Quiver Gauge Models in F-Theory on Local Tetrahedron

We study a class of 4D $\mathcal{N}=1$ supersymmetric GUT- type models in the framework of the Beasley-Heckman-Vafa theory. We first review general results on MSSM and supersymmetric GUT; and we describe useful tools on 4D quiver gauge theories in F- theory set up. Then we study the effective supersymmetric gauge theory in the 7-brane wrapping 4-cycles in F-theory on local elliptic CY4s based on a complex tetrahedral surface $\mathcal{T}$ and its blown ups $\mathcal{T}_{n}$. The complex 2d geometries $\mathcal{T}$ and $\mathcal{T}_{n}$ are \emph{non planar} projective surfaces that extend the projective plane $\mathbb{P}^{2}$ and the del Pezzos. Using the power of toric geometry encoding the toric data of the base of the local CY4, we build a class of \emph{4D} $\mathcal{N}=1$ non minimal GUT- type models based on $\mathcal{T}$ and $\mathcal{T}_{n}$. An explicit construction is given for the SU$(5) $ GUT-type model.