README.md

metadata

license: cc-by-sa-4.0
pretty_name: Weight Systems Defining Five-Dimensional IP Lattice Polytopes
configs:
  - config_name: non-reflexive
    data_files:
      - split: full
        path: non-reflexive/*.parquet
  - config_name: reflexive
    data_files:
      - split: full
        path: reflexive/*.parquet
tags:
  - physics
  - math

Weight Systems Defining Five-Dimensional IP Lattice Polytopes

This dataset contains all weight systems defining five-dimensional reflexive and non-reflexive IP lattice polytopes, instrumental in the study of Calabi-Yau fourfolds in mathematics and theoretical physics. The data was compiled by Harald Skarke and Friedrich Schöller in arXiv:1808.02422. More information is available at the Calabi-Yau data website. The dataset can be explored using the search frontend. See below for a short mathematical exposition on the construction of polytopes.

Please cite the paper when referencing this dataset:

@article{Scholler:2018apc,
    author = {Schöller, Friedrich and Skarke, Harald},
    title = "{All Weight Systems for Calabi-Yau Fourfolds from Reflexive Polyhedra}",
    eprint = "1808.02422",
    archivePrefix = "arXiv",
    primaryClass = "hep-th",
    doi = "10.1007/s00220-019-03331-9",
    journal = "Commun. Math. Phys.",
    volume = "372",
    number = "2",
    pages = "657--678",
    year = "2019"
}

Dataset Details

The dataset consists of two subsets: weight systems defining reflexive (and therefore IP) polytopes and weight systems defining non-reflexive IP polytopes. Each subset is split into 4000 files in Parquet format. Rows within each file are sorted lexicographically by weights.

Each row in the dataset represents a polytope and contains the six weights defining it, along with the vertex count, facet count, and lattice point count. The reflexive dataset also includes the Hodge numbers $h^{1,1}h^\{1,1\}$, $h^{1,2}h^\{1,2\}$, and $h^{1,3}h^\{1,3\}$ of the corresponding Calabi-Yau manifold, and the lattice point count of the dual polytope.

For any Calabi-Yau fourfold, the Euler characteristic $\chi \backslash chi$ and the Hodge number $h^{2,2}h^\{2,2\}$ can be derived as follows:

$$\chi =48+6(h^{1,1}-h^{1,2}+h^{1,3})\backslash chi\; =\; 48\; +\; 6\; (h^\{1,1\}\; -\; h^\{1,2\}\; +\; h^\{1,3\})$$

$$h^{2,2}=44+4h^{1,1}-2h^{1,2}+4h^{1,3}h^\{2,2\}\; =\; 44\; +\; 4\; h^\{1,1\}\; -\; 2\; h^\{1,2\}\; +\; 4\; h^\{1,3\}$$

This dataset is licensed under the CC BY-SA 4.0 license.

Data Fields

• weight0 to weight5: Weights of the weight system defining the polytope.
• vertex_count: Vertex count of the polytope.
• facet_count: Facet count of the polytope.
• point_count: Lattice point count of the polytope.
• dual_point_count: Lattice point count of the dual polytope (only for reflexive polytopes).
• h11: Hodge number $h^{1,1}h^\{1,1\}$ (only for reflexive polytopes).
• h12: Hodge number $h^{1,2}h^\{1,2\}$ (only for reflexive polytopes).
• h13: Hodge number $h^{1,3}h^\{1,3\}$ (only for reflexive polytopes).

Usage

The dataset can be used without downloading it entirely, thanks to the streaming capability of the datasets library. The following Python code snippet demonstrates how to stream the dataset and print the first five rows:

from datasets import load_dataset

dataset = load_dataset("cy-data/ws-5d", name="reflexive", split="full", streaming=True)

for row in dataset.take(5):
    print(row)

When cloning the Git repository with Git Large File Storage (LFS), data files are stored both in the Git LFS storage directory and in the working tree. To avoid occupying double the disk space, use a filesystem that supports copy-on-write, and run the following commands to clone the repository:

# Initialize Git LFS
git lfs install

# Clone the repository without downloading LFS files immediately
GIT_LFS_SKIP_SMUDGE=1 git clone https://huggingface.co/datasets/cy-data/ws-5d

# Change to the repository directory
cd ws-5d

# Test deduplication (optional)
git lfs dedup --test

# Download the LFS files
git lfs fetch

# Create working tree files as clones of the files in the Git LFS storage directory using
# copy-on-write functionality
git lfs dedup

Construction of Polytopes

This is an introduction to the mathematics involved in the construction of polytopes relevant to this dataset. For more details and precise definitions, consult the paper arXiv:1808.02422 and references therein.

