F-Theory Realizations of Exact Mssm Matter Spectra

F-theory is remarked by its powerful phenomenological model building potential due to geometric descriptions of compactifications. It translates physics quantities in the effective low energy theory to mathematical objects extracted from the geometry of the compactifications. The connection is built upon identifying the varying axio-dilaton field in type IIB supergravity theory with the complex structure modulus of an elliptic curve, that serves as the fiber of an elliptic fibration. This allows us to capture the non-perturbative back-reactions of seven branes onto the compactification space B3 of an elliptically fibered Calabi–Yau fourfold Y4. The ingredients of Standard model physics, including gauge symmetries, charged matter, and Yukawa couplings, are then encoded beautifully by Y4\u27s singularity structures in codimensions one, two, and three, respectively. Moreover, many global consistency conditions, including the D3-tadpole cancellation, can be reduced to simple criteria in terms of the intersection numbers of base divisors. In this thesis, we focus on searching for explicit models in the language of F-theory geometry that admit exact Minimal Supersymmetric Standard Model (MSSM) matter spectra. We first present a concrete realization of the Standard Model (SM) gauge group with Z2 matter parity, which admits three generations of chiral fermions. The existence of this discrete symmetry beyond the SM gauge group forbids proton decay. We then construct a family of O(1015) F-theory vacua. These are the largest currently known class of globally consistent string constructions that admit exactly three chiral families and gauge coupling unification. We advance to study the vector-like spectra in 4d F-theory SMs. The 4-form gauge background G4 controls the chiral spectra. This is the field strength of 3-form gauge potential C3, which impacts the vector-like spectra. It is well known that these massless zero modes are counted by line bundle cohomologies over matter curves induced by the F-theory gauge background. In order to understand the line bundle cohomology\u27s dependence on the moduli of the compactification geometry, we pick a simple geometry and create the database consisted of matter curves, the line bundles and the vector-like spectra. We analyze this database by machine learning techniques and ugain full understanding it via the Brill-Nother theory. Subsequently, we present the appearance of root bundles and how they enter as significant ingredients of realistic F-theory geometries. The algebraic geometry approaches to root bundles allow combinatoric descriptions, which facilitate the analyze of statistics on the vector-like spectra at the end of this thesis