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

TL;DR

This paper addresses the problem of bounding the number of diffeomorphism classes of smooth, simply connected Calabi-Yau threefolds with torsion-free cohomology that arise from the Kreuzer-Skarke list, focusing on Picard numbers up to $h\le 6$. It develops a broad suite of basis-invariant data—GCD invariants, tensor-power invariants, and polynomial invariants—together with an algorithm that tests for $GL(h,\mathbb{Z})$ basis transformations mapping the intersection form and second Chern class data between manifolds, and complements this with exact symbolic checks for small $h$. The authors obtain exact counts for $h=1,2,3$ ($4$, $27$, $183$ classes respectively) and derive tight lower and upper bounds for $h=4,5,6$, aided by a systematic search and invariants to partition the data; they additionally extrapolate to the full favourable KS list, predicting an astronomical number of FRSTs and diffeomorphism classes, on the order of $10^{497}$ FRSTs and between $10^{396}$ and $10^{401}$ diffeomorphism classes. The work demonstrates that Wall-type invariants, combined with computationally tractable basis-independent quantities and an intelligent search strategy, can effectively bound the complexity of CY diffeomorphism class counts and provides a data-rich foundation for further exploration at higher Picard numbers and deeper invariant theory.

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.

Figures (7)

• Figure 1: The bipartite graph denoting the index pattern for the $\delta=4$, $h=3$ 'coincidental' invariant, firstly displayed undirected and then directed. Levi-Civita-type vertices are coloured red. Uniquely, for $h=3$ both the Levi-Civita and array vertices are trivalent.)
• Figure 2: Neural network that learns the required $\operatorname{GL}(h,\mathbb Z)$-transformation $P$.
• Figure 3: Plot corresponding to Table \ref{['tab:Results']} on a logarithmic scale. Here, we plot the lower (from the invariants) and upper (from the systematic search) bounds on the number of distinct manifolds. We also plot the bare numbers of polytopes and triangulations, as well as the number of numerically distinct triangulations (i.e. triangulations with exactly the same topological data).
• Figure 4: Average number of distinct manifolds per polytope from Table \ref{['tab:Results']}.
• Figure 5: The solid lines are upper bounds on the number of diffeomorphism classes per distinct triangulation, for $h \leq 6$, identified by the algorithm introduced Section \ref{['sec:alg']} as $k_{\text{max}}$ is increased. For $h=2,3,4,5,6$, $k_{\text{max}}=\infty,\infty,15,12,5$. Lower bounds for each value of $h$ are indicated with dashed lines in the appropriate colour.