Calabi-Yau Research Articles

Movable cones of complete intersections of multidegree one on products of projective spaces

We study Calabi–Yau manifolds which are complete intersections of hypersurfaces of multidegree 1 in an m-fold product of n-dimensional projective spaces. Using the theory of Coxeter groups, we show that the birational automorphism group of such a Calabi–Yau manifold X is infinite and a free product of copies of Z. Moreover, we give an explicit description of the boundary of the movable cone Mov¯(X). In the end, we consider examples for the general and non-general case and picture the movable cone and the fundamental domain for the action of Bir(X).

Symplectic Cuts and Open/Closed Strings I

This paper introduces a concrete relation between genus zero closed Gromov–Witten invariants of Calabi–Yau threefolds and genus zero open Gromov–Witten invariants of a Lagrangian A-brane in the same threefold. Symplectic cutting is a natural operation that decomposes a symplectic manifold (X,ω) with a Hamiltonian U(1) action into two pieces glued along an invariant divisor. In this paper we study a quantum uplift of the cut construction defined in terms of equivariant gauged linear sigma models. The nexus between closed and open Gromov–Witten invariants is a quantum Lebesgue measure associated to a choice of cut, that we introduce and study. Integration of this measure recovers the equivariant quantum volume of the whole CY3, thereby encoding closed Gromov–Witten invariants. Conversely, the monodromies of the quantum measure around cycles in Kähler moduli space encode open Gromov–Witten invariants of a Lagrangian A-brane associated to the cut. Both in the closed and the open string sector we find a remarkable interplay between worldsheet instantons and semiclassical volumes regularized by equivariance. This leads to equivariant generating functions of GW invariants that extend smoothly across the entire moduli space, and which provide a unifying description of standard GW potentials. The latter are recovered in the non-equivariant limit in each of the different phases of the geometry.

Modular Calabi-Yau fourfolds and connections to M-theory fluxes

In this work, we study the local zeta functions of Calabi-Yau fourfolds. This is done by developing arithmetic deformation techniques to compute the factor of the zeta function that is attributed to the horizontal four-form cohomology. This, in turn, is sensitive to the complex structure of the fourfold. Focusing mainly on examples of fourfolds with a single complex structure parameter, it is demonstrated that the proposed arithmetic techniques are both applicable and consistent. We present a Calabi-Yau fourfold for which a factor of the horizontal four-form cohomology further splits into two pieces of Hodge type (4, 0) + (2, 2) + (0, 4) and (3, 1) + (1, 3). The latter factor corresponds to a weight-3 modular form, which allows expressing the value of the periods in terms of critical values of the L-function of this modular form, in accordance with Deligne’s conjecture. The arithmetic considerations are related to M-theory Calabi-Yau fourfold compactifications with background four-form fluxes. We classify such background fluxes according to their Hodge type. For those fluxes associated to modular forms, we express their couplings in the low-energy effective action in terms of L-function values.

Machine learning complete intersection Calabi-Yau 3-folds

Gaussian process regression, kernel support vector regression, the random forest, extreme gradient boosting, and the generalized linear model algorithms are applied to data of complete intersection Calabi-Yau threefolds. It is shown that Gaussian process regression is the most suitable for learning the Hodge number h2,1 in terms of h1,1. The performance of this regression algorithm is such that the Pearson correlation coefficient for the validation set is R2=0.9999999995 with a root mean square error RMSE=0.0002895011. As for the train set, these two parameters are as follows: R2=0.9999999994 and RMSE=0.0002854348. The training error and the cross-validation error of this regression are 1×10−9 and 1.28×10−7, respectively. Learning the Hodge number h1,1 in terms of h2,1 yields R2=1.000000 and RMSE=7.395731×10−5 for the validation set of the Gaussian process regression. Published by the American Physical Society 2024

Deformations of some local Calabi-Yau manifolds

We study deformations of certain crepant resolutions of isolated rational Gorenstein singularities. After a general discussion of the deformation theory, we specialize to dimension $3$ and consider examples which are good (log) resolutions as well as the case of small resolutions. We obtain some partial results on the classification of canonical threefold singularities that admit good crepant resolutions. Finally, we study a noncrepant example, the blowup of a small resolution whose exceptional set is a smooth curve.

