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We construct, to second order, the effective action of General Relativity forsmall excitations generated by a vector field and use it to study conformalsymmetry in the boundary of AdS$_3$. By requiring finiteness of the boundaryeffective action(s) for certain asymptotic transformations, we derive thewell-known Virasoro algebra and central charge associated with the boundary ofAdS$_3$. The bulk action for these transformations can be arbitrarily small.

Effective Action, Boundary Conditions, and Virasoro Algebra for AdS$_3$