Tetrahedron in F-theory Compactification

TL;DR

This paper explores complex tetrahedral surfaces and their blow-ups, linking them to del Pezzo surfaces, and discusses their potential in engineering gauge symmetries and matter couplings in F-theory GUT models.

Contribution

It introduces a family of blown-up tetrahedral surfaces related to del Pezzo surfaces and analyzes their toric structures for F-theory model building.

Findings

Blown-up tetrahedral surfaces are toric and exhibit U(1) x U(1) symmetry.

These surfaces can localize fields and interactions for GUT models.

The study connects geometric structures with gauge symmetry enhancements.

Abstract

Complex tetrahedral surface $\mathcal{T}$ is a non planar projective surface that is generated by four intersecting complex projective planes $CP^{2}$. In this paper, we study the family $\{\mathcal{T}_{m}\} $ of blow ups of $\mathcal{T}$ and exhibit the link of these $\mathcal{T}_{m}$s with the set of del Pezzo surfaces $dP_{n}$ obtained by blowing up n isolated points in the $CP^{2}$. The $\mathcal{T}_{m}$s are toric surfaces exhibiting a $U(1) \times U(1) $ symmetry that may be used to engineer gauge symmetry enhancements in the Beasley-Heckman-Vafa theory. The blown ups of the tetrahedron have toric graphs with faces, edges and vertices where may localize respectively fields in adjoint representations, chiral matter and Yukawa tri-fields couplings needed for the engineering of F- theory GUT models building.