  Boundaries in gauge field theories are known to be the locus of a wealth of
interesting phenomena, as illustrated for example by the holographic principle
or by the AdS/CFT and bulk-boundary correspondences. In particular, it has been
acknowledged for quite some time that boundaries can break gauge invariance,
and thereby turn gauge degrees of freedom into physical ones. There is however
no known systematic way of identifying these degrees of freedom and possible
associated boundary observables. Following recent work by Donnelly and Freidel,
we show that this can be achieved by extending the covariant Hamiltonian
formalism so as to make it gauge-invariant under arbitrary large gauge
transformations. This can be done at the expense of extending the phase space
by introducing new boundary fields, which in turn determine new boundary
symmetries and observables. We present the general framework behind this
construction, and find the conditions under which it can be applied to an
arbitrary Lagrangian. By studying the examples of Abelian Chern-Simons theory
and first order three-dimensional gravity, we then show that the new boundary
observables satisfy the known corresponding Kac-Moody affine algebras. This
shows that this new extended phase space formulation does indeed properly
describe the dynamical boundary degrees of freedom, and gives credit to the
results which have been previously derived in the case of diffeomorphism
symmetry. We expect that this systematic understanding of the boundary
symmetries will play a major role for the quantization of gravity in finite
regions.
