Calabi-Yau Threefolds from Vex Triangulations

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.

💡 Research Summary

This paper investigates the birational geometry and Kähler moduli space of Calabi–Yau (CY) threefold hypersurfaces that arise from four‑dimensional reflexive polytopes via toric constructions. While it has long been known that fine, regular, star triangulations (FRSTs) of such polytopes produce toric varieties whose anticanonical hypersurfaces are smooth CY threefolds (Batyrev’s construction), the authors point out that FRSTs do not exhaust the birational equivalence classes of either the ambient toric varieties or the CY hypersurfaces. Consequently, the Kähler moduli space obtained from FRSTs is only a proper subset of the full extended Kähler cone.

The authors formulate three equivalent classification problems: (1) enumerate birational equivalence classes of toric varieties; (2) enumerate all fine regular simplicial fans that can be built from a fixed set of rays; (3) enumerate all fine regular triangulations of a given vector configuration. They develop a detailed dictionary translating concepts between triangulation theory and toric geometry (Table 1). In triangulation theory, a vector configuration is a set of rays; a regular subdivision corresponds to a height vector ω that makes each cone a lower face of the lifted polytope. The collection of all regular subdivisions forms the secondary fan, whose cones are in bijection with fine regular triangulations. Gale duality is used to relate the ray matrix to the linear relations among rays, which greatly simplifies the computation of the secondary fan.

On the toric side, a set of rays Σ(1) defines a simplicial fan Σ, which in turn defines a Q‑factorial toric variety V_Σ. The divisor class group and the torus‑invariant divisors are described via the GLSM charge matrix Q. The Kähler cone of V_Σ, denoted Γ(V_Σ), sits inside the effective cone and is a subcone of the Kähler cone of any anticanonical CY hypersurface X⊂V_Σ.

The paper then introduces “vex triangulations,” a class of regular triangulations that are not star triangulations: they may fail to cover the convex hull of the polytope or may not be star with respect to the origin. Such fans are generally non‑weak‑Fano but remain Gorenstein. The central geometric result (Proposition 3) proves that for any fixed 4‑dimensional reflexive polytope, every fine regular simplicial fan—whether coming from an FRST or a vex triangulation—produces a non‑weak‑Fano Gorenstein toric variety whose anticanonical hypersurface is a smooth, birational CY threefold. Thus vex triangulations enlarge the set of smooth toric CY threefolds beyond those obtainable from FRSTs.

To demonstrate the practical impact, the authors apply their algorithms to the Kreuzer–Skarke (KS) database of 473,800,776 reflexive 4‑polytopes. They exhaustively enumerate all fine regular triangulations for the subset with h^{1,1}≤7, obtaining a total of 24,023,940 triangulations. Remarkably, over 70 % of these are vex triangulations, and the proportion of vex triangulations grows rapidly with the number of rays. By adapting the counting methods of previous FRST studies, they estimate an upper bound of 10^{979} fine regular triangulations for the entire KS database—far exceeding the known number of FRSTs.

The authors also analyze the implications for the Kähler moduli space. Each fine regular triangulation defines a toric variety V_Σ with its own Kähler cone Γ(V_Σ). The moving cone Mov(V_Σ) (the union of all such cones) is shown to be contained in the extended Kähler cone of the corresponding CY threefold (Equation 4.1, Proposition 4). Consequently, vex triangulations fill previously missing regions of the extended Kähler cone, providing toric realizations of “non‑toric phases” that were earlier accessible only via non‑toric birational models. This has direct relevance for string compactifications, where different chambers of the Kähler cone correspond to distinct physical phases related by flop transitions.

In summary, the paper makes several key contributions:

1. It clarifies the equivalence between triangulation classification, fan classification, and birational classification of toric varieties.
2. It introduces and systematically constructs vex triangulations, showing that they are abundant and computationally tractable.
3. It proves that all fine regular triangulations of a 4‑dimensional reflexive polytope yield smooth, birational CY threefolds, extending Batyrev’s FRST result.
4. It provides exhaustive enumeration for low h^{1,1} and a rigorous upper bound for the full KS database, revealing an astronomically large landscape of CY threefolds.
5. It demonstrates that vex triangulations map out substantial new regions of the extended Kähler moduli space, thereby enriching both mathematical classifications of CY manifolds and the toolkit for string phenomenology.

Overall, the work bridges a gap between combinatorial triangulation theory and toric geometry, delivering new computational tools and a vastly expanded catalogue of smooth Calabi–Yau threefolds with explicit toric descriptions.