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We study 2d and 3d gravity theories on spacetimes with causal (timelike ornull) codimension one boundaries while allowing for variations in the positionof the boundary. We construct the corresponding solution phase space andspecify boundary degrees freedom by analysing boundary (surface) chargeslabelling them. We discuss Y and W freedoms and change of slicing in thesolution space. For D dimensional case we find D+1 surface charges, which aregeneric functions over the causal boundary. We show that there exist solutionspace slicings in which the charges are integrable. For the 3d case thereexists an integrable slicing where charge algebra takes the form of Heisenberg\oplus\ {\cal A}_3 where {\cal A}_3 is two copies of Virasoro at Brown-Henneauxcentral charge for AdS_3 gravity and BMS_3 for the 3d flat space gravity.

Symmetries at Causal Boundaries in 2D and 3D Gravity