Efficient Algorithm for Generating Homotopy Inequivalent Calabi-Yaus

TL;DR

The paper introduces an efficient algorithm for exploring inequivalent Calabi-Yau threefold hypersurfaces in toric varieties by reducing redundancy in triangulation enumeration, significantly decreasing computational effort.

Contribution

It proposes a novel method to generate distinct triangulations using interior height vectors, avoiding exhaustive enumeration of all fine, regular, star triangulations.

Findings

Reduces the number of operations needed to generate triangulations by orders of magnitude.

Provides a method to directly generate the support of the secondary subfan for sampling.

Demonstrates the approach on the Kreuzer-Skarke database.

Abstract

We present an algorithm for efficiently exploring inequivalent Calabi-Yau threefold hypersurfaces in toric varieties. A direct enumeration of fine, regular, star triangulations (FRSTs) of polytopes in the Kreuzer-Skarke database is foreseeably impossible due to the large count of distinct FRSTs. Moreover, such an enumeration is needlessly redundant because many such triangulations have the same restrictions to 2-faces and hence, by Wall's theorem, lead to equivalent Calabi-Yau threefolds. We show that this redundancy can be circumvented by finding a height vector in the strict interior of the intersection of the secondary cones associated with each 2-face triangulation. We demonstrate that such triangulations are generated with orders of magnitude fewer operations than the naive approach of generating all FRSTs and selecting only those differing on 2-faces. Similar methods are also presented to directly generate (the support of) the secondary subfan of all fine triangulations, relevant for random sampling of FRSTs.