AdS$_3$ Einstein Gravity and Boundary Description: Pedagogical Review

TL;DR

This review explores the boundary description of AdS$_3$ Einstein gravity, highlighting its topological nature, connection to 2D Schwarzian theory, and implications for holography and quantum phenomena.

Contribution

It provides a comprehensive pedagogical overview of the boundary dynamics of AdS$_3$ gravity, including derivation of the boundary theory and analysis of its quantum and modular properties.

Findings

Boundary theory is a 2D Schwarzian theory.

Partition function is one-loop exact.

Loss of modular symmetry in boundary theory.

Abstract

We review the various aspects of the 3D Einstein gravity theory with a negative cosmological constant and its boundary description. We also explore its connections to CFTs, modular symmetry, and holography. It is worth noting that this particular theory is topological in nature, which means that all the physical degrees of freedom are located on the boundary. Additionally, we can derive the boundary description on a torus, which takes the form of a 2D Schwarzian theory. This observation suggests that the relevant degrees of freedom for the theory can be described using this 2D theory. Because of the renormalizability of the 3D gravity theory, one can probe the quantum regime. This suggests that it is possible to investigate quantum phenomena. Unlike the conventional CFTs, when considering the AdS$_3$ background, the boundary theory loses modular symmetry. This represents a departure from the usual behavior of CFT and is quite intriguing. The Weyl transformation induces anomaly in CFTs, and we indicate that applying this transformation to the 2D Schwarzian theory leads to similar results. Summing over all geometries with the asymptotic AdS$_3$ boundary condition is equivalent to summing over a modular group. The partition function is one-loop exact and therefore an analytical expression from the summation. This theory holds potential applications in Quantum Information and is a recurring theme in the study of holography, where gravitational theories are connected with CFTs.