Extrinsic curvature in two-dimensional causal dynamical triangulation

Causal dynamical triangulation (CDT) is a nonperturbative quantization of general relativity. Horava-Lifshitz gravity, on the other hand, modifies general relativity to allow for perturbative quantization. Past work has given rise to the speculation that Horava-Lifshitz gravity might correspond to the continuum limit of CDT. In this paper we add another piece to this puzzle by applying the CDT quantization prescription directly to Horava-Lifshitz gravity in two dimensions. We derive the continuum Hamiltonian, and we show that it matches exactly the Hamiltonian derived from canonically quantizing the Horava-Lifshitz action. Unlike the standard CDT case, here the introduction of a foliated lattice does not impose further restriction on the configuration space and, as a result, lattice quantization does not leave any imprint on continuum physics as expected.