Enumerating Calabi-Yau Manifolds: Placing bounds on the number of diffeomorphism classes in the Kreuzer-Skarke list

TL;DR

This paper establishes bounds on the number of diffeomorphism classes of Calabi-Yau threefolds derived from the Kreuzer-Skarke list, using topological invariants and basis-independent measures, suggesting a vast diversity of such manifolds.

Contribution

It introduces new basis-independent invariants and methods to bound the number of diffeomorphism classes in the Kreuzer-Skarke list of Calabi-Yau threefolds.

Findings

Lower bounds on the number of classes using new invariants

Upper bounds by identifying basis transformations

Conjecture of up to 10^{400} distinct diffeomorphism classes

Abstract

The diffeomorphism class of simply-connected smooth Calabi-Yau threefolds with torsion-free cohomology is determined via certain basic topological invariants: the Hodge numbers, the triple intersection form, and the second Chern class. In the present paper, we shed some light on this classification by placing bounds on the number of diffeomorphism classes present in the set of smooth Calabi-Yau threefolds constructed from the Kreuzer-Skarke list of reflexive polytopes up to Picard number six. The main difficulty arises from the comparison of triple intersection numbers and divisor integrals of the second Chern class up to basis transformations. By using certain basis-independent invariants, some of which appear here for the first time, we are able to place lower bounds on the number of classes. Upper bounds are obtained by explicitly identifying basis transformations, using constraints related to the index of line bundles. Extrapolating our results, we conjecture that the favourable entries of the Kreuzer-Skarke list of reflexive polytopes leads to some $10^{400}$ diffeomorphically distinct Calabi-Yau threefolds.