Further Bounding the Kreuzer-Skarke Landscape

TL;DR

This work tackles the problem of bounding how many diffeomorphism classes of Calabi–Yau threefolds can be produced via Batyrev’s FRST construction from the Kreuzer–Skarke 4D reflexive polytopes. The authors leverage Wall’s theorem to replace a diffeomorphism count by counting 2-face equivalence classes of FRSTs, and then refine upper bounds by exact counts on large 2-faces and by bounding the rest of the dataset, focusing on polytopes with $h^{1,1}\ge 300$, especially $\Delta^\

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 [1]'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.

Figures (10)

• Figure 1: Diagram of the 'lifting' procedure defining regular triangulations. The points $p_1$, $p_2$, $p_3$, and $p_4$ are embedded into $\mathbb{R}^3$ and then lifted by heights $\omega_1=1.1$, $\omega_2=0.2$, $\omega_3=0.9$, and $\omega_4=0.3$. The convex hull of the lifted point configuration is a $3$-simplex whose lower faces are plotted in blue. Projecting out the lifted coordinate generates the regular triangulation plotted in black. Figure modified from macfadden2023efficient.
• Figure 2: The number of fine triangulations of right lattice triangles $T_{2,n}, T_{3,n},$ and $T_{7,n}$ as a function of the height $n$. In particular, $T_{2,84}$, $T_{3,84}$, and $T_{7,84}$ correspond to $f_8$, $f_9$, and $f_{10}$ respectively.
• Figure 3: Improvements in the upper bounds on the number of 2-face equivalence classes for polytopes with $h^{1,1} \geq 300$, comparing those computed in Demirtas:2020dbm (red) with those found in this work (blue).
• Figure 4: Two fine regular triangulations of $P_{2,4}$ being patched to a single triangulation of $P_{4,4}$. This triangulation is fine but not regular. This example was originally found by Francisco Santos and appears in Kaibel_Ziegler_2003.
• Figure 5: The primary subdivisions of $f_2,\dots,f_7$. Primary subdivisions of $f_8$ and $f_9$ are described in the text. The vertices $v_1,\dots,v_5$ of $\Delta_{491}^\circ$ are numbered as the columns of (\ref{['eq:491']}).

Theorems & Definitions (4)

• Lemma 3.1: Proposition 3.4 in Kaibel_Ziegler_2003
• Lemma 3.2: Lemma 3.5 in Kaibel_Ziegler_2003
• Lemma 3.3: Lemma 3.3 in Kaibel_Ziegler_2003
• Lemma B.1