A Hexagon Model for 3D Lorentzian Quantum Cosmology

TL;DR

This paper introduces a 3D Lorentzian quantum gravity model using triangulations with flat tori, revealing combinatorial structures, exponential entropy scaling, and explicit Teichmüller parameter expressions, facilitating analytic studies of quantum gravity phenomena.

Contribution

It presents a novel triangulation-based model of 3D Lorentzian quantum gravity with explicit geometric and combinatorial formulations, enabling detailed analytic investigations.

Findings

Combinatorics of the transfer matrix resembles vicious walkers on a lattice.

Model entropy scales exponentially with volume.

Explicit formulas for Teichmüller parameters in terms of triangulation data.

Abstract

We formulate a dynamically triangulated model of three-dimensional Lorentzian quantum gravity whose spatial sections are flat two-tori. It is shown that the combinatorics involved in evaluating the one-step propagator (the transfer matrix) is that of a set of vicious walkers on a two-dimensional lattice with periodic boundary conditions and that the entropy of the model scales exponentially with the volume. We also give explicit expressions for the Teichm\"uller parameters of the spatial slices in terms of the discrete parameters of the 3d triangulations, and reexpress the discretized action in terms of them. The relative simplicity and explicitness of this model make it ideally suited for an analytic study of the conformal-factor cancellation observed previously in Lorentzian dynamical triangulations and of its relation to alternative, reduced phase space quantizations of 3d gravity.