Equivalent Quantisations of (2+1)-Dimensional Gravity

For spacetimes with the topology $\text{IR}\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}{T}^2$, the action of (2+1)-dimensional gravity with negative cosmological constant $\text{La}$ is written uniquely in terms of the time-independent traces of holonomies around two intersecting noncontractible paths on $T^2$. The holonomy parameters are related to the moduli on slices of constant mean curvature by a time-dependent canonical transformation which introduces an effective Hamiltonian. The quantisation of the two classically equivalent formulations differs by terms of order $O({\mathrm{\hslash }}^3)$, negligible for small $\mathrm{\mid }\text{La}\mathrm{\mid }$.