  Computing the exact spectrum of charged massless matter is a crucial step
towards understanding the effective field theory describing F-theory vacua in
four dimensions. In this work we further develop a coherent framework to
determine the charged massless matter in F-theory compactified on elliptic
fourfolds, and demonstrate its application in a concrete example. The gauge
background is represented, via duality with M-theory, by algebraic cycles
modulo rational equivalence. Intersection theory within the Chow ring allows us
to extract coherent sheaves on the base of the elliptic fibration whose
cohomology groups encode the charged zero-mode spectrum. The dimensions of
these cohomology groups are computed with the help of modern techniques from
algebraic geometry, which we implement in the software gap. We exemplify this
approach in models with an Abelian and non-Abelian gauge group and observe
jumps in the exact massless spectrum as the complex structure moduli are
varied. An extended mathematical appendix gives a self-contained introduction
to the algebro-geometric concepts underlying our framework.
