Lagrangian torus fibration of quintic Calabi-Yau hypersurfaces I: Fermat quintic case

Gauge theory and G2-geometry on Calabi-Yau links

The $7$-dimensional link $K$ of a weighted homogeneous hypersurface on the round $9$-sphere in $\mathbb{C}^5$ has a nontrivial null Sasakian structure which is contact Calabi-Yau, in many cases. It admits a canonical co-closed $\rm G_2$-structure $φ$ induced by the Calabi-Yau $3$-orbifold basic geometry. We distinguish these pairs $(K,φ)$ by the Crowley-Nordström $\mathbb{Z}_{48}$-valued $ν$ invariant, for which we prove odd parity and provide an algorithmic formula. We describe moreover a natural Yang-Mills theory on such spaces, with many important features of the torsion-free case, such as …

Energy functionals for Calabi-Yau metrics

We identify a set of "energy" functionals on the space of metrics in a given Kaehler class on a Calabi-Yau manifold, which are bounded below and minimized uniquely on the Ricci-flat metric in that class. Using these functionals, we recast the problem of numerically solving the Einstein equation as an optimization problem. We apply this strategy, using the "algebraic" metrics (metrics for which the Kaehler potential is given in terms of a polynomial in the projective coordinates), to the Fermat quartic and to a one-parameter family of quintics that includes the Fermat and conifold quintics. We …

Neural Network Approximations for Calabi-Yau Metrics

Ricci flat metrics for Calabi-Yau threefolds are not known analytically. In this work, we employ techniques from machine learning to deduce numerical flat metrics for the Fermat quintic, for the Dwork quintic, and for the Tian-Yau manifold. This investigation employs a single neural network architecture that is capable of approximating Ricci flat Kaehler metrics for several Calabi-Yau manifolds of dimensions two and three. We show that measures that assess the Ricci flatness of the geometry decrease after training by three orders of magnitude. This is corroborated on the validation set, where …

Special Lagrangian submanifolds and Algebraic complexity one Torus Actions

In the first part of this paper we consider compact algebraic manifolds M^2n with an algebraic (n-1)-Torus action. We show that there is a T-invariant meromorphic section $σ$ of the canonical bundle of M. Any such $σ$ defines a divisor D. On the complement M'=M-D we have a trivialization of the canonical bundle and a T-action. If the first cohomology of M' vanishes, then results of [2] imply that there is a Special Lagrangian (SLag) fibration on M'. We study how the fibers compactify in M and give examples of SLag fibrations on M', including some cases there the first cohomology of M' doesn't …

Calabi-Yau三维流形上的特征值与特征形式

Eigenvalues and eigenforms on Calabi-Yau threefolds

本文提出一种数值算法，通过输入近似度量计算Calabi-Yau流形上Laplace-de Rham算子的特征值与特征形式，并在Fermat五次三维流形和$\mathbb{P}^{3}$上验证了算法的有效性。

Calabi-Yau超曲面上线丛的Laplacian数值谱

Numerical spectra of the Laplacian for line bundles on Calabi-Yau hypersurfaces

本文首次数值计算了Calabi-Yau三重流形上取值于线丛的Dolbeault Laplacian谱，并通过三个例子验证了算法的准确性，为弦紧致化中Yukawa耦合计算提供了必要工具。