum equations for that sector. Our findings display a flux-tadpole-to-stabilized-moduli ratio that is marginally above the bound proposed by the Tadpole Conjecture, and we discuss several effects that would take us below such a bound.

Non-Kähler Calabi-Yau geometry and pluriclosed flow

Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow in terms of Hermitian metrics on holomorphic Courant algebroids, implying new global existence results, in particular on all complex non-Kähler surfaces of Kodaira dimension κ≥0. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric, which in turn gives a classification of generalized Kähler structures on these spaces.

Relationship between two Calabi–Yau orbifolds arising as hyper–surfaces in a quotient of the same weighted projective space

In this article we consider a question: what is the relation between two Calabi-Yau manifolds of two different Berglund–Hubsch types if they appear as hyper–surfaces in the quotient of the same weighted projective space. We show that these manifolds are connected by a special change of coordinates, which we call the resonance transformation.

Affine characters at negative level and elliptic genera of non-critical strings

We study the elliptic genera of the non-critical strings of six dimensional superconformal field theories from the point of view of the strings’ worldsheet theory. We formulate a general ansatz for these in terms of characters of the affine Lie algebra associated to the 6d gauge group at negative level, and provide ample evidence for the validity of this ansatz for 6d theories obtained via F-theory compactification on elliptically fibered Calabi-Yau manifolds over a Hirzebruch base. We obtain novel closed form results for many elliptic genera in terms of our ansatz, and show that our results specialize consistently when moving along Higgsing trees.

On K3 fibred Calabi–Yau threefolds in weighted scrolls

The aim of this paper is to classify mildly singular Calabi–Yau threefolds fibred in low-degree weighted K3 surfaces and embedded as anticanonical hypersurfaces in weighted scrolls, extending results of Mullet. We also study projective degenerations, revisiting an example due to Gross and Ruan. Finally we briefly discuss the general question of embedding a projective fibration into a weighted scroll.

Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors

Abstract Cluster varieties come in pairs: for any 𝒳 {\mathcal{X}} cluster variety there is an associated Fock–Goncharov dual 𝒜 {\mathcal{A}} cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry. Particularly, we show that the mirror to the 𝒳 {\mathcal{X}} cluster variety is a degeneration of the Fock–Goncharov dual 𝒜 {\mathcal{A}} cluster variety and vice versa. To do this, we investigate how the cluster scattering diagram of Gross, Hacking, Keel and Kontsevich compares with the canonical scattering diagram defined by Gross and Siebert to construct mirror duals in arbitrary dimensions. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi–Yau varieties obtained as blow-ups of toric varieties.

A symbol and coaction for higher-loop sunrise integrals

We construct a symbol and coaction for the ll-loop sunrise family of integrals, both for equal-mass and generic-mass cases. These constitute the first concrete examples of symbols and coactions for integrals involving Calabi-Yau threefolds and higher. In order to achieve a symbol of finite length, we recast the differential equations satisfied by these integrals in a unipotent form. We augment the integrals in a natural way by including ratios of maximal cuts \tau_iτi. We discuss the relationship of this construction to constructions of symbols and coactions for multiple polylogarithms and elliptic multiple polylogarithms, and its connection to notions of transcendental weight.

Global Critical Chart for Local Calabi–Yau Threefolds

Abstract In this paper, we investigate Keller’s deformed Calabi–Yau completion of the derived category of coherent sheaves on a smooth variety. In particular, for an $n$-dimensional smooth variety $Y$, we describe the derived category of the total space of an $\omega _{Y}$-torsor as a certain deformed $(n+1)$-Calabi–Yau completion of the derived category of $Y$. As an application, we investigate the geometry of the derived moduli stack of compactly supported coherent sheaves on a local curve, that is, a Calabi–Yau threefold of the form $\textrm{Tot}_{C}(N)$, where $C$ is a smooth projective curve and $N$ is a rank two vector bundle on $C$. We show that the derived moduli stack is equivalent to the derived critical locus of a function on a certain smooth moduli space. This result will be used by the first author and Naoki Koseki in their joint work on Higgs bundles and Gopakumar–Vafa invariants.

On the Hasse invariant of Hilbert modular varieties mod p

Let F be a totally real field and let S denote the geometric special fiber of a Hilbert modular variety associated to F, at a prime unramified in F. We show that the order of vanishing of the Hasse invariant on S is equal to the largest integer m such that the smallest piece of the conjugate filtration lies in the mth piece of the Hodge filtration. This result is a direct analogue of Ogus' on families of Calabi–Yau varieties in positive characteristic (see [15]). We also show that the order of vanishing at a point is the same as the codimension of the Ekedahl-Oort stratum containing it.

