Holonomies and Boundary Symmetries in Discrete BF Formulation of Jackiw-Teitelboim Gravity

Abstract

We develop a fully discrete and non-perturbative realization of two--dimensional Jackiw--Teitelboim (JT) gravity as an $\mathrm{sl}(2,\mathbb{R})$ BF theory. 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 within a framework that does not require an explicit Schwarzian action.