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Boundaries in gauge field theories are known to be the locus of a wealth ofinteresting phenomena, as illustrated for example by the holographic principleor by the AdS/CFT and bulk-boundary correspondences. In particular, it has beenacknowledged for quite some time that boundaries can break gauge invariance,and thereby turn gauge degrees of freedom into physical ones. There is howeverno known systematic way of identifying these degrees of freedom and possibleassociated boundary observables. Following recent work by Donnelly and Freidel,we show that this can be achieved by extending the covariant Hamiltonianformalism so as to make it gauge-invariant under arbitrary large gaugetransformations. This can be done at the expense of extending the phase spaceby introducing new boundary fields, which in turn determine new boundarysymmetries and observables. We present the general framework behind thisconstruction, and find the conditions under which it can be applied to anarbitrary Lagrangian. By studying the examples of Abelian Chern-Simons theoryand first order three-dimensional gravity, we then show that the new boundaryobservables satisfy the known corresponding Kac-Moody affine algebras. Thisshows that this new extended phase space formulation does indeed properlydescribe the dynamical boundary degrees of freedom, and gives credit to theresults which have been previously derived in the case of diffeomorphismsymmetry. We expect that this systematic understanding of the boundarysymmetries will play a major role for the quantization of gravity in finiteregions.

Edge modes and corner ambiguities in 3d Chern-Simons theory and gravity