Quasi-local holographic dualities in non-perturbative 3d quantum gravity

TL;DR

This paper investigates quasi-local holographic dualities in 3d quantum gravity using the $Ponzano-Regge$ state-sum to relate finite-region bulk quantum geometry to boundary 2d theories. It shows how boundary spin-network data induce 2d statistical models, exemplified by the isotropic $6$-vertex model at $j= frac{1}{2}$ on a square lattice, and derives a boundary action for a twisted torus that, in a stationary-phase regime, reproduces the expected semi-classical partition function and reveals a $BMS_3$-like structure in the $ abla$-to-zero limit. The study discusses two distinct large-scale limits and their interplay and highlights non-local topological insertions that encode bulk non-contractible cycles without summing over bulk topologies. Together these results provide a concrete, non-perturbative realization of holography in quasi-local regions and suggest avenues toward holographic renormalization in discrete gravity. The framework offers concrete tools for probing bulk-boundary correspondence in finite regions and for comparing non-perturbative gravity with perturbative AdS3/CFT2 insights.

Abstract

We present a line of research aimed at investigating holographic dualities in the context of three dimensional quantum gravity within finite bounded regions. The bulk quantum geometrodynamics is provided by the Ponzano-Regge state-sum model, which defines 3d quantum gravity as a discrete topological quantum field theory (TQFT). This formulation provides an explicit and detailed definition of the quantum boundary states, which allows a rich correspondence between quantum boundary conditions and boundary theories, thereby leading to holographic dualities between 3d quantum gravity and 2d statistical models as used in condensed matter. After presenting the general framework, we focus on the concrete example of the coherent twisted torus boundary, which allows for a direct comparison with other approaches to 3d/2d holography at asymptotic infinity. We conclude with the most interesting questions to pursue in this framework.

Figures (6)

• Figure 1: The 6 possible arrow configurations at a vertex for the 6-vertex model. The statistical weights $\omega$ associated to the vertex configurations are $\omega(I)= \omega(II) = a$, $\omega(III)= \omega(IV) = b$ and $\omega(V)= \omega(VI) = c$. For the correspondence with the spin network evaluation defined the Ponzano-Regge partition function for an homogeneous spin $j=\frac{1}{2}$ on the boundary lattice, the arrow direction translates into the sign of the magnetic index $m$ living on the edge. Indeed, for a spin $j=\frac{1}{2}$, the magnetic index can only take two values, $m=\pm \frac{1}{2}$.
• Figure 2: Discretization of the twisted solid torus for parameters $N_t=3$, $N_x=6$ and a shift $N_\gamma=1$. $(a)$: On the left hand side, we draw the cylinder with base the 2-disk of radius $a$ and with Euclidean time extension $\beta$. To get the twisted solid torus, we identify the top and the bottom of the cylinder up to the discrete shift $i\rightarrow i+N_{\gamma}$ for all lattice sites $i$, which causes the twist $\gamma = 2\pi \frac{N_\gamma}{N_x}$ in the gluing. $(b)$: On the right hand side, we draw the boundary lattice associated to the discretization of the solid torus. We attach a spin $T$ (resp. $L$) to each horizontal (resp. vertical) edges on the boundary, and we attach to each boundary vertex $v$ an intertwiner as defined by formula \ref{['eq_coh']}.
• Figure 3: Three plaquettes - faces - of the boundary discretization dual to 3 neighboring vertices. The dotted line are the links of the dual lattice. In red, the dihedral angle $\psi_{l}$ along the link $l$ relates the group elements $G_{s(l)}$ and $G_{t(l)}$ living on the two corresponding plaquettes, which are vertical neighbors. In blue, the dihedral angle $\psi_{l'}^{h}$ along the horizontal link $l'$ relates the group elements $G_{s(l')}$ and $G_{t(l')}$ .
• Figure 4: Plots of $\frac{\log(|\mathcal{A}(1)|)}{\log(|\mathcal{A}(n)|)}$ for $n$ running from $1$ to $\frac{N_x-1}{2}$ with the parameters $N_t=20$, $L=8$, $T=8$. The four plots correspond to $N_x=50,100,200,400$ (red, blue, green, orange). The $x$-axis corresponds to $\frac{2n}{N_x-1}$, which runs from 0 to 1.
• Figure 5: Plot of ${\frac{-N_t}{2}}\log{|}2T_{N_{x}}(a_{n})-2){|}$ in terms of the continuous angle variable $x=n\psi\in\left[\frac{2\pi}{N_x},\pi-\frac{\pi}{N_x}\right]\subset[0,\pi]$ for the odd lattice size $N_{x}=41$ and $N_{t}=1$ and for spins $T=5$ and $L=5,10,20,100$. As $L$ increases and thus the ratio $\frac{T}{L}$ goes to 0, the curves gets more and more curved and goes to the limit function (lowest curve).