(2 + 1) covariant lattice theory and 't Hooft's formulation

We show that 't Hooft's representation of (2 + 1)-dimensional gravity in terms of flat polygonal tiles is closely related to a gauge-fixed version of the covariant Hamiltonian lattice theory. 't Hooft's gauge is remarkable in that it leads to a Hamiltonian which is a linear sum of vertex Hamiltonians, each of which is defined modulo . A cyclic Hamiltonian implies that `time' is quantized. However, it turns out that this Hamiltonian is constrained . If one chooses an internal time and solves this constraint for the `physical Hamiltonian', the result is not a cyclic function. Even if one quantizes following Dirac, the `internal time' observable does not acquire a discrete spectrum. We also show that in Euclidean three-dimensional lattice gravity, `space' can be either discrete or continuous depending on the choice of quantization. Finally, we propose a generalization of 't Hooft's gauge for Hamiltonian lattice formulations of topological gravity dimension four.