Jackiw-Teitelboim Gravity from Holonomies: Discrete BF Formulation and Boundary Symmetries

TL;DR

This work develops a fully discrete, non-perturbative BF formulation of two-dimensional JT gravity using lattice holonomies in $\mathrm{SL}(2,\mathbb{R})$ and adjoint dilatons. The bulk is topological, with all physical content encoded at the boundary, where affine Kac–Moody and Brown–Henneaux Virasoro structures arise from admissible boundary conditions; a precise OPE dictionary connecting the lattice to continuum CFT data is established and validated by a controlled continuum limit. Quantization proceeds via boundary monodromy sectors, yielding a Virasoro algebra with central charge $c=12k/\alpha$ in each sector and an entropy $S=2\pi\sqrt{\mathcal{C}}=2\pi p$ determined purely by gauge-invariant boundary data through the dilaton Casimir $\mathcal{C}=p^2$. The entropy result agrees with Cardy in the continuum limit, and the framework naturally links to holonomy discretizations in three dimensions via dimensional reduction, clarifying the boundary origin of JT gravity dynamics and its Schwarzian limit as an effective description of the same discrete boundary data.

Abstract

We develop a fully discrete and non-perturbative formulation of two-dimensional Jackiw-Teitelboim (JT) gravity within the BF framework. Using group-valued holonomies and Lie-algebra--valued dilatons, the bulk theory is shown to be purely topological, with all physical information encoded at the boundary. We analyze admissible discrete boundary conditions and derive the corresponding asymptotic symmetry algebras directly at the lattice level, including an affine Kac-Moody symmetry and its Brown-Henneaux reduction to a Virasoro algebra, together with the associated Virasoro-dilaton structure. A precise operator product expansion (OPE) dictionary is established by taking the controlled continuum limit of the discrete Poisson brackets. Beyond asymptotic symmetries, we provide an effective boundary description and a representation-theoretic quantization organized by monodromy sectors. Within this discrete framework, black hole entropy follows from gauge-invariant holonomy data and is expressed in terms of the dilaton Casimir, reproducing the Bekenstein--Hawking result without invoking a fundamental Schwarzian action.