Lorentzian 3d Gravity with Wormholes via Matrix Models

TL;DR

This work establishes a non-perturbative bridge between 3D Lorentzian quantum gravity and a hermitian two-matrix model with ABAB interaction, showing that the gravitational transfer matrix can be written as the logarithm of a matrix-model partition function. The authors map sandwich geometries to ABAB-generated 2D graphs, uncovering a two-phase structure: a weak-gravity phase with propagating 2D universes and a strong-coupling phase featuring spatial wormholes; touching-interactions in the matrix model realize wormhole-like degeneracies in the 3D geometry. The continuum limit is discussed through criticality in the ABAB model, with the gravitational coupling acting as a finite scale and Lorentzian gravity corresponding to a phase with suppressed wormholes. Overall, the paper provides an analytic framework for higher-dimensional quantum gravity via matrix models, connects to 2D gravity/Liouville theory insights, and outlines a program to extract the continuum Hamiltonian and study wormholes in the Lorentzian regime.

Abstract

We uncover a surprising correspondence between a non-perturbative formulation of three-dimensional Lorentzian quantum gravity and a hermitian two-matrix model with ABAB-interaction. The gravitational transfer matrix can be expressed as the logarithm of a two-matrix integral, and we deduce from the known structure of the latter that the model has two phases. In the phase of weak gravity, well-defined two-dimensional universes propagate in proper time, whereas in the strong-coupling phase the spatial hypersurfaces disintegrate into many components connected by wormholes.

Figures (9)

• Figure 1: Pyramids and tetrahedra can be used to discretize 3d Lorentzian space-times. We show the three types of fundamental building blocks and their location with respect to the spatial hypersurfaces of constant integer-$t$.
• Figure 2: A piece of a typical quadrangulation at $t+1/2$. The three types of squares made from solid and dashed lines arise as sections of the (4,1)-, (1,4)- and (2,2)-building blocks.
• Figure 3: Matrix-model representation of the building blocks at $t+1/2$. The gluing rules for the squares are determined by the Gaussian integrations, $\langle A_{ij}A_{kl}\rangle =\delta_{il} \delta_{jk}$, $\langle B_{ij}B_{kl}\rangle =\delta_{il} \delta_{jk}$, and $\langle A_{ij}B_{kl}\rangle =0$.
• Figure 4: Vertices of $\phi^4$-graphs dual to spatial quadrangulations.
• Figure 5: Examples of matrix-model configurations at $t+1/2$ which are not allowed in the original Lorentzian gravity model and which result in geometries with wormholes at time $t$. Shrinking the dashed links to zero, one obtains the two-geometries at the bottom. The thick dashed lines in the quadrangulations at the top are contracted to touching points or to points along one-dimensional wormholes.