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The study of vector-like spectra in 4-dimensional F-theory compactificationsinvolves root bundles, which are important for understanding the QuadrillionF-theory Standard Models (F-theory QSMs) and their potential implications inphysics. Recent studies focused on a superset of physical root bundles whosecohomologies encode the vector-like spectra for certain matter representations.It was found that more than 99.995\% of the roots in this superset for thefamily $B_3( \Delta_4^\circ )$ of $\mathcal{O}(10^{11})$ different F-theory QSMgeometries had no vector-like exotics, indicating that this scenario is highlylikely.  To study the vector-like spectra, the matter curves in the F-theory QSMs wereanalyzed. It was found that each of them can be deformed to nodal curve that isidentical across all spaces in $B_3( \Delta^\circ )$. Therefore, from studyinga few nodal curves, one can probe the vector-like spectra of a large fractionof F-theory QSMs. To this end, the cohomologies of all limit roots weredetermined, with line bundle cohomology on rational nodal curves playing amajor role. A computer algorithm was used to enumerate all limit roots andanalyze the global sections of all tree-like limit roots. For the remainingcircuit-like limit roots, the global sections were manually determined. Theseresults were organized into tables, which represent -- to the best knowledge ofthe author -- the first arithmetic steps towards Brill-Noether theory of limitroots.

Root bundles: Applications to F-theory Standard Models