Deriving the non-perturbative gravitational dual of quantum Liouville theory from BCFT operator algebra

TL;DR

This work provides a constructive, non-perturbative holographic dual for an irrational CFT by discretizing the Liouville path-integral on arbitrary surfaces into a 3D bulk state-sum built from hyperbolic tetrahedra and quantum $6j$ symbols of $\mathcal{U}_q(sl(2,\mathbb{R}))$. The key idea is a shrinkable boundary that closes holes in the BCFT triangulation, yielding a strange correlator $Z_{\text{Liouville}} = \langle \Omega | \Psi_{\mathcal{U}_q(sl(2,\mathbb{R}))} \rangle$, which converges and non-perturbatively encodes bulk geometries; in the large-$c$ limit, the bulk amplitude recovers the Einstein–Hilbert action $S_{EH}$ on hyperbolic space. The construction yields an exact holographic tensor network for Liouville theory, with bulk edges corresponding to geodesic data and a Renormalisation operator $U(\lambda)$ implementing bulk coarse-graining via pentagon and orthogonality relations. The approach connects to TV/Teichmüller TQFT and provides a principled, non-perturbative sum over geometries for an irrational CFT, while offering a framework to explore AdS/BCFT, black-hole physics, and potential generalizations to other irrational RCFTs and higher dimensions.

Abstract

We demonstrate that, by utilizing the boundary conformal field theory (BCFT) operator algebra of the Liouville CFT, one can express its path-integral on any Riemann surface as a three dimensional path-integral with appropriate boundary conditions, generalising the recipe for rational CFTs \cite{Hung:2019bnq, Brehm:2021wev, Chen:2022wvy, Cheng:2023kxh}. This serves as a constructive method for deriving the \textit{quantum} holographic dual of the CFT, which reduces to Einstein gravity in the large central charge limit. As a byproduct, the framework provides an explicit discrete state-sum of a 3D non-chiral topological theory constructed from quantum $6j$ symbols of $\mathcal{U}_q(sl(2,\mathbb{R}))$ with non-trivial boundary conditions, representing a long-sought non-perturbative discrete formulation of 3D pure gravity with negative cosmological constant, at least within a class of three manifolds. This constitutes the first example of an exact holographic tensor network that reproduces a known irrational CFT with a precise quantum gravitational interpretation.

Figures (12)

• Figure 1: One choice of tiling of the plane by triangles whose angles are slightly chipped.
• Figure 2: The top left picture is a particular choice of triangulation of the CFT partition function. These triangles and their dual graphs correspond to open structure coefficients and conformal blocks respectively. The dashed vertex corresponds to 3-point block in Racah gauge absorbing the Plancherel measure as normalisation. The solid line vertex corresponds to 3-point block in block gauge.
• Figure 3: The triangles with contracted red dots describe the wave-function $|\Psi \rangle$. Each of these triangles are $6j$ symbols, and they satisfy the pentagon relation (top) and the orthogonality condition (bottom) which serves to connect wave-functions of different triangulations via local moves. In equation form, the pentagon relation is given by \ref{['pentagonappendix']}, while the orthogonality condition is expressed in \ref{['eq:ortho']}.
• Figure 4: Crossing relations of the conformal blocks.
• Figure 5: (a) Graphical representation of the associativity for boundary four point conformal blocks. (b) Example of reducing a graph with a loop to a graph containing a bubble with 2 legs (area circled by dashed lines and shaded red) via crossing relations. The blue dashed arrows indicate that those pairs of edges actually are identified. We label the edges to match with equation (\ref{['eq:Ebrane']}).