We thank the referee for raising this interesting point.

See, for instance, the exposition in [20, 21] for a discussion of the various possible boundary conditions in gravity and their virtues.

These equations can also be derived from the solutions of the Einstein-Hamilton-Jacobi equation along paths of steepest descent together with a notion of time which emerges from the geometry of the configuration space via the length of these paths measured with respect to the DeWitt metric [32].

The reduced phase space quantization in the York gauge for ($2+1$)-dimensional gravity on spacetimes with compact spatial slices without any asymptotic boundary (and hence no standard holographic dual) was done in [35, 36].

See, for instance, [38, 39], and references therein. For a definition in terms of the areal radial coordinate, see the classic treatments in [17, 18].

They satisfy the surface deformation algebra under the Poisson bracket [8, 29, 30]: $$\begin{gathered} \{{\mathcal{H}}_{\perp }(x),{\mathcal{H}}_{\perp }(x^{\prime })\}=(g^{ij}{\mathcal{H}}_j(x)+g^{ij}{\mathcal{H}}_j(x^{\prime })){\delta }_{,i}(x,x^{\prime }), \\ \{{\mathcal{H}}_i(x),{\mathcal{H}}_{\perp }(x^{\prime })\}={\mathcal{H}}_{\perp }(x){\delta }_{,i}(x,x^{\prime }), \\ \{{\mathcal{H}}_i(x),{\mathcal{H}}_j(x^{\prime })\}={\mathcal{H}}_i(x){\delta }_{,j}(x,x^{\prime })+{\mathcal{H}}_j(x){\delta }_{,i}(x,x^{\prime }), \end{gathered}$$where the derivatives on the $\delta$ functions are all with respect to the first argument.

We thank E. Witten for pointing this out.

For spatial metrics $g_{ij}$ which approach the ${\mathrm{AdS}}_d$ spatial metric $\frac{\mathrm{d}{r}^2}{r^2}+r^2\mathrm{d}{\mathrm{\Omega }}_{d-1}^2$, the falloff condition on ${\xi }_{\perp }$ that ensures that the boundary contribution is zero is ${\xi }_{\perp }\sim r^{-(d-1)/2-\epsilon }$.

The above analysis implies that the matrix of Poisson brackets ${\mathcal{M}}_{AB}=\{{\mathcal{C}}_A,{\mathcal{C}}_B\}$ is invertible, where ${\mathcal{C}}_A$ collectively denotes all constraints and gauge conditions. To analyze the Hamiltonian dynamics, one can alternatively work with the original phase but with the Poisson bracket replaced by the Dirac bracket constructed out of the above matrix ${\mathcal{M}}_{AB}$ in the standard way.

Given any metric $g_{ij}^{\prime }$ whose Ricci scalar $R^{\prime }$ approaches $2\mathrm{\Lambda }$ near the boundary, we can always find a unique $\varphi$ such that ${\tilde{g}}_{ij}={\mathrm{e}}^{2\varphi }{g}_{ij}^{\prime }$ has constant Ricci scalar curvature $2\mathrm{\Lambda }$ everywhere. This $\varphi$ is the unique solution of the Yamabe equation $D^{\prime 2}\varphi =\frac{1}{2}{R}^{\prime }-\mathrm{\Lambda }{\mathrm{e}}^{2\varphi }$. See [38, 45, 46, 47, 48] for more details.

The more general boundary conditions which include a $\phi$ dependence are related to the Brown-Henneaux boundary gravitons [18]. We could also take a nonzero constant value for ${\tilde{K}}_{\theta \phi }=J/2$, which would correspond to a rotating BTZ solution. We do not consider this generalization here though all our results can be extended to this case with minor modifications.

We thank Alok Laddha for bringing Kuchar’s paper to our attention.

We thank O. Aharony for raising this point.