  Gauge theories possess non-local features that, in the presence of
boundaries, inevitably lead to subtleties. In this article, we continue our
study of a unified solution based on a geometric tool operating on field-space:
a connection form. We specialize to the $D+1$ formulation of Yang-Mills
theories on configuration space, and we precisely characterize the gluing of
the Yang-Mills field across regions. In the $D+1$ formalism, the
connection-form splits the electric degrees of freedom into their
pure-radiative and Coulombic components, rendering the latter as conjugate to
the pure-gauge part of the gauge potential. Regarding gluing, we obtain a
characterization for topologically simple regions through closed formulas.
These formulas exploit the properties of a generalized Dirichlet-to-Neumann
operator defined at the gluing surface; through them, we find only the
radiative components and the local charges are relevant for gluing. Finally, we
study the gluing into topologically non-trivial regions in 1+1 dimensions. We
find that in this case, the regional radiative modes do not fully determine the
global radiative mode (Aharonov-Bohm phases). For the global mode takes a new
contribution from the kernel of the gluing formula, a kernel which is
associated to non-trivial cohomological cycles. In no circumstances do we find
a need for postulating new local degrees of freedom at boundaries. $\text{The
partial results of these notes have been completed and substantially clarified
in a more recent, comprehensive article from October 2019. (titled: "The
quasilocal degrees of freedom of Yang-Mills theory").}$