Generating Triangulations and Fibrations with Reinforcement Learning

We apply reinforcement learning (RL) to generate fine regular star triangulations of reflexive polytopes, that give rise to smooth Calabi-Yau (CY) hypersurfaces. We demonstrate that, by simple modifications to the data encoding and reward function, one can search for CYs that satisfy a set of desirable string compactification conditions. For instance, we show that our RL algorithm can generate triangulations together with holomorphic vector bundles that satisfy anomaly cancellation and poly-stability conditions in heterotic compactification. Furthermore, we show that our algorithm can be used to search for reflexive subpolytopes together with compatible triangulations that define fibration structures of the CYs.

Symmetries of Fano varieties

Abstract Prokhorov and Shramov proved that the BAB conjecture, which Birkar later proved, implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension. This property in particular gives an upper bound on the size of finite semi-simple groups (i.e., those with no nontrivial normal abelian subgroups) acting faithfully on 𝑛-dimensional complex Fano varieties, and this bound only depends on 𝑛. We investigate the geometric consequences of an action by a certain semi-simple group: the symmetric group. We give an effective upper bound for the maximal symmetric group action on an 𝑛-dimensional Fano variety. For certain classes of varieties – toric varieties and Fano weighted complete intersections – we obtain optimal upper bounds. Finally, we draw a connection between large symmetric actions and boundedness of varieties, by showing that the maximally symmetric Fano fourfolds form a bounded family. Along the way, we also show analogues of some of our results for Calabi–Yau varieties and log terminal singularities.

Derived equivalence and homological projective duality in GLSM

Brane transport provides a way to realize functors between the categories of B-branes of different phases in gauged linear sigma model (GLSM). When appropriately designed, these functors induce derived equivalence between Calabi–Yau manifolds. In some cases, brane transport can also be used to extract the homological projective dual (HPD) of certain projective embeddings. We describe the GLSMs realizing derived equivalence between different geometries via brane transport with emphasis on the nonabelian models, and how to construct the GLSMs where brane transport gives rise to the embedding of HPD category into the derived category of universal hyperplane section. The latter gives rise to an explanation of the relationship between GLSM and HPD, and also a noncommutative geometric interpretation to HPD.

Orientifold Calabi-Yau threefolds: divisor exchanges and multi-reflections

Using the Kreuzer-Skarke database of 4-dimensional reflexive polytopes, we systematically constructed a new database of orientifold Calabi-Yau threefolds with h1,1(X) ≤ 12. Our approach involved non-trivial ℤ2 involutions, incorporating both divisor exchanges and multi-divisor reflections acting on the Calabi-Yau threefolds. Each proper involution results in an orientifold Calabi-Yau threefolds and we constructed 320, 386, 067 such examples. We developed a novel algorithm that significantly reduces the complexity of determining all the fixed loci under the involutions, and clarifies the types of O-planes. Our results show that under proper involutions, the majority of cases end up with O3/O7-plane systems, and most of these further admit a naive Type IIB string vacua. Additionally, a new type of free action was determined. We also computed the smoothness and the splitting of Hodge numbers in the ℤ2-orbifold limit for these orientifold Calabi-Yau threefolds.

The mirror Clemens–Schmid sequence

We introduce a four-term long exact sequence that relates the cohomology of a smooth variety admitting a projective morphism onto a projective base to the cohomology of the open set obtained by removing the preimage of a general linear section. We show that this sequence respects the perverse Leray filtration and induces exact sequences of mixed Hodge structures on its graded pieces. We conjecture that this exact sequence should be thought of as mirror to the Clemens–Schmid sequence, which describes the cohomology of degenerations. We exhibit this mirror relationship explicitly for all Type II and many Type III degenerations of K3 surfaces. In three dimensions, we show that for Tyurin degenerations of Calabi–Yau threefolds our conjecture is a consequence of existing mirror conjectures, and we explicitly verify our conjecture for a more complicated degeneration of the quintic threefold.

