  Motivated by the appearance of fractional powers of line bundles in studies
of vector-like spectra in 4d F-theory compactifications, we analyze the
structure and origin of these bundles. Fractional powers of line bundles are
also known as root bundles and can be thought of as generalizations of spin
bundles. We explain how these root bundles are linked to inequivalent F-theory
gauge potentials of a $G_4$-flux.
  While this observation is interesting in its own right, it is particularly
valuable for F-theory Standard Model constructions. In aiming for MSSMs, it is
desired to argue for the absence of vector-like exotics. We work out the root
bundle constraints on all matter curves in the largest class of currently-known
F-theory Standard Model constructions without chiral exotics and gauge coupling
unification. On each matter curve, we conduct a systematic "bottom"-analysis of
all solutions to the root bundle constraints and all spin bundles. Thereby, we
derive a lower bound for the number of combinations of root bundles and spin
bundles whose cohomologies satisfy the physical demand of absence of
vector-like pairs.
  On a technical level, this systematic study is achieved by a well-known
diagrammatic description of root bundles on nodal curves. We extend this
description by a counting procedure, which determines the cohomologies of
so-called limit root bundles on full blow-ups of nodal curves. By use of
deformation theory, these results constrain the vector-like spectra on the
smooth matter curves in the actual F-theory geometry.