Polytopes

A polytope is the convex hull of a finite set of points in $nn$-dimensional Euclidean space, ${\mathbb{R}}^n\backslash mathbb\{R\}^n$. This means it is the smallest convex shape that contains all these points. The minimal collection of points that define a particular polytope are its vertices. Familiar examples of polytopes include triangles and rectangles in two dimensions, and cubes and octahedra in three dimensions.

A polytope is considered an IP polytope (interior point polytope) if the origin of ${\mathbb{R}}^n\backslash mathbb\{R\}^n$ is in the interior of the polytope, not on its boundary or outside it.

For any IP polytope $\mathrm{\nabla }\backslash nabla$, its dual polytope ${\mathrm{\nabla }}^{\ast }\backslash nabla^*$ is defined as the set of points $\mathbf{y}\backslash mathbf\{y\}$ satisfying

$$\mathbf{x}\cdot \mathbf{y}\ge -1\text{for all }\mathbf{x}\in \mathrm{\nabla }\mathrm{.}\backslash mathbf\{x\}\; \backslash cdot\; \backslash mathbf\{y\}\; \backslash ge\; -1\; \backslash quad\; \backslash text\{for\; all\; \}\; \backslash mathbf\{x\}\; \backslash in\; \backslash nabla\; \backslash ;.$$

This relationship is symmetric: the dual of the dual of a polytope is the polytope itself, i.e., ${\mathrm{\nabla }}^{\ast \ast }=\mathrm{\nabla }\backslash nabla^\{**\}\; =\; \backslash nabla$.

Weight Systems

Weight systems provide a means to describe simple polytopes known as simplexes. More broadly, combined weight systems, which are collections of individual weight systems, can describe any polytope. A combined weight system is a matrix consisting of real numbers. The construction process is outlined as follows:

Consider a polytope in ${\mathbb{R}}^n\backslash mathbb\{R\}^n$ with vertex count $kk$, where $kk$ is bigger than $nn$. It is possible to position $nn$ of these vertices at arbitrary (linearly independent) locations through a linear transformation. The placement of the remaining $k-nk\; -\; n$ vertices is then determined. Their positions are the defining properties of a polytope. To specify these positions independently of the applied linear transformation, one can use the following system of equations. If ${\mathbf{v}}_0,{\mathbf{v}}_1,\dots {\mathbf{v}}_{k-1}\backslash mathbf\{v\}\_0,\; \backslash mathbf\{v\}\_1,\; \backslash dots\; \backslash mathbf\{v\}\_\{k-1\}$ are the vertices of the polytope, these relations fix $k-nk\; -\; n$ vertices in terms of the other $nn$:

$$\sum _{i=0}^{k-1}{q}_i^{(j)}{\mathbf{v}}_i=0\text{for }0\le j\le k-n-1,\backslash sum\_\{i=0\}^\{k-1\}\; q\_i^\{(j)\}\; \backslash mathbf\{v\}\_i\; =\; 0\; \backslash quad\; \backslash text\{for\; \}\; 0\; \backslash le\; j\; \backslash le\; k\; -\; n\; -\; 1\; \backslash ;,$$

where $q_i^{(j)}q\_i^\{(j)\}$ is the matrix of real numbers, the combined weight system. In cases where $k=n+1k\; =\; n\; +\; 1$, $jj$ is limited to the value zero, reducing the matrix to a single weight system $q_{i}q\_i$. In this scenario, the polytope is a simplex, and the equation simplifies to:

$$\sum _{i=0}^n{q}_i{\mathbf{v}}_i=0\mathrm{.}\backslash sum\_\{i=0\}^n\; q\_i\; \backslash mathbf\{v\}\_i\; =\; 0\; \backslash ;.$$

It is important to note that scaling all weights in a weight system by a common factor results in an equivalent weight system that defines the same polytope.

For this dataset, the focus is on a specific construction of lattice polytopes described in subsequent sections.

Lattice Polytopes

A lattice polytope is a polytope with vertices at the points of a regular grid, or lattice. Using linear transformations, any lattice polytope can be transformed so that its vertices have integer coordinates, hence they are also referred to as integral polytopes.

The dual of a lattice with points $LL$ is the lattice consisting of all points $\mathbf{y}\backslash mathbf\{y\}$ that satisfy

$$\mathbf{x}\cdot \mathbf{y}\in \mathbb{Z}\text{for all }\mathbf{x}\in L\mathrm{.}\backslash mathbf\{x\}\; \backslash cdot\; \backslash mathbf\{y\}\; \backslash in\; \backslash mathbb\{Z\}\; \backslash quad\; \backslash text\{for\; all\; \}\; \backslash mathbf\{x\}\; \backslash in\; L\; \backslash ;.$$

Reflexive polytopes are a specific type of lattice polytope characterized by having a dual that is also a lattice polytope, with vertices situated on the dual lattice. These polytopes play a central role in the context of this dataset.

The weights of a lattice polytope are always rational. This characteristic enables the rescaling of a weight system so that its weights become integers without any common divisor. This rescaling has been performed in this dataset.

