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Toric Elliptic Fibrations and F-Theory Compactifications

Volker Braun

TL;DR

This work analyzes toric elliptic fibrations for F-theory by exploiting the polytope-encoded geometry of Calabi–Yau hypersurfaces in toric varieties. It develops a framework (fibered polytopes, Weierstrass models, fiber-divisor-graphs) to extract discriminant data, Kodaira types, and gauge groups, and applies it to a large catalog of flat fibrations over $\mathbb{P}^2$ to map gauge content and moduli transitions. A key result is the identification of $SU(10)$ in a detailed Weierstrass example, the demonstration of non-split discriminant components leading to sub-maximal gauge algebras (e.g., $F_4$ in a non-split case), and the clarification of how flatness, resolution, and monodromy shape the physical spectrum. The paper also discusses global constraints from 6D anomaly cancellation, illustrating why naive upper bounds (such as $SU(27)$) must be refined by accounting for codimension-two degenerations and tensor multiplets, thereby linking intricate geometry to consistent low-energy physics.

Abstract

The 102581 flat toric elliptic fibrations over P^2 are identified among the Calabi-Yau hypersurfaces that arise from the 473800776 reflexive 4-dimensional polytopes. In order to analyze their elliptic fibration structure, we describe the precise relation between the lattice polytope and the elliptic fibration. The fiber-divisor-graph is introduced as a way to visualize the embedding of the Kodaira fibers in the ambient toric fiber. In particular in the case of non-split discriminant components, this description is far more accurate than previous studies. The discriminant locus and Kodaria fibers groups of all 102581 elliptic fibrations are computed. The maximal gauge group is SU(27), which would naively be in contradiction with 6-dimensional anomaly cancellation.

Toric Elliptic Fibrations and F-Theory Compactifications

TL;DR

This work analyzes toric elliptic fibrations for F-theory by exploiting the polytope-encoded geometry of Calabi–Yau hypersurfaces in toric varieties. It develops a framework (fibered polytopes, Weierstrass models, fiber-divisor-graphs) to extract discriminant data, Kodaira types, and gauge groups, and applies it to a large catalog of flat fibrations over to map gauge content and moduli transitions. A key result is the identification of in a detailed Weierstrass example, the demonstration of non-split discriminant components leading to sub-maximal gauge algebras (e.g., in a non-split case), and the clarification of how flatness, resolution, and monodromy shape the physical spectrum. The paper also discusses global constraints from 6D anomaly cancellation, illustrating why naive upper bounds (such as ) must be refined by accounting for codimension-two degenerations and tensor multiplets, thereby linking intricate geometry to consistent low-energy physics.

Abstract

The 102581 flat toric elliptic fibrations over P^2 are identified among the Calabi-Yau hypersurfaces that arise from the 473800776 reflexive 4-dimensional polytopes. In order to analyze their elliptic fibration structure, we describe the precise relation between the lattice polytope and the elliptic fibration. The fiber-divisor-graph is introduced as a way to visualize the embedding of the Kodaira fibers in the ambient toric fiber. In particular in the case of non-split discriminant components, this description is far more accurate than previous studies. The discriminant locus and Kodaria fibers groups of all 102581 elliptic fibrations are computed. The maximal gauge group is SU(27), which would naively be in contradiction with 6-dimensional anomaly cancellation.

Paper Structure

This paper contains 25 sections, 57 equations, 14 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Toric Elliptic Fibrations
  3. Toric Morphisms
  4. Homogeneous Coordinates
  5. Fibrations of Polytopes
  6. Torus Fibrations
  7. Flat Fibrations
  8. Kodaira vs. Miranda
  9. Toric Fibrations and Polytopes
  10. Classification of Fibered Polytopes
  11. Weierstrass Models
  12. An Example of a Toric Elliptic Fibration
  13. Fibration of the Polytope
  14. Weierstrass Model
  15. Gauge Group
  16. ...and 10 more sections

Figures (14)

  • Figure 1: Toric fibration of the Hirzebruch surface $F_3$ over ${\mathop{ {\mathbb{P}}}\nolimits}^1$.
  • Figure 2: Blowup of ${\mathbb{C}}^2/\mathbb{Z}_2$.
  • Figure 3: The distribution of flat toric elliptic fibrations with base ${\mathop{ {\mathbb{P}}}\nolimits}^2$.
  • Figure 4: The region of heights $h^{11}+h^{21}\leq 150$ in \ref{['fig:bigplot']}.
  • Figure 5: The fan of ${\mathop{ {\mathbb{P}}}\nolimits}^2$.
  • ...and 9 more figures

Theorems & Definitions (1)

  • Definition 1: Fiber-Divisor-Graph