All Weight Systems for Calabi-Yau Fourfolds from Reflexive Polyhedra

TL;DR

This paper provides a comprehensive classification of weight systems in five dimensions, identifying over 322 billion systems that generate reflexive polytopes and Calabi-Yau fourfolds, significantly advancing the understanding of their geometric structures.

Contribution

It presents the first complete classification of sextuples of weights for all reflexive polytopes in five dimensions, revealing the vast landscape of Calabi-Yau fourfolds.

Findings

Identified 322,383,760,930 weight systems in five dimensions.

Found 185,269,499,015 weight systems directly produce reflexive polytopes.

Generated 532,600,483 distinct Hodge number sets for Calabi-Yau fourfolds.

Abstract

For any given dimension $d$, all reflexive $d$-polytopes can be found (in principle) as subpolytopes of a number of maximal polyhedra that are defined in terms of $(d+1)$-tuples of integers (weights), or combinations of $k$-tuples of weights with $k<d+1$. We present the results of a complete classification of sextuples of weights pertaining to the construction of all reflexive polytopes in five dimensions. We find 322 383 760 930 such weight systems. 185 269 499 015 of them give rise directly to reflexive polytopes and thereby to mirror pairs of Calabi-Yau fourfolds. These lead to 532 600 483 distinct sets of Hodge numbers.