Typically, the dual of a lattice polytope defined by a weight system is not a lattice polytope. However, our interest lies in a different construction than simply considering polytopes defined by (combined) weight systems, as described above. In this construction, they are just the starting point. We start with the polytope $\mathrm{\nabla }\backslash nabla$, arising from a weight system as previously described. Then, we define the polytope $\mathrm{\Delta }\backslash Delta$ as the convex hull of the intersection of ${\mathrm{\nabla }}^{\ast }\backslash nabla^*$ with the points of the dual lattice. In the context of this dataset, the polytope $\mathrm{\Delta }\backslash Delta$ is referred to as ‘the polytope’. Correspondingly, ${\mathrm{\Delta }}^{\text{\hspace{0.17em}⁣}\ast }\backslash Delta^\{\backslash !*\}$ is referred to as ‘the dual polytope’. The lattice of $\mathrm{\Delta }\backslash Delta$ is taken to be the coarsest lattice possible, such that $\mathrm{\nabla }\backslash nabla$ is a lattice polytope, i.e., the lattice generated by the vertices of $\mathrm{\nabla }\backslash nabla$. This construction is exemplified in the following sections.

A weight system is considered an IP weight system if the corresponding $\mathrm{\Delta }\backslash Delta$ is an IP polytope; that is, the origin is within its interior. Since only IP polytopes have corresponding dual polytopes, this condition is essential for the polytope $\mathrm{\Delta }\backslash Delta$ to be classified as reflexive.

Two Dimensions

In two dimensions, all IP weight systems define reflexive polytopes and every vertex of ${\mathrm{\nabla }}^{\ast }\backslash nabla^*$ lies on the dual lattice, making $\mathrm{\Delta }\backslash Delta$ and ${\mathrm{\nabla }}^{\ast }\backslash nabla^*$ identical. There are exactly three IP weight systems that define two-dimensional polytopes (polygons). Each polytope is reflexive and has three vertices and three facets (edges):

weight system	number of points of $\mathrm{\nabla }\backslash nabla$	number of points of ${\mathrm{\nabla }}^{\ast }\backslash nabla^*$
(1, 1, 1)	4	10
(1, 1, 2)	5	9
(1, 2, 3)	7	7

We will now construct these polytopes from their corresponding weight system. Fixing the first two vertices of the polytopes

$${\mathbf{v}}_0=(1,0)\text{and}{\mathbf{v}}_1=(0,1),\backslash mathbf\{v\}\_0\; =\; (1,\; 0)\; \backslash quad\; \backslash text\{and\}\; \backslash quad\; \backslash mathbf\{v\}\_1\; =\; (0,\; 1)\; \backslash ;,$$

one can obtain the position of the third vertex by solving the weight system equation from before:

$${\mathbf{v}}_2=-\frac{q_0{\mathbf{v}}_0+q_1{\mathbf{v}}_1}{q_2}\mathrm{.}\backslash mathbf\{v\}\_2\; =\; -\; \backslash frac\{q\_0\; \backslash mathbf\{v\}\_0\; +\; q\_1\; \backslash mathbf\{v\}\_1\}\{q\_2\}\; \backslash ;.$$

The resulting polytopes and their duals are depicted below. Lattice points are indicated by dots.

One may notice that a simpler description could be obtained by fixing ${\mathbf{v}}_2=(1,0)\backslash mathbf\{v\}\_2\; =\; (1,\; 0)$ instead of ${\mathbf{v}}_0\backslash mathbf\{v\}\_0$, which would avoid fractional vertex coordinates. However, this approach would not illustrate the general case in higher dimensions, where this is not possible since there is not always a weight equal to 1.

General Dimension

In higher dimensions, the situation becomes more complex. Not all IP polytopes are reflexive, and generally, $\mathrm{\Delta }\mathrm{\ne }{\mathrm{\nabla }}^{\ast }\backslash Delta\; \backslash neq\; \backslash nabla^*$.

This example shows the construction of the three-dimensional polytope $\mathrm{\Delta }\backslash Delta$ with weight system (2, 3, 4, 5) and its dual ${\mathrm{\Delta }}^{\text{\hspace{0.17em}⁣}\ast }\backslash Delta^\{\backslash !*\}$. Lattice points lying on the polytopes are indicated by dots. $\mathrm{\Delta }\backslash Delta$ has 7 vertices and 13 lattice points, ${\mathrm{\Delta }}^{\text{\hspace{0.17em}⁣}\ast }\backslash Delta^\{\backslash !*\}$ also has 7 vertices, but 16 lattice points.

The counts of reflexive single-weight-system polytopes by dimension $nn$ are:

$nn$	reflexive single-weight-system polytopes
2	3
3	95
4	184,026
5	(this dataset) 185,269,499,015