Boundaries in gauge theories are a delicate issue. Arbitrary boundary choices
enter the calculation of charges via Noether's second theorem, obstructing the
assignment of unambiguous physical charges to local gauge symmetries. Replacing
the arbitrary boundary choice with new degrees of freedom suggests itself. But,
concretely, such boundary degrees of freedom are spurious---i.e. they are not
part of the original field content of the theory---and have to disappear upon
gluing. How should we fit them into what we know about field-theory? We resolve
these issues in a unified and geometric manner, by introducing a connection
1-form, ϖ, in the field-space of Yang-Mills theory. Using this geometric
tool, a modified version of symplectic geometry---here called `horizontal'---is
possible. Independently of boundary conditions, this formalism bestows to each
region a physical notion of charge: the horizontal Noether charge. The
horizontal gauge charges always vanish, while global charges still arise for
reducible configurations characterized by global symmetries. The field-content
itself is used as a reference frame to distinguish `gauge' and `physical'; no
new degrees of freedom, such as group-valued edge modes, are required.
Different choices of reference fields give different ϖ's, which are
cousins of gauge-fixing like the Higgs-unitary and Coulomb gauges. But the
formalism extends well beyond gauge-fixings, for instance by avoiding the
Gribov problem. For one choice of ϖ, would-be Goldstone modes arising
from the condensation of matter degrees of freedom play precisely the role of
the known group-valued edge modes, but here they arise as preferred coordinates
in field space, rather than new fields. For another choice, in the Abelian
case, ϖ recovers the Dirac dressing of the electron.Comment: 71 pages, 3 appendices, 9 figures. Summary of the results at the
beginning of the paper. v2: numerous improvements in the presentation, and
introduction of new references, taking colleague feedback into accoun