Article
Mathematics
Lorenzo Foscolo, Mark Haskins, Johannes Nordstrom
Summary: A new analytic method has been developed to construct complete noncompact Ricci-flat 7-manifolds, specifically G(2)-manifolds, and establish a connection with the Cheeger-Fukaya-Gromov theory. The construction involves a complete noncompact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M -> B satisfying a necessary topological condition, leading to a 1-parameter family of circle-invariant complete G(2)-metrics. The constructed G(2)-metrics have controlled asymptotic geometry at infinity, providing new insights into the existence of continuous families of complete noncompact G(2)-metrics of arbitrarily high dimension.
DUKE MATHEMATICAL JOURNAL
(2021)
Article
Mathematics
Tristan C. Collins, Bin Guo, Freid Tong
Summary: The study explores degenerations of Ricci-flat Kahler metrics as the Kahler class degenerates, showing convergence to incomplete smooth metrics under certain assumptions. Singular Calabi-Yau metrics with asymptotically conical behavior are constructed on certain quasi-projective varieties, with metric geometry homeomorphic to the topology of the singular variety. These results are then applied to examine geometric transitions between Calabi-Yau manifolds.
MATHEMATISCHE ANNALEN
(2022)
Article
Mathematics
Lothar Schiemanowski
Summary: A natural approach to constructing nearly G(2) manifolds is to resolve nearly G(2) spaces with isolated conical singularities by gluing in asymptotically conical G(2) manifolds modeled on the same cone. However, we show that in many cases, such a resolution does not exist. This is based on topological results for asymptotically conical G(2) manifolds and asymptotically conical Calabi-Yau 6-manifolds.
JOURNAL OF GEOMETRIC ANALYSIS
(2023)
Article
Mathematics
Martin de Borbon, Cristiano Spotti
Summary: We construct asymptotically conical Calabi-Yau metrics with cone singularities along a compact simple normal crossing divisor, which includes the minimal resolution of 2D quotient singularities for any finite subgroup Gamma subset of U(2) acting freely on the threesphere. This generalized construction extends Kronheimer's smooth ALE gravitational instantons to more complex geometric backgrounds.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2021)
Article
Mathematics
Jakob Stein
Summary: We provide a detailed description of the behavior of Calabi-Yau instantons and monopoles on Calabi-Yau 3-folds with asymptotically conical geometry and SU(2)(2) acting with co-homogeneity one. We discover new families of invariant instantons in gauge theory on the smoothing and small resolution of the conifold, and on the canonical bundle of CP1 x CP1. We classify the moduli-spaces of instantons and monopoles satisfying a curvature decay condition, and observe the expected bubbling phenomena in these families of instantons.
JOURNAL OF GEOMETRIC ANALYSIS
(2023)
Article
Astronomy & Astrophysics
Wei Cui, Xin Gao, Juntao Wang
Summary: Generalized complete intersection Calabi-Yau manifold (gCICY) is a new construction of Calabi-Yau manifolds. The standard algebraic method to generate new gCICYs is laborious, leading to unknown numbers and classifications. This paper explores the use of neural networks for progress in this area. Results showed high accuracy in classifying existing gCICYs and predicting new ones not used in training, demonstrating the effectiveness of machine learning in this field.
Article
Mathematics
Leonid Chekhov, Marta Mazzocco, Vladimir Rubtsov
Summary: This paper studies a class of quantum del Pezzo surfaces, introduces the generalized Sklyanin-Painleve algebra and characterizes its properties. The algebra includes various known algebras and manifolds as special cases.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics
Tristan C. Collins, Adam Jacob, Yu-Shen Lin
Summary: The paper explores the existence of special Lagrangian submanifolds on log Calabi-Yau manifolds with complete Ricci-flat Kahler metric constructed by Tian and Yau. It is shown that in complex dimension 2, special Lagrangian torus fibrations exist, confirming conjectures made by various scholars. Additionally, the paper identifies singular fibers in specific cases, confirming predictions and conjectures.
