New Calabi-Yau Manifolds from Genetic Algorithms

TL;DR

This paper introduces a genetic algorithm to generate reflexive polytopes, leading to the discovery of new Calabi-Yau manifolds and expanding the known landscape of these geometric structures.

Contribution

It demonstrates the effectiveness of genetic algorithms in generating reflexive polytopes, including in higher dimensions, and identifies new Calabi-Yau four-folds with previously unknown properties.

Findings

Successfully reproduces known reflexive polytopes in 2D and 3D

Constructs new 5D reflexive polytopes with minimal vertices

Identifies new Calabi-Yau four-folds and Hodge numbers

Abstract

Calabi-Yau manifolds can be obtained as hypersurfaces in toric varieties built from reflexive polytopes. We generate reflexive polytopes in various dimensions using a genetic algorithm. As a proof of principle, we demonstrate that our algorithm reproduces the full set of reflexive polytopes in two and three dimensions, and in four dimensions with a small number of vertices and points. Motivated by this result, we construct five-dimensional reflexive polytopes with the lowest number of vertices and points. By calculating the normal form of the polytopes, we establish that many of these are not in existing datasets and therefore give rise to new Calabi-Yau four-folds. In some instances, the Hodge numbers we compute are new as well.