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Non-toric bases for elliptic Calabi-Yau threefolds and 6D F-theory vacua

Washington Taylor, Yi-Nan Wang

TL;DR

The paper develops a combinatorial, algorithmic framework to classify smooth base surfaces supporting elliptic Calabi–Yau threefolds beyond toric geometry, enabling systematic enumeration of bases with $h^{1,1}(S)<8$ and those yielding large $h^{2,1}(X)$. By encoding base geometry in the effective cone and employing blow-up rules, Diophantine characterizations, and Cremona-type transformations, the authors construct large families of bases and quantify their F-theory content via non-Higgsable clusters and related data. They report 6511 bases for $h^{2,1}(X)\ge 150$, including 2640 non-toric ones, and identify 15 new Calabi–Yau threefolds not present in Kreuzer–Skarke, with the largest new pair $(h^{1,1},h^{2,1})=(29,299)$. The results suggest that non-toric bases do not dramatically expand the landscape at large $h^{2,1}$ and that toric methods continue to capture a representative portion of elliptic CY3-folds, with implications for 6D universality in F-theory.

Abstract

We develop a combinatorial approach to the construction of general smooth compact base surfaces that support elliptic Calabi-Yau threefolds. This extends previous analyses that have relied on toric or semi-toric structure. The resulting algorithm is used to construct all classes of such base surfaces $S$ with $h^{1, 1} (S) < 8$ and all base surfaces over which there is an elliptically fibered Calabi-Yau threefold $X$ with Hodge number $h^{2, 1} (X) \geq 150$. These two sets can be used todescribe all 6D F-theory models that have fewer than seven tensor multiplets or more than 150 neutral scalar fields respectively in their maximally Higgsed phase. Technical challenges to constructing the complete list of base surfaces for all Hodge numbers are discussed.

Non-toric bases for elliptic Calabi-Yau threefolds and 6D F-theory vacua

TL;DR

The paper develops a combinatorial, algorithmic framework to classify smooth base surfaces supporting elliptic Calabi–Yau threefolds beyond toric geometry, enabling systematic enumeration of bases with and those yielding large . By encoding base geometry in the effective cone and employing blow-up rules, Diophantine characterizations, and Cremona-type transformations, the authors construct large families of bases and quantify their F-theory content via non-Higgsable clusters and related data. They report 6511 bases for , including 2640 non-toric ones, and identify 15 new Calabi–Yau threefolds not present in Kreuzer–Skarke, with the largest new pair . The results suggest that non-toric bases do not dramatically expand the landscape at large and that toric methods continue to capture a representative portion of elliptic CY3-folds, with implications for 6D universality in F-theory.

Abstract

We develop a combinatorial approach to the construction of general smooth compact base surfaces that support elliptic Calabi-Yau threefolds. This extends previous analyses that have relied on toric or semi-toric structure. The resulting algorithm is used to construct all classes of such base surfaces with and all base surfaces over which there is an elliptically fibered Calabi-Yau threefold with Hodge number . These two sets can be used todescribe all 6D F-theory models that have fewer than seven tensor multiplets or more than 150 neutral scalar fields respectively in their maximally Higgsed phase. Technical challenges to constructing the complete list of base surfaces for all Hodge numbers are discussed.

Paper Structure

This paper contains 18 sections, 4 theorems, 33 equations, 10 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Complex surfaces as F-theory bases
  3. F-theory on elliptic Calabi-Yau threefolds
  4. Algebraic geometry of rational complex surfaces
  5. The general strategy and some simple examples
  6. Identifying new curves after a blow-up
  7. Types of blowups
  8. Generating the set of $(-1)$-curves on $S'$
  9. Working example: (generalized) del Pezzo surfaces
  10. Geometric obstructions
  11. Finiteness of curves
  12. The algorithm
  13. Classification of all bases with rk(Pic$(S))<8$
  14. Classification of all bases for elliptic Calabi-Yau threefolds with $h^{2,1}\geq 150$
  15. Statistics of bases giving generic EFS CY3's with large $h^{2,1}$
  16. ...and 3 more sections

Key Result

Lemma 1

A negative divisor class on $S$ is always rigid. For a non-negative effective divisor class $D$ on $S$, there exists a representative $D\in|D|$ that passes through any $([D]^2-[K_S]\cdot [D])/2$ points in $S$. \newlabell:passthrough0

Figures (10)

  • Figure 1: Loop of irreducible effective curves on the Hirzebruch surface $\mathbb{F}_n$, corresponding to irreducible toric divisors associated with rays in the toric fan. $[F] = [F']$ correspond to the same divisor class, and $[S_0] =[S_\infty] + n F$. The self-intersections of each curve are labeled beside the divisor class
  • Figure 2: Blowing up Hirzebruch surfaces. For each surface $\mathbb{F}_{n\geq 0}$, there are only two different ways to blow up it: blow up on the curve of negative self-intersection or blow up at a generic point. For $\mathbb{F}_{12}$, the (-12)-curve cannot be blown up, as the base of an elliptically fibered Calabi-Yau threefold is not allowed to have curves with self-intersection lower than $-12$Classbasis. The results of blowing up are represented as the self-intersections of the (cyclic) sequence of toric divisors in the blown up surface.
  • Figure 3: The illustration of some of the blow up process discussed in this section, starting from $\mathbb{F}_1$.
  • Figure 4: Two geometric configurations with the same intersection structure and vector representations of curves, but which can be blown up to give surfaces with distinct intersection structure
  • Figure 5: The geometry of the blow-up points $a$,$b$,$c$,$d$,$e$,$f$,$g$,$h$,$i$, of the configuration in Table \ref{['t:3I3']}.
  • ...and 5 more figures

Theorems & Definitions (4)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Proposition 1