Quantizing Horava-Lifshitz Gravity via Causal Dynamical Triangulations

TL;DR

This work extends causal dynamical triangulations by incorporating discrete curvature-squared terms from $(2+1)$-dimensional projectable Hořava-Lifshitz gravity into the Regge action, and analyzes the resulting nonperturbative quantum theory via Markov chain Monte Carlo on foliated triangulations. A discrete HL action is derived using Gauss–Codazzi decomposition and volume-sharing discretization for $R_2^2$ and $K^2$, yielding an imaginary-time action with couplings $k_0,k_3,\lambda,\alpha$ evaluated on $S^2\times S^1$ lattices with $a=1$, $\eta=1$. The simulations reveal three extended phases (C, D, E) with phase behavior influenced by the curvature-squared terms, and observables such as the spectral dimension $d_s(\sigma)$ and the discrete 2-volume $N_2^{SL}$ show semiclassical, Euclidean de Sitter–like geometries across these phases. The results indicate compatibility between HL gravity and CDT nonperturbative quantization, suggesting a pathway to explore renormalization-group flows and universality classes connecting HL gravity and CDT.

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 Horava-Lifshitz gravity. Focusing on an ensemble of spacetimes whose spacelike hypersurfaces are 2-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 Horava-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 2-sphere.

Figures (10)

• Figure 1: From left to right: the $(3,1)$-, $(2,2)$-, and $(1,3)$-tetrahedra. The abscissa indicates the number of vertices on an initial triangulated spacelike hypersurface, and the ordinate indicates the number of vertices on the next triangulated spacelike hypersurface.
• Figure 2: An embedding in three dimensions of two $(3,1)$-tetrahedra (solid black) joined by three $(2,2)$- tetrahedra (thin black) all sharing a common edge. A vector perpendicular to the triangular base of a $(3,1)$-tetrahedron rotates through an angle $\pi- 2\theta_{L}^{(3,1)}- 3 \theta_{L}^{(2,2)}$ as it is parallel transported across the common edge.
• Figure 3: Depictions of representative spacetimes showing the number $N_{2}^{SL}$ of spacelike triangles as a function of discrete time $\tau$. \ref{['3d601000']} Phase A ($k_{0}=6.00$, $k_{3}=1.85$, $\lambda=1.00$, $\alpha=0.00$) \ref{['3d101000']} Phase C ($k_{0}=1.00$, $k_{3}=0.75$, $\lambda=1.00$, $\alpha=0.00$)
• Figure 4: The ensemble average spectral dimension $\langle d_{s}\rangle$ as a function of diffusion time $\sigma$. \ref{['sd601000']} Phase A ($k_{0}=6.00$, $k_{3}=1.85$, $\lambda=1.00$, $\alpha=0.00$) \ref{['sd101000']} Phase C ($k_{0}=1.00$, $k_{3}=0.75$, $\lambda=1.00$, $\alpha=0.00$)
• Figure 5: \ref{['criticalsurface']} The critical surface as explored thus far. \ref{['phasediagram']} The critical surface projected onto the $\lambda$-$\alpha$ plane showing phases C, D, and E respectively in blue circles, magenta squares, and orange diamonds.