Results from an Algebraic Classification of Calabi-Yau Manifolds

TL;DR

This paper systematically classifies Calabi-Yau threefolds using an algebraic approach based on reflexive polyhedra, identifying a large set of spaces with specific topological features, including three-generation examples.

Contribution

It introduces a new inductive algebraic method for classifying CY3 spaces via reflexive polyhedra and enumerates a comprehensive set of such spaces with detailed topological data.

Findings

Classified over 180,000 CY3 spaces with reflexive polyhedra.

Identified 212 three-generation CY3 spaces with K3 sections.

Mapped CY3 spaces to their Hodge number pairs.

Abstract

We present results from an inductive algebraic approach to the systematic construction and classification of the `lowest-level' CY3 spaces defined as zeroes of polynomial loci associated with reflexive polyhedra, derived from suitable vectors in complex projective spaces. These CY3 spaces may be sorted into `chains' obtained by combining lower-dimensional projective vectors classified previously. We analyze all the 4242 (259, 6, 1) two- (three-, four-, five-) vector chains, which have, respectively, K3 (elliptic, line-segment, trivial) sections, yielding 174767 (an additional 6189, 1582, 199) distinct projective vectors that define reflexive polyhedra and thereby CY3 spaces, for a total of 182737. These CY3 spaces span 10827 (a total of 10882) distinct pairs of Hodge numbers h_11, h_12. Among these, we list explicitly a total of 212 projective vectors defining three-generation CY3 spaces with K3 sections, whose characteristics we provide.