On Holomorphic Factorization in Asymptotically AdS 3D Gravity

TL;DR

This paper explores the holographic correspondence for pure AdS3 gravity, identifying the gravitational sector within SL(2,C) Chern-Simons theory and discussing the holomorphic factorization of the gravity partition function.

Contribution

It characterizes the gravitational configurations as boundary projective structures and links the phase space to the cotangent bundle over Schottky space, proposing a holomorphic factorization of the partition function.

Findings

Configurations with metric interpretation are parameterized by boundary projective structures.

The asymptotic phase space is the cotangent bundle over Schottky space.

The gravity partition function exhibits holomorphic factorization.

Abstract

This paper studies aspects of ``holography'' for Euclidean signature pure gravity on asymptotically AdS 3-manifolds. This theory can be described as SL(2,C) CS theory. However, not all configurations of CS theory correspond to asymptotically AdS 3-manifolds. We show that configurations that do have the metric interpretation are parameterized by the so-called projective structures on the boundary. The corresponding asymptotic phase space is shown to be the cotangent bundle over the Schottky space of the boundary. This singles out a ``gravitational'' sector of the SL(2,C) CS theory. It is over this sector that the path integral has to be taken to obtain the gravity partition function. We sketch an argument for holomorphic factorization of this partition function.