Hamiltonian surface charges in 3d Einstein gravity

Hamiltonian surface charges are introduced in terms of the covariant phase space formalism, then they are computed for Einstein gravity in three spacetime dimensions. In general, this gives a different result depending on whether one uses the second-order (metric) or first-order (Cartan or Chern-Simons) formulation. This is studied explicitly for three different spacetime examples, only in one of which the difference between formalisms becomes apparent.3