Quasi-local holographic dualities in non-perturbative 3d quantum gravity I - Convergence of multiple approaches and examples of Ponzano-Regge statistical duals

TL;DR

The paper develops a non-perturbative, quasi-local holographic framework for 3d quantum gravity using the Ponzano–Regge model on a twisted solid torus, emphasizing how boundary spin-network states generate dual boundary theories. It reviews multiple non-perturbative and perturbative approaches to 3d gravity, clarifies boundary conditions and actions, and derives exact PR amplitudes for boundary quadrangulations, revealing a twist-dependent pole structure that mirrors BMS$_3$ features in the flat limit. In the $j= frac{1}{2}$ boundary regime the PR amplitudes map onto the 6-vertex model, establishing a precise link between quantum gravity boundary data and integrable statistical systems; the $J=0$ $s$-channel case provides an exactly solvable benchmark and highlights finiteness at finite boundaries with pole behavior emerging in asymptotic limits. The work lays groundwork for subsequent parts that treat large-spin coherent states and more general boundary states, aiming to connect non-perturbative gravity with boundary sigma-models, Liouville-type theories, and BMS$_3$ representations, thereby enriching holographic insights in a background-independent, topological setting.

Abstract

This is the first of a series of papers dedicated to the study of the partition function of three-dimensional quantum gravity on the twisted solid torus with the aim to deepen our understanding of holographic dualities from a non-perturbative quantum gravity perspective. Our aim is to compare the Ponzano-Regge model for non-perturbative three-dimensional quantum gravity with the previous perturbative calculations of this partition function. We begin by reviewing the results obtained in the past ten years via a wealth of different approaches, and then introduce the Ponzano--Regge model in a self-contained way. Thanks to the topological nature of three-dimensional quantum gravity we can solve exactly for the bulk degrees of freedom and identify dual boundary theories which depend on the choice of boundary states, that can also describe finite, non-asymptotic boundaries. This series of papers aims precisely at the investigation of the role played by the different quantum boundary conditions leading to different boundary theories. Here, we will describe the spin network boundary states for the Ponzano-Regge model on the twisted torus and derive the general expression for the corresponding partition functions. We identify a class of boundary states describing a tessellation with maximally fuzzy squares for which the partition function can be explicitly evaluated. In the limit case of a large, but finely discretized, boundary we find a dependence on the Dehn twist angle characteristic for the BMS3 character. We furthermore show how certain choices of boundary states lead to known statistical models as dual field theories-but with a twist.

Figures (15)

• Figure 1: Example of the background triangulation with $N_x=6$ and $N_t=2$. The effect of the twist $N_\gamma$ appear when we identified $A_i$ and $C_i$ through $A_{i}=C_{i+N_\gamma}$. Each prism is triangulated with tree tetrahedra, that can be construct by considering a diagonal per vertical faces of the prism. In the right panel we draw a prism triangulated with three tetrahedra, draw in red, blue and white.
• Figure 2: We have depicted in red a boundary 3-cell and the associated boundary 2-cell is in deep red. The set of boundary 2-cells provides a cellular decomposition of the boundary manifold. The dual of the boundary 3-cell is denoted by $O_1$, and the dual of the boundary 2-cell is denoted by $B_1$. The dual edge of a boundary 2-cell emanating from the node $O_1$, dual to the red boundary 3-cell, is depicted as a dashed dark red line. The dual of the boundary edge $e$ in solid dark red line.
• Figure 3: Decomposition of a 4-valent intertwiner into two 3-valent ones along one of the diagonals. The recoupling spin associated to the length of the diagonal is denoted by $j$. $\theta$ is the dihedral angle between the two triangles.
• Figure 4: Oriented square lattice on the twisted torus. The twist angle is $\gamma=2\pi \frac{N_{\gamma}}{N_{x}}$. Starting from the vertex $(t,x)$, the edge on the right is associated to $g^{h}_{t,x}$ and the edge below to $g^{v}_{t,x}$. All the spins are fixed to $j$. The horizontal periodic condition is without twist.
• Figure 5: The three channels for splitting a 4-valent intertwiner into two 3-valents ones linked by an intermediate link carrying a spin $J$.