Emergent potentials and non-perturbative open topological strings

We show that integrating out M2 branes ending on M5 branes inside Calabi-Yau manifolds captures non-perturbative open topological string physics. The integrating out is performed using a contour integral in complexified Schwinger proper time. For the resolved conifold, this contour can be extended to include the zero pole, which we argue captures the ultraviolet completion of the integrating out and yields the tree-level polynomial terms in the free energy. This is a manifestation of the Emergence Proposal, and provides further evidence for it. Unlike the case of closed strings, where the emergent terms are kinetic terms in the action, for these open strings it is tree-level potential terms which are emergent. This provides a first quantitative example of the proposal that classical tree-level potentials in string theory emerge from integrating out co-dimension one states.

Metric flows with neural networks

Abstract We develop a general theory of flows in the space of Riemannian metrics induced by neural network (NN) gradient descent. This is motivated in part by recent advances in approximating Calabi–Yau metrics with NNs and is enabled by recent advances in understanding flows in the space of NNs. We derive the corresponding metric flow equations, which are governed by a metric neural tangent kernel (NTK), a complicated, non-local object that evolves in time. However, many architectures admit an infinite-width limit in which the kernel becomes fixed and the dynamics simplify. Additional assumptions can induce locality in the flow, which allows for the realization of Perelman’s formulation of Ricci flow that was used to resolve the 3d Poincaré conjecture. We demonstrate that such fixed kernel regimes lead to poor learning of numerical Calabi–Yau metrics, as is expected since the associated NNs do not learn features. Conversely, we demonstrate that well-learned numerical metrics at finite-width exhibit an evolving metric-NTK, associated with feature learning. Our theory of NN metric flows therefore explains why NNs are better at learning Calabi–Yau metrics than fixed kernel methods, such as the Ricci flow.

Spinor–Vector Duality and Mirror Symmetry

Mirror symmetry was first observed in worldsheet string constructions, and was shown to have profound implications in the Effective Field Theory (EFT) limit of string compactifications, and for the properties of Calabi–Yau manifolds. It opened up a new field in pure mathematics, and was utilised in the area of enumerative geometry. Spinor–Vector Duality (SVD) is an extension of mirror symmetry. This can be readily understood in terms of the moduli of toroidal compactification of the Heterotic String, which includes the metric the antisymmetric tensor field and the Wilson line moduli. In terms of the toroidal moduli, mirror symmetry corresponds to mappings of the internal space moduli, whereas Spinor–Vector Duality corresponds to maps of the Wilson line moduli. In the past few of years, we demonstrated the existence of Spinor–Vector Duality in the effective field theory compactifications of string theories. This was achieved by starting with a worldsheet orbifold construction that exhibited Spinor–Vector Duality and resolving the orbifold singularities, hence generating a smooth, effective field theory limit with an imprint of the Spinor–Vector Duality. Just like mirror symmetry, the Spinor–Vector Duality can be used to study the properties of complex manifolds with vector bundles. Spinor–Vector Duality offers a top-down approach to the “Swampland” program, by exploring the imprint of the symmetries of the ultra-violet complete worldsheet string constructions in the effective field theory limit. The SVD suggests a demarcation line between (2,0) EFTs that possess an ultra-violet complete embedding versus those that do not.

Smooth asymptotics for collapsing Calabi–Yau metrics

AbstractWe prove that Calabi–Yau metrics on compact Calabi–Yau manifolds whose Kähler classes shrink the fibers of a holomorphic fibration have a priori estimates of all orders away from the singular fibers. To this end, we prove an asymptotic expansion of these metrics in terms of powers of the fiber diameter, with ‐order remainders that satisfy uniform ‐estimates with respect to a collapsing family of background metrics. The constants in these estimates are uniform not only in the sense that they are independent of the fiber diameter, but also in the sense that they only depend on the constant in the estimate for known from previous work of the second‐named author. For , the new estimates are proved by blowup and contradiction, and each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.

Boundedness of elliptic Calabi–Yau varieties with a rational section

Boundedness of elliptic Calabi–Yau varieties with a rational section

Group-invariant machine learning on the Kreuzer-Skarke dataset

We use supervised machine learning to predict Hodge numbers for Calabi-Yau threefolds encoded by reflexive polyhedra. The Hodge number is invariant to the order of the vertices and the swapping of axes. Incorporating these properties, i.e. the invariance of column and row permutations for a matrix containing the polyhedron's vertices, promises better performance for Hodge number prediction. On a medium-sized subset of the Kreuzer-Skarke dataset, we train and evaluate approaches with different degrees of invariance. Our comparison shows that machine learning models incorporating symmetries actually outperform models that do not, with our best model achieving almost 97% accuracy.

