Calabi-Yau threefolds with large h^{2, 1}

TL;DR

The paper addresses the problem of classifying Calabi–Yau threefolds that are elliptically fibered with a section and possess large $h^{2,1}$, using the Weierstrass description to connect base geometry, gauge tuning, and matter content in F-theory. It systematically constructs all such threefolds with $h^{2,1}\ge 350$ by tuning Weierstrass coefficients over bases $\mathbb{F}_{12},\mathbb{F}_{8},\mathbb{F}_{7}$ and their blow-ups, revealing a connected moduli space of EFS CY3s and identifying three apparently new manifolds with $(h^{1,1},h^{2,1})$ around 371–363. The work combines toric and non-toric base analysis, anomaly cancellation, and explicit monomial counting in Weierstrass models to verify consistency and to map the landscape; it also discusses obstacles to a complete enumeration of all EFS CY3s and to extending the program to fourfolds. Overall, the results support the view that large-Hodge CY3s are dominated by elliptic fibrations and demonstrate the feasibility of a constructive, modular classification via base-tuned Weierstrass data, while highlighting technical barriers that remain for a full global classification.

Abstract

We carry out a systematic analysis of Calabi-Yau threefolds that are elliptically fibered with section ("EFS") and have a large Hodge number h^{2, 1}. EFS Calabi-Yau threefolds live in a single connected space, with regions of moduli space associated with different topologies connected through transitions that can be understood in terms of singular Weierstrass models. We determine the complete set of such threefolds that have h^{2, 1} >= 350 by tuning coefficients in Weierstrass models over Hirzebruch surfaces. The resulting set of Hodge numbers includes those of all known Calabi-Yau threefolds with h^{2, 1} >= 350, as well as three apparently new Calabi-Yau threefolds. We speculate that there are no other Calabi-Yau threefolds (elliptically fibered or not) with Hodge numbers that exceed this bound. We summarize the theoretical and practical obstacles to a complete enumeration of all possible EFS Calabi-Yau threefolds and fourfolds, including those with small Hodge numbers, using this approach.

Figures (5)

• Figure 1: Clusters of intersecting curves that must carry a nonabelian gauge group factor. For each cluster the corresponding gauge algebra is noted and the gauge algebra and number of charged matter hypermultiplet are listed in Table \ref{['t:clusters']}
• Figure 2: A general F-theory base $B_2$ is formed by a sequence of blow-ups on a Hirzebruch surface $\mathbb{F}_m$. In this example, three generic points are blown up sequentially on $\mathbb{F}_{12}$, and a fourth blow-up point is chosen to be on the exceptional divisor from the third blow-up. These points are all blown up on fibers in such a way that a global $\mathbb{C}^*$ structure is preserved. The final base $\beta$ enters the discussion in the text in several places.
• Figure 3: Monomials in the generic Weierstrass model over $\mathbb{F}_{12}$ are of the form $f_{k, m}z^kw^m, g_{k, m}z^kw^m$, and can be associated with points depicted above in the lattice $N^*$ dual to the lattice $N$ carrying the rays in the toric fan for $\mathbb{F}_{12}$. Circles denote monomials in $f$, and dots denote monomials in $g$. Blowing up a generic point in $\mathbb{F}_{12}$ can be described in a local coordinate system by setting all monomials below the red line to vanish. As described in the text, an $SU(2)$ gauge group cannot be tuned on the exceptional divisor from the blow-up without forcing the monomial coefficient $g_{0, 7}$ to vanish, which makes it impossible to form a Calabi-Yau due to a (4, 6) vanishing on the divisor $S$.
• Figure 4: In this paper we explicitly construct all elliptically fibered Calabi-Yau threefolds with section having $h^{2, 1}\geq 350$. The Hodge numbers of these threefolds are shown here, with the detailed construction explained in the bulk of the text. Black points represent generic elliptic fibrations over different bases $B_2$, and colored points represent tuned Weierstrass models over these bases with enhanced gauge groups. The three purple data points appear to be new Calabi-Yau manifolds not found in the Kreuzer-Skarke database (see §\ref{['sec:new']}). All elliptically fibered Calabi-Yau threefolds with section are connected by geometric transitions associated with tuning Weierstrass moduli over a particular base ("Higgsing/unHiggsing") and/or blowing up and down points in the base (corresponding to tensionless string transitions in the physical F-theory context). Note that the point $(10, 376)$, corresponding to generic elliptic fibrations over $\mathbb{F}_7, \mathbb{F}_8$, is connected to the other threefolds shown through a sequence of blow-up and blow-down transitions on the base that pass through the set of threefolds with smaller Hodge numbers $h^{2, 1}< 350$. Note also that there are two distinct constructions that give the Hodge numbers $(19, 355)$; in addition to an untuned Weierstrass model with generic gauge group $G_2 \times SU(2)$ there is a tuning of the generic $(15, 375)$ Weierstrass model with a gauge group $SU(2) \times SU(3) \times SU(2)$.
• Figure 5: Blowing up on the top $-2$ curve ($C_4$) on a $(-1, -2, -2, -2, -1)$ chain results in a divisor structure giving a non-toric base, with a $(-3, -2, -2)$ non-Higgsable cluster (case (C) in the text). In the limit of moduli space where the intersection points of the two $-1$ curves with the $-3$ curve coincide, the fiber becomes $(-2, -1, -3, -2, -2, -1)$ and the base becomes toric (case (B)).