avermeersch/calabi-yau-threefolds

---
name: Reflexive Polyhedra of Calabi-Yau Threefolds
date: '2023-09-28'
license: other
domain: Physics
tags:
- Calabi-Yau
- Toric Geometry
- String Theory
- Polyhedra
- Geometry
- Physics
pretty_name: Calabi-Yau 3-Folds
size_categories:
- 1K<n<10K
---

# Dataset Card for Reflexive Polyhedra of Calabi-Yau Threefolds

## Table of Contents

- [Dataset Description](#dataset-description)
- [General Information](#general-information)
- [Dataset Origin](#dataset-origin)
- [Dataset Characteristics](#dataset-characteristics)
- [Schema](#schema)
- [Data Fields](#data-fields)
- [Data Format](#data-format)
- [Usage](#usage)
- [Getting Started](#getting-started)
- [Machine Learning Applications](#machine-learning-applications)
- [Citations](#citations)

## Dataset Description

### General Information

Calabi-Yau threefolds are a special class of smooth, compact three-dimensional spaces that have become fundamental objects in both mathematics and theoretical physics. In the context of string theory, they serve as the internal dimensions over which strings compactify, leading to a four-dimensional effective theory. The geometry of these threefolds is closely related to many physical phenomena, including the number of particle generations, gauge symmetries, and the cosmological constant. This dataset encompasses all 4319 reflexive polyhedra in 3 dimensions, offering a comprehensive view of potential Calabi-Yau geometries. The reflexive polyhedra serve as dual representations of these threefolds and are crucial in understanding their topological and geometric properties.

### Dataset Origin

The dataset is derived from the original work documented in [hep-th/9805190](https://arxiv.org/abs/hep-th/9805190). While the original dataset was in a PALP-compatible structure, this version has been converted to a nested JSON format to better accommodate machine learning applications. The PALP-compatible version of the dataset can be accessed at [CYk3](http://hep.itp.tuwien.ac.at/~kreuzer/CY/CYk3.html).

## Dataset Characteristics

### Schema

The dataset is presented in a nested JSON format, with each entry containing both metadata and a matrix representing the vertices of the corresponding polyhedron.

### Data Fields

- `M1`, `M2`: These are point and vertex numbers in the M lattice, which is a mathematical lattice in the context of toric geometry. This lattice serves as the foundational geometric space from which the polyhedron is constructed.

- `N1`, `N2`: Similar to the M lattice, these are point and vertex numbers in the N lattice, which is dual to the M lattice. The N lattice provides a different but equally important geometric perspective for understanding the polyhedron.

- `Pic`: The Picard number is a topological invariant that measures the rank of the Néron-Severi group of a manifold. In the context of Calabi-Yau threefolds, it helps to determine the number of independent 2-cycles, which has physical implications like the number of U(1) gauge fields in the effective theory.

- `Cor`: The correction term is a specific mathematical entity that adjusts the Picard number to account for certain topological peculiarities. The Picard numbers of a polyhedron and its dual add up to \( 20 + \text{correction} \).

- `Matrix`: A 3xN matrix containing the coordinates of the vertices of the polyhedron. Each row represents a dimension in 3D space, and each column represents a vertex.

### Data Format

Each entry in the dataset is structured as follows:

```json
{
"M1": ...,
"M2": ...,
"N1": ...,
"N2": ...,
"Pic": ...,
"Cor": ...,
"Matrix": [
[...],
[...],
[...]
]
}
```

## Usage

### Getting Started

To access the dataset using the Hugging Face `datasets` library, the following Python code can be used:

```python
from datasets import load_dataset

dataset = load_dataset("calabi-yau-threefolds")
```

### Machine Learning Applications

This dataset provides rich opportunities for various machine learning tasks:

- Geometric deep learning for topological invariant prediction.
- Unsupervised learning techniques for polyhedra clustering.
- Graph neural networks to model vertex connections.

### Citations

For dataset usage, please cite the original paper using the following BibTeX entry:

```bibtex
@misc{kreuzer1998classification,
title={Classification of Reflexive Polyhedra in Three Dimensions},
author={M. Kreuzer and H. Skarke},
year={1998},
eprint={hep-th/9805190},
archivePrefix={arXiv},
primaryClass={hep-th}
}
```

原始信息汇总

数据集卡片：Calabi-Yau三维反射多面体

数据集描述

一般信息

Calabi-Yau三维流形是一类特殊的平滑、紧致三维空间，已成为数学和理论物理中的基本对象。在弦理论的背景下，它们作为弦紧化的内部维度，导致四维有效理论。这些三维流形的几何与许多物理现象密切相关，包括粒子代数、规范对称性和宇宙常数。该数据集包含所有4319个三维反射多面体，提供了潜在Calabi-Yau几何的全面视图。反射多面体作为这些三维流形的对偶表示，对于理解它们的拓扑和几何性质至关重要。

数据集来源

该数据集源自hep-th/9805190中记录的原始工作。虽然原始数据集采用PALP兼容结构，但此版本已转换为嵌套JSON格式，以更好地适应机器学习应用。PALP兼容版本的数据集可在CYk3访问。

数据集特征

模式

数据集以嵌套JSON格式呈现，每个条目包含元数据和表示相应多面体顶点的矩阵。

数据字段

• M1, M2：这是M格中的点和顶点编号，M格是环面几何学中的数学格。该格作为构建多面体的基本几何空间。

• N1, N2：类似于M格，这些是N格中的点和顶点编号，N格是M格的对偶。N格提供了理解多面体的不同但同样重要的几何视角。

• Pic：Picard数是一个拓扑不变量，测量流形的Néron-Severi群的秩。在Calabi-Yau三维流形的背景下，它有助于确定独立2-循环的数量，这在物理上有诸如有效理论中U(1)规范场的数量等影响。

• Cor：修正项是一个特定的数学实体，用于调整Picard数以考虑某些拓扑特殊性。多面体及其对偶的Picard数之和为 ( 20 + ext{修正} )。

• Matrix：一个3xN矩阵，包含多面体顶点的坐标。每行表示3D空间中的一个维度，每列表示一个顶点。

数据格式

数据集中的每个条目结构如下：

json { "M1": ..., "M2": ..., "N1": ..., "N2": ..., "Pic": ..., "Cor": ..., "Matrix": [ [...], [...], [...] ] }

使用

入门

使用Hugging Face datasets库访问数据集，可以使用以下Python代码：

python from datasets import load_dataset

dataset = load_dataset("calabi-yau-threefolds")

机器学习应用

该数据集为各种机器学习任务提供了丰富的机会：

• 用于拓扑不变量预测的几何深度学习。
• 用于多面体聚类的无监督学习技术。
• 用于建模顶点连接的图神经网络。

引用

使用数据集时，请使用以下BibTeX条目引用原始论文：

bibtex @misc{kreuzer1998classification, title={Classification of Reflexive Polyhedra in Three Dimensions}, author={M. Kreuzer and H. Skarke}, year={1998}, eprint={hep-th/9805190}, archivePrefix={arXiv}, primaryClass={hep-th} }