Cohomologies of coherent sheaves and massless spectra in F-theory

In this PhD thesis we investigate the significance of Chow groups for zero mode counting and anomaly cancellation in F-theory vacua. The major part of this thesis focuses on zero mode counting. We explain that elements of Chow group describe a subset of gauge backgrounds and give rise to a line bundle on each matter curve. The sheaf cohomologies of these line bundles are found to encode the chiral and anti-chiral localised zero modes in this compactification. Therefore, it is of prime interest to compute these sheaf cohomologies. Unfortunately, the line bundles in question are in general non-pullback line bundles. In particular, this is the case for the hypercharge flux employed in F-theory models of grand unified theories (GUTs). Consequently, existing methods, such as the cohomCalg-algorithm, cannot be applied. In collaboration with the mathematician Mohamed Barakat, we have therefore implemented algorithms which determine the sheaf cohomologies of all coherent sheaves on toric varieties. These algorithms are provided by the gap-package SheafCohomologiesOnToricVarieties which extends the homalg_project of Mohamed Barakat. We exemplify these algorithms in explicit (toy-)models of F-theory GUTs. As a spin-off of this analysis, we proved that in an entire class of F-theory vacua, the matter surface fluxes satisfy a number of relations in the Chow ring, which we related to anomaly cancellation. Based on this evidence we conjecture that the well-known anomaly cancellation conditions in F-theory -- typically phrased as intersections in the cohomology ring -- can be extended even to relations in the Chow ring