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On Holomorphic Factorization in Asymptotically AdS 3D Gravity

Kirill Krasnov

TL;DR

The paper analyzes holography in Euclidean pure AdS3 gravity by recasting the theory as SL(2,C) Chern–Simons, identifying the gravitational sector with a restricted CS phase space linked to boundary projective structures. It shows the asymptotic phase space is the cotangent bundle over Schottky space and demonstrates holomorphic factorization of the gravity partition function at genus zero, arising from a decomposition into Liouville-like boundary data and holomorphic/anti-holomorphic CS blocks. The results rely on Fefferman–Graham expansion, Schottky uniformization, and a detailed parameterization of asymptotic CS connections, laying groundwork for a boundary CFT interpretation that is likely distinct from Liouville theory. The analysis provides a concrete framework for a holomorphic factorization structure in non-compact CS gravity and ties bulk configurations to boundary conformal structures via projective data and quadratic differentials.

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.

On Holomorphic Factorization in Asymptotically AdS 3D Gravity

TL;DR

The paper analyzes holography in Euclidean pure AdS3 gravity by recasting the theory as SL(2,C) Chern–Simons, identifying the gravitational sector with a restricted CS phase space linked to boundary projective structures. It shows the asymptotic phase space is the cotangent bundle over Schottky space and demonstrates holomorphic factorization of the gravity partition function at genus zero, arising from a decomposition into Liouville-like boundary data and holomorphic/anti-holomorphic CS blocks. The results rely on Fefferman–Graham expansion, Schottky uniformization, and a detailed parameterization of asymptotic CS connections, laying groundwork for a boundary CFT interpretation that is likely distinct from Liouville theory. The analysis provides a concrete framework for a holomorphic factorization structure in non-compact CS gravity and ties bulk configurations to boundary conformal structures via projective data and quadratic differentials.

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.

Paper Structure

This paper contains 13 sections, 4 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Overview and main results
  3. Actions
  4. The Asymptotic Phase Space
  5. Asymptotically AdS spaces via Schottky groups
  6. The Fefferman-Graham asymptotic expansion
  7. The Banados, Skenderis-Solodukhin and Rooman-Spindel metrics
  8. The asymptotic structure of the CS connections
  9. General parameterization of the connections
  10. The partition function
  11. CS and WZW actions
  12. Chern-Simons connections
  13. Projective structures

Key Result

Lemma 1

If $u, v$ are two linearly independent solutions of the Fuchs equation (Fuchs), then also satisfies the same equation with respect to $z$.

Theorems & Definitions (4)

  • Lemma 1
  • Lemma 2
  • Lemma 1'
  • Lemma 2'