Quantizing Hořava-Lifshitz gravity via causal dynamical triangulations

Abstract

We extend the discrete Regge action of causal dynamical triangulations to include discrete versions of the curvature squared terms appearing in the continuum action of $(2+1)$-dimensional projectable Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz gravity. Focusing on an ensemble of spacetimes whose spacelike hypersurfaces are two-spheres, we employ Markov chain Monte Carlo simulations to study the path integral defined by this extended discrete action. We demonstrate the existence of known and novel macroscopic phases of spacetime geometry, and we present preliminary evidence for the consistency of these phases with solutions to the equations of motion of classical Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz gravity. Apparently, the phase diagram contains a phase transition between a time-dependent de Sitter-like phase and a time-independent phase. We speculate that this phase transition may be understood in terms of deconfinement of the global gravitational Hamiltonian integrated over a spatial two-sphere.