Root bundles: Applications to F-theory Standard Models

The study of vector-like spectra in 4-dimensional F-theory compactifications involves root bundles, which are important for understanding the Quadrillion F-theory Standard Models (F-theory QSMs) and their potential implications in physics. Recent studies focused on a superset of physical root bundles whose cohomologies encode the vector-like spectra for certain matter representations. It was found that more than 99.995\% of the roots in this superset for the family $B_3( \Delta_4^\circ )$ of $\mathcal{O}(10^{11})$ different F-theory QSM geometries had no vector-like exotics, indicating that this scenario is highly likely. To study the vector-like spectra, the matter curves in the F-theory QSMs were analyzed. It was found that each of them can be deformed to nodal curve that is identical across all spaces in $B_3( \Delta^\circ )$. Therefore, from studying a few nodal curves, one can probe the vector-like spectra of a large fraction of F-theory QSMs. To this end, the cohomologies of all limit roots were determined, with line bundle cohomology on rational nodal curves playing a major role. A computer algorithm was used to enumerate all limit roots and analyze the global sections of all tree-like limit roots. For the remaining circuit-like limit roots, the global sections were manually determined. These results were organized into tables, which represent -- to the best knowledge of the author -- the first arithmetic steps towards Brill-Noether theory of limit roots.