A Hamiltonian Lattice Theory for Homogeneous Curved Spacetimes in 2+1 Dimensions

We propose an exact Hamiltonian lattice theory for (2+1)-dimensional spacetimes with homogeneous curvature. By gauging away the lattice we find a generalization of the ``polygon representation’’ of (2+1)-dimensional gravity. We compute the holonomies of the Lorentz connection ${\bf A}_i = \omega_i^a {\bf L}_a + e_i^a {\bf K}_a$ and find that the cycle conditions are satisfied only in the limit $\mathrm{\Lambda }\to 0$. This implies that, unlike in (2+1)-dimensional Einstein gravity, the connection $\mathbf{A}$ is not flat. If one modifies the theory by taking the cycle conditions as constraints, then one finds that the constraints algebra is first-class only if the Poisson bracket structure is deformed. This suggests that a finite theory of quantum gravity would require either a modified action including higher-order curvature terms, or a deformation of the commutator structure of the metric observables.