Non-minimal elliptic threefolds at infinite distance. Part I. Log Calabi-Yau resolutions

We study infinite-distance limits in the complex structure moduli space of elliptic Calabi-Yau threefolds. In F-theory compactifications to six dimensions, such limits include infinite-distance trajectories in the non-perturbative open string moduli space. The limits are described as degenerations of elliptic threefolds whose central elements exhibit non-minimal elliptic fibers, in the Kodaira sense, over curves on the base. We show how these non-crepant singularities can be removed by a systematic sequence of blow-ups of the base, leading to a union of log Calabi-Yau spaces glued together along their boundaries. We identify criteria for the blow-ups to give rise to open chains or more complicated trees of components and analyse the blow-up geometry. While our results are general and applicable to all non-minimal degenerations of Calabi-Yau threefolds in codimension one, we exemplify them in particular for elliptic threefolds over Hirzebruch surface base spaces. We also explain how to extract the gauge algebra for F-theory probing such reducible asymptotic geometries. This analysis is the basis for a detailed F-theory interpretation of the associated infinite-distance limits that will be provided in a companion paper [1].

The classification of automorphisms and quotients of Calabi-Yau threefolds of type A

The aim of the paper is to investigate the only two families FGA of Calabi-Yau 3-folds A/G with A an abelian 3-fold and G≤Aut(A) a finite group acting freely: one is constructed in [11] and the other is presented here. We provide a complete classification of the automorphism group of X∈FGA. Additionally, we construct and classify the quotients X/ϒ for any ϒ≤Aut(X). Specifically, for those groups ϒ that preserve the volume form of X then X/ϒ admits a desingularization Y which is a Calabi-Yau 3-fold: we compute the Hodge numbers and the fundamental group of these Y, thereby determining all topological in-equivalent Calabi-Yau 3-folds obtained in this way.

Introduction to Algebraic Deformation Theory and The Case of k-Linear Categories

Deformation theory is a branch of mathematics which studies how mathematical objects, such as algebraic varieties, schemes, algebras, or categories, can be deformed “continuously” depending on a space of parameter while preserving certain algebraic or geometric structures. Algebraic deformation theory, which was pioneered by Murray Gerstenhaber in 1960s-1970s, established its role as a cornerstone in modern mathematics and theoretical physics. This theory provides a powerful framework for understanding the subtle variations and deformations of mathematical and physical objects depending on a parameter space. Lying at the interface of algebra, geometry and topology this theory has been being studied extensively worldwide and obtained many applications in various areas of mathematics and theoretical physics, such as the study of Calabi-Yau manifolds, mirror symmetry, quantum physics. In such context, this article aims to introduce this important mathematical theory to the community of Vietnamese mathematicians in a hope to bring more attention of Vietnamese mathematicians and math students to this vibrant research area. We also expect that this topic will be taught at universities in Vietnam in the near future.

Algorithms for representations of quiver Yangian algebras

In this note, we aim to review algorithms for constructing crystal representations of quiver Yangians in detail. Quiver Yangians are believed to describe an action of the BPS algebra on BPS states in systems of D-branes wrapping toric Calabi-Yau three-folds. Crystal modules of these algebras originate from molten crystal models for Donaldson-Thomas invariants of respective three-folds. Despite the fact that this subject was originally at the crossroads of algebraic geometry with effective supersymmetric field theories, equivariant toric action simplifies applied calculations drastically. So the sole pre-requisite for this algorithm’s implementation is linear algebra. It can be easily taught to a machine with the help of any symbolic calculation system. Moreover, these algorithms may be generalized to toroidal and elliptic algebras and exploited in various numerical experiments with those algebras. We illustrate the work of the algorithms in applications to simple cases of Ysl2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extrm{Y}\\left({\\mathfrak{sl}}_2\\right) $$\\end{document}, Ygl̂1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extrm{Y}\\left({\\hat{\\mathfrak{gl}}}_1\\right) $$\\end{document} and Ygl̂11\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extrm{Y}\\left({\\hat{\\mathfrak{gl}}}_{\\left.1\\right|1}\\right) $$\\end{document}.