Level crossings, attractor points and complex multiplication

We study the complex structure moduli dependence of the scalar Laplacian eigenmodes for one-parameter families of Calabi-Yau n-folds in ℙn+1. It was previously observed that some eigenmodes get lighter while others get heavier as a function of these moduli, which leads to eigenvalue crossing. We identify the cause for this behavior for the torus. We then show that at points in a sublocus of complex structure moduli space where Laplacian eigenmodes cross, the torus has complex multiplication. We speculate that the generalization to arbitrary Calabi-Yau manifolds could be that level crossing is related to rank one attractor points. To test this, we compute the eigenmodes numerically for the quartic K3 and the quintic threefold, and match crossings to CM and attractor points in these varieties. To quantify the error of our numerical methods, we also study the dependence of the numerical spectrum on the quality of the Calabi-Yau metric approximation, the number of points sampled from the Calabi-Yau variety, the truncation of the eigenbasis, and the distance from degeneration points in complex structure moduli space.

Deep learning for K3 fibrations in heterotic/Type IIA string duality

The development of Large Language Models, such as the recently released GPT-4, has revolutionized the field of Machine Learning and opened new avenues for interdisciplinary research. Prompt Engineering, a methodology for designing effective input prompts to guide these AI models, will emerge as a powerful tool for accelerating research efforts. In this study, we leverage Prompt Engineering with GPT-4 to address the problem of predicting K3 Fibrations in Calabi-Yau manifolds embedded in toric varieties with a single weight system. Out of the 184,026 weights spaces previously discovered, 101,495 remained unclassified concerning the presence of a K3 projection, thereby providing an opportunity for machine learning to bridge this gap. By utilizing an ensemble of Deep Neural Networks, we are able to predict the existence of the K3 fibration. Furthermore, we assess the potential of these models to predict the spectrum of possible Hodge Numbers and other properties of reflexive polytopes. These results not only demonstrate the utility of AI in studying the heterotic/Type IIA string duality in F-Theory but also serve as a stepping stone for further machine learning integration into traditional scientific workflows.

Asymptotic accelerated expansion in string theory and the Swampland

We study whether the universal runaway behaviour of stringy scalar potentials towards infinite field distance limits can produce an accelerated expanding cosmology à la quintessence. We identify a loophole to some proposed bounds that forbid such asymptotic (at parametric control) accelerated expansion in 4d mathcal{N} = 1 supergravities, by considering several terms of the potential competing asymptotically. We then analyse concrete string theory examples coming from F-theory flux compactifications on Calabi-Yau fourfolds, extending previous results by going beyond weak string coupling to different infinite distance limits in the complex structure moduli space. We find some potential candidates to yield asymptotic accelerated expansion with a flux potential satisfying gamma =frac{leftVert nabla VrightVert }{V}<sqrt{2} along its gradient flow. However, whether this truly describes an accelerated expanding cosmology remains as an open question until full moduli stabilization including the Kahler moduli is studied. Finally, we also reformulate the condition for forbidding asymptotic accelerated expansion as a convex hull de Sitter conjecture which resembles a convex hull scalar WGC for the membranes generating the flux potential. This provides a pictorial way to quickly determine the asymptotic gradient flow trajectory in multi-moduli setups and the value of γ along it.

Laplace Homotopy Perturbation Method (Lhpm) For Solving Systems Of N-Dimensional Non-Linear Partial Differential Equation

In this research, we proposed coupling the Laplace transform method and the homotopy Perturbation Method (LHPM). We employed the fusion of the Laplace method to make up for the shortcomings of other semi-analytical approaches like the homotopy perturbation method, variation iteration method, and the Adomian decomposition method. We aim to obtain an approximate and semi-analytic solution of the n-dimensional system of nonlinear partial differential equations. N-dimensional partial differential equations with nonlinear terms are known as nonlinear partial differential equations. They have been used to solve mathematical problems like the Poincaré conjecture and the Calabi-Yau conjecture and describe physical systems, from gravity to fluid dynamics. Therefore, we proffer a semi-analytic solution in the form of a Taylor multivariate series of displacements x, y, and time t using the proposed method. A side-by-side comparison was carried out to compare the exact solution with the new solution using 3-dimensional graphs, and thus the graph analysis followed. Results show excellent agreement, and the emergence of this method as a viable alternative demonstrates its viability by requiring fewer computations and being much easier and more convenient than others, making it suitable for widespread use in engineering as well.

CYJAX: A package for Calabi-Yau metrics with JAX

We present the first version of CYJAX, a package for machine learning Calabi–Yau metrics using JAX. It is meant to be accessible both as a top-level tool and as a library of modular functions. CYJAX is currently centered around the algebraic ansatz for the Kähler potential which automatically satisfies Kählerity and compatibility on patch overlaps. As of now, this implementation is limited to varieties defined by a single defining equation on one complex projective space. We comment on some planned generalizations.More documentation can be found at: https://cyjax.readthedocs.io.The code is available at: https://github.com/ml4physics/cyjax.

On the collapsing of Calabi–Yau manifolds and Kähler–Ricci flows

Abstract We study the collapsing of Calabi–Yau metrics and of Kähler–Ricci flows on fiber spaces where the base is smooth. We identify the collapsed Gromov–Hausdorff limit of the Kähler–Ricci flow when the divisorial part of the discriminant locus has simple normal crossings. In either setting, we also obtain an explicit bound for the real codimension-2 Hausdorff measure of the Cheeger–Colding singular set and identify a sufficient condition from birational geometry to understand the metric behavior of the limiting metric on the base.