Calabi-Yau Threefolds from Vex Triangulations

TL;DR

<3-5 sentence high-level summary>This paper expands the study of Calabi–Yau threefolds embedded as toric hypersurfaces by moving beyond the traditional reliance on fine regular star triangulations (FRSTs) to include vex triangulations, thereby broadening the toric birational landscape. It develops a dual viewpoint from triangulation theory and toric geometry to classify fine regular fans of 4D reflexive polytopes via the secondary fan, and proves that all fine regular triangulations yield smooth CY hypersurfaces that lie in the same birational class. The authors apply this framework to the Kreuzer–Skarke database, exhaustively counting FRSTs and vex triangulations with $h^{1,1}\le7$ and showing vex triangulations dominate at larger $h^{1,1}$, while providing an upper bound $\le 10^{979}$ on total FRSTs in KS. They demonstrate that vex triangulations generate many new CY topologies and map substantial, previously inaccessible regions of the Kähler moduli space, thereby enriching both mathematics and string-theory model-building avenues.

Abstract

We study the birational geometry (i.e., Kähler moduli space) of Calabi--Yau (CY) threefold hypersurfaces in toric varieties arising from four-dimensional reflexive polytopes. In particular, it has been observed that the birational classes of these geometries are not exhausted by toric hypersurfaces arising from fine, regular, star triangulations (FRSTs). We begin by introducing a classification problem: enumeration of birational classes of toric varieties, which is equivalent to enumeration of certain triangulations/fans. We consider this problem from the complementary perspectives of triangulation theory and toric geometry, reviewing both theories in detail; this culminates in an explanation of how to generate all fine regular triangulations of a vector configuration (i.e., fine regular simplicial fans). We then apply this theory to the Kreuzer--Skarke (KS) database, where we encounter both FRSTs and vex triangulations. We study the non-weak-Fano toric varieties arising from vex triangulations, along with their CY hypersurfaces. In particular, we show that all fine regular triangulations of a fixed 4D reflexive polytope give rise to smooth birational CY hypersurfaces, extending Batyrev's result from FRSTs to vex triangulations. We exhaustively enumerate all $24,023,940$ fine regular triangulations in the KS database with $h^{1,1}\leq 7$, of which over $70\%$ are vex triangulations, and provide an upper bound of $10^{979}$ for fine regular triangulations in the entire KS database. We conclude that vex triangulations of four-dimensional reflexive polytopes give rise to a large number of smooth Calabi--Yau threefolds and importantly provide toric descriptions for novel regions in the Kähler moduli space.

Figures (9)

• Figure 1: Subdivisions and triangulations of a point configuration and a vector configuration.
• Figure 2: 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 $\omega_1=1.1$, $\omega_2=0.3$, $\omega_3=0.2$, and $\omega_4=0.9$. 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 3: The two triangulations $\mathscr{T}$ and $\mathscr{T}'$ of the square $\mathbf{A}_\text{PC} = \{(1,1),(1,2),(2,1),(2,2)\}$. Flipping $\mathscr{T}\leftrightarrow\mathscr{T}'$ can be thought as coarsening a triangulation into the subdivision $\mathscr{S}$ and then refining this subdivision in 'the other way'.
• Figure 4: A triangulation $\mathscr{T}$ of a point configuration. Two simplices $\sigma$ and $\sigma'$ of dimensions $0$ and $1$ respectively are plotted in blue. In green, the $\text{link}_\mathscr{T}(\sigma)$ and $\text{link}_\mathscr{T}(\sigma')$ are plotted. In purple, $\text{st}_\mathscr{T}(\sigma)$ and $\text{st}_\mathscr{T}(\sigma')$ are plotted. Note the link is a subcomplex of the star, so green colored regions also correspond to the star.
• Figure 5: Left: regular triangulations $\mathscr{T}$ of a point configuration $\mathbf{A}$, labeled with heights $\omega_j$ which generate $\mathscr{T}$. Right: the associated vector configuration $\mathbf{A}\setminus0$ (not the homogenization), lifted with corresponding heights.

Theorems & Definitions (21)

• Definition 1
• Definition 2
• Proposition 1
• Proposition 2
• proof : Proof of \ref{['thm:pc_to_vc']}
• Lemma 1: De_Loera2010-ss, Lemma 4.4.9
• Proposition 3
• Proposition 4
• proof
• Lemma 2: cls, Lemma 8.3.6