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Complete Intersection Fibers in F-Theory

Volker Braun, Thomas W. Grimm, Jan Keitel

TL;DR

Global F-theory compactifications with complete-intersection fibers extend the model-building landscape beyond hypersurfaces. The authors develop a general algorithm to transform CI elliptic fibers into Weierstrass form, enabling Jacobian computations and toric Mordell-Weil analyses across thousands of nef partitions from 3D reflexive polytopes. They uncover new toric MW groups, including $\mathbb{Z}_4$ torsion and higher ranks, and demonstrate explicit models with features such as a distinctively charged $\mathbf{10}$ representation in an $SU(5)$ GUT and a discrete $\mathbb{Z}_4$ symmetry, illustrating richer gauge-structure possibilities. The work broadens the F-theory toolkit, linking fiber geometry to non-Abelian and discrete gauge content and opening avenues for exploring topologies and phenomenology in more general ambient spaces.

Abstract

Global F-theory compactifications whose fibers are realized as complete intersections form a richer set of models than just hypersurfaces. The detailed study of the physics associated with such geometries depends crucially on being able to put the elliptic fiber into Weierstrass form. While such a transformation is always guaranteed to exist, its explicit form is only known in a few special cases. We present a general algorithm for computing the Weierstrass form of elliptic curves defined as complete intersections of different codimensions and use it to solve all cases of complete intersections of two equations in an ambient toric variety. Using this result, we determine the toric Mordell-Weil groups of all 3134 nef partitions obtained from the 4319 three-dimensional reflexive polytopes and find new groups that do not exist for toric hypersurfaces. As an application, we construct several models that cannot be realized as toric hypersurfaces, such as the first toric SU(5) GUT model in the literature with distinctly charged 10 representations and an F-theory model with discrete gauge group Z_4 whose dual fiber has a Mordell-Weil group with Z_4 torsion.

Complete Intersection Fibers in F-Theory

TL;DR

Global F-theory compactifications with complete-intersection fibers extend the model-building landscape beyond hypersurfaces. The authors develop a general algorithm to transform CI elliptic fibers into Weierstrass form, enabling Jacobian computations and toric Mordell-Weil analyses across thousands of nef partitions from 3D reflexive polytopes. They uncover new toric MW groups, including torsion and higher ranks, and demonstrate explicit models with features such as a distinctively charged representation in an GUT and a discrete symmetry, illustrating richer gauge-structure possibilities. The work broadens the F-theory toolkit, linking fiber geometry to non-Abelian and discrete gauge content and opening avenues for exploring topologies and phenomenology in more general ambient spaces.

Abstract

Global F-theory compactifications whose fibers are realized as complete intersections form a richer set of models than just hypersurfaces. The detailed study of the physics associated with such geometries depends crucially on being able to put the elliptic fiber into Weierstrass form. While such a transformation is always guaranteed to exist, its explicit form is only known in a few special cases. We present a general algorithm for computing the Weierstrass form of elliptic curves defined as complete intersections of different codimensions and use it to solve all cases of complete intersections of two equations in an ambient toric variety. Using this result, we determine the toric Mordell-Weil groups of all 3134 nef partitions obtained from the 4319 three-dimensional reflexive polytopes and find new groups that do not exist for toric hypersurfaces. As an application, we construct several models that cannot be realized as toric hypersurfaces, such as the first toric SU(5) GUT model in the literature with distinctly charged 10 representations and an F-theory model with discrete gauge group Z_4 whose dual fiber has a Mordell-Weil group with Z_4 torsion.

Paper Structure

This paper contains 20 sections, 72 equations, 4 figures, 18 tables.

Table of Contents

  1. Introduction
  2. Koszul and Residues
  3. Weierstrass Form for Complete Intersections
  4. Basic Algorithm
  5. Sections of Line Bundles
  6. The Second Differential
  7. An Algorithm to Compute Relations
  8. Kodaira Map
  9. Two Exceptions
  10. Classifying Toric Mordell-Weil Groups
  11. Complete Intersections in Toric Varieties
  12. Nef Partitions of 3d Lattice Polytopes
  13. Toric Mordell-Weil Groups
  14. Results for Elliptic Curves of Codimension Two
  15. Examples
  16. ...and 5 more sections

Figures (4)

  • Figure 1: Histogram of the number of nef partitions of the $4319$ reflexive polytopes in three dimensions.
  • Figure 2: Histogram of the number of polytopes that have a given number of nef partitions. There are $3090$ reflexive three-dimensional polytopes that do not admit a nef partition. The reflexive polytope with PALP id $214$ has the most nef partitions, namely $21$.
  • Figure 3: Histogram of the number of toric sections for the $3118$ nef partitions of three-dimensional reflexive polytopes that are not direct products.
  • Figure 4: Histogram of the toric Mordell-Weil rank for the nef partitions of three-dimensional reflexive polytopes. The $326$ complete intersections that are either a direct product or do not have a toric section are excluded.