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Hamiltonian analysis of a topological theory in the presence of boundaries

We perform the canonical Hamiltonian analysis of a topological gauge theory, that can be seen both as a theory defined on a four dimensional spacetime region with boundaries --the bulk theory--, or as a theory defined on the boundary of the region --the boundary theory--. In our case the bulk theory is given by the 4-dimensional $U(1)$ Pontryagin action and the boundary one is defined by the $U(1)$ Chern-Simons action. We analyse the conditions that need to be imposed on the bulk theory so that the total Hamiltonian, smeared constraints and generators of gauge transformations be well defined (differentiable) for generic boundary conditions. We pay special attention to the interplay between the constraints and boundary conditions in the bulk theory on the one side, and the constraints in the boundary theory, on the other side. We illustrate how both theories are equivalent, despite the different canonical variables and constraint structure, by explicitly showing that they both have the same symmetries, degrees of freedom and observables.