A unified geometric framework for boundary charges and dressings:
non-Abelian theory and matter
Boundaries in gauge theories are a delicate issue. We approach it through a
geometric formalism based on the space of field configurations (field-space).
Our main geometric tool is a connection 1-form on field-space, $\varpi$, which
we introduce and explore in detail. Using it, we upgrade the presymplectic
structure of Yang-Mills theory with boundaries to be invariant even under
field-dependent gauge transformations. The gauge charges always vanish, while
global charges still arise for configurations with global symmetries. The
formalism is powerful because it is relational: the field-content itself is
used as a reference frame to distinguish `gauge' and `physical'; no new degrees
of freedom (e.g. group-valued edge modes) are required. Different choices of
reference fields give different $\varpi$'s, which are related to gauge-fixings
such as Higgs unitary and Coulomb gauge. But the formalism extends well beyond
gauge-fixings, for instance by avoiding the Gribov problem. For one choice of
$\varpi$, Goldstone modes arising from the condensation of matter degrees of
freedom play precisely the role of known edge modes. For another choice,
$\varpi$ is related to non-Abelian analogues of the Dirac dressing of the
electron. Applied to the Lorentz symmetry of vielbein gravity, our formalism
explains the origin of previously known Lorentz-invariant constructions.
Lastly, we understand dressings as field-space Wilson lines for $\varpi$. We
use the construction to unify the Lavelle-McMullan dressing with the
gauge-invariant fields of Gribov-Zwanziger and Vilkovisky-DeWitt, and to put
forward a notion of `historical dressing'.