DUKE MATHEMATICAL JOURNAL
(2021)
Article
Astronomy & Astrophysics
Harold Erbin, Riccardo Finotello
Summary: This study predicts the Hodge numbers of Calabi-Yau 3-folds using machine learning, showing that neural networks can improve accuracy compared to existing literature. Through exploratory data analysis, design of validation procedures and baseline models, and comparison of ML models, the study successfully enhanced accuracy in predicting Hodge numbers.
Article
Computer Science, Artificial Intelligence
Magdalena Larfors, Andre Lukas, Fabian Ruehle, Robin Schneider
Summary: In this paper, we introduce the use of neural networks to compute numerical Ricci-flat Calabi-Yau metrics for complete intersection and Kreuzer-Skarke manifolds, and present the cymetric package for implementing these techniques. We develop methods for point-sampling on these manifolds and train the neural networks using a custom loss function. Our results demonstrate that volumes and line bundle slopes can be accurately computed from the resulting Ricci-flat metrics, and we also apply our approach to compute an approximate Hermitian-Yang-Mills connection on a specific line bundle.
MACHINE LEARNING-SCIENCE AND TECHNOLOGY
(2022)
Article
Physics, Particles & Fields
James Gray, Juntao Wang
Summary: Non-simply connected Calabi-Yau threefolds are important in string theory and can be described by quotienting a simply connected Calabi-Yau variety by a freely acting discrete symmetry. This paper classifies cyclic symmetries descending from linear actions on the ambient spaces of these threefolds, presenting a list of 129 symmetries/non-simply connected Calabi-Yau threefolds.
JOURNAL OF HIGH ENERGY PHYSICS
(2022)
Article
Mathematics
Daisuke Inoue
Summary: In this paper, new examples of derived equivalent Calabi-Yau 3-folds with Picard number greater than one are constructed, and their mirror Calabi-Yau manifolds are studied.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics, Applied
Tristan Bozec, Damien Calaque, Sarah Scherotzke
Summary: This study demonstrates the connection between relative Calabi-Yau structures on noncommutative moment maps and (quasi-)bisymplectic structures. The authors also apply this connection to investigate the Poisson structures on the moduli spaces of representations of deformed multiplicative preprojective algebras.
FORUM OF MATHEMATICS SIGMA
(2023)
Article
Mathematics
Martin De Borbon, Cristiano Spotti
Summary: In this paper, we prove the existence of a Ricci-flat Kahler metric g(RF) with cone angle 2pi beta(j) along a given finite collection of lines Lj subset of CP2, satisfying certain natural conditions. This metric asymptotically approaches a polyhedral Kahler cone at each multiple point. Additionally, we discuss a Chern-Weil formula that relates the energy of gRF to a logarithmic Euler characteristic, with points weighted according to the volume density of the metric.
JOURNAL OF DIFFERENTIAL GEOMETRY
(2023)
Article
Mathematics
David Fernandez, Estanislao Herscovich
Summary: In this article, it is proven that double quasi-Poisson algebras naturally lead to pre-Calabi-Yau algebras, with the higher multiplications indexed by even integers not vanishing but being given by cyclical expressions involving Bernoulli numbers as coefficients. This is in contrast to the pre-Calabi-Yau algebra constructed in previous works, which had higher multiplications that vanish.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2022)
Article
Physics, Mathematical
Ronan J. Conlon, Rafe Mazzeo, Frederic Rochon
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2015)
Article
Mathematics
Ronan J. Conlon, Hans-Joachim Hein
GEOMETRIC AND FUNCTIONAL ANALYSIS
(2015)
Article
Mathematics
Ronan J. Conlon, Hans-Joachim Hein
DUKE MATHEMATICAL JOURNAL
(2013)
Article
Mathematics
Ronan J. Conlon, Alix Deruelle
JOURNAL OF DIFFERENTIAL GEOMETRY
(2020)
