Further Bounding the Kreuzer-Skarke Landscape

TL;DR

This paper significantly tightens the upper bound on the number of diffeomorphism classes of Calabi-Yau threefolds derived from Batyrev's construction, using combinatorial bounds on triangulations of reflexive polytopes.

Contribution

It proves a new upper bound of 10^{296} on the number of CY diffeomorphism classes, improving previous bounds, by analyzing 2-face equivalence classes of triangulations.

Findings

Upper bound of 10^{296} on CY diffeomorphism classes.

Lower bound of 10^{276} on 2-face equivalence classes.

Bounded the number of 2-face equivalence classes for polytopes with h^{1,1} ≥ 300.

Abstract

Batyrev's construction provides a map from fine, regular, star triangulations (FRSTs) of 4D reflexive polytopes to smooth Calabi-Yau threefolds (CYs). We prove that there are at most $10^{296}$ diffeomorphism classes of CYs produced in this manner, improving arXiv:2008.01730's upper bound of $10^{428}$. To show this, we make use of the fact that any two FRSTs with the same 2-face restrictions give rise to diffeomorphic CYs and bound the number of such '2-face equivalence classes' for all polytopes with Hodge number $h^{1,1} \geq 300$. We also put a lower bound of $10^{276}$ on the number of 2-face equivalence classes, but emphasize that this is not a lower bound on the number of diffeomorphism classes of CYs, as distinct 2-face equivalence classes may give rise to diffeomorphic threefolds.