Counting Calabi-Yau Threefolds

TL;DR

The paper advances the enumeration of Calabi–Yau threefolds by applying Wall's theorem to reduce topological classification to Wall data, and then enriching this with arithmetic, algebraic, and GV-invariants to distinguish equivalence classes. It develops a systematic, invariant-based pipeline to separate toric phases (via FRST classifications of 4D reflexive polytopes) and non-toric phases (via flop-induced birational families), delivering exact counts for simply connected toric hypersurfaces with $1\le h^{1,1}\le5$, plus ten Wall-data classes for non-simply connected cases and provisional counts for non-toric phases at $h^{1,1}=2$. The methodology combines rational-geometry invariants from $\mathcal{K},\mathcal{I},\mathcal{H}$, Hessian and Jacobian data, finite-field point counts, and GV sequences to certify equivalences or inequivalences, with an explicit protocol to deduce Mori cones when possible. The results sharpen the understanding of the Calabi–Yau landscape, provide a concrete, computable framework for distinguishing diffeomorphism classes, and raise questions about potential single invariants and broader applicability to higher $h^{1,1}$ and non-toric realizations.

Abstract

We enumerate topologically-inequivalent compact Calabi-Yau threefold hypersurfaces. By computing arithmetic and algebraic invariants and the Gopakumar-Vafa invariants of curves, we prove that the number of distinct simply connected Calabi-Yau threefold hypersurfaces resulting from triangulations of four-dimensional reflexive polytopes is 4, 27, 183, 1,184 and 8,036 at $h^{1,1}$ = 1, 2, 3, 4, and 5, respectively. We also establish that there are ten equivalence classes of Wall data of non-simply connected Calabi-Yau threefolds from the Kreuzer-Skarke list. Finally, we give a provisional count of threefolds obtained by enumerating non-toric flops at $h^{1,1} =2$.

Figures (1)

• Figure 1: A flowchart of our analysis for toric phases. Everything inside a rectangle is provably inequivalent to everything inside a different rectangle, and all phases within a blue oval are equivalent to each other.