The Kreuzer-Skarke Axiverse

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We study the topological properties of Calabi-Yau threefold hypersurfaces at large $h^{1,1}$.

• We obtain two million threefolds $X$ by triangulating polytopes from the Kreuzer-Skarke list, including all polytopes with $240 \le h^{1,1}\le 491$.
• We show that the K\"ahler cone of $X$ is very narrow at large $h^{1,1}$, and as a consequence, control of the $\alpha^{\prime}$ expansion in string compactifications on $X$ is correlated with the presence of ultralight axions.
• If every effective curve has volume $\ge 1$ in string units, then the typical volumes of irreducible effective curves and divisors, and of $X$ itself, scale as $(h^{1,1})^p$, with $3\lesssim p \lesssim 7$ depending on the type of cycle in question.