# Holographic formulation of 3D metric gravity with finite boundaries

## TL;DR

This paper develops holographic boundary theories for linearized 3D gravity with finite boundaries, deriving effective actions that resemble Liouville theory coupled to boundary curvature, providing new insights into holography in finite regions.

## Contribution

It introduces a method to derive holographic boundary theories from 3D gravity dynamics for finite boundaries, revealing Liouville-like structures with boundary Ricci coupling.

## Key findings

- Boundary theories are Liouville-like with Ricci scalar coupling.
- Effective boundary actions derived from 3D gravity dynamics.
- Examples illustrate the structure and properties of these boundary theories.

## Abstract

In this work we construct holographic boundary theories for linearized 3D gravity, for a general family of finite or quasi-local boundaries. These boundary theories are directly derived from the dynamics of 3D gravity by computing the effective action for a geometric boundary observable, which measures the geodesic length from a given boundary point to some centre in the bulk manifold. We identify the general form for these boundary theories and find that these are Liouville like with a coupling to the boundary Ricci scalar. This is illustrated with various examples, which each offer interesting insights into the structure of holographic boundary theories.