Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometric aspects of the AdS/CFT correspondence

Michael T. Anderson

TL;DR

The paper develops a rigorous mathematical framework for the Euclidean AdS/CFT correspondence through conformally compact (AH) Einstein metrics, focusing on the Fefferman–Graham expansion and the Dirichlet–Neumann data that connect boundary conformal structure to bulk geometry. It analyzes global existence and uniqueness of AH fillings, the role of positive boundary scalar curvature in preventing cusp Degenerations, and the renormalized action that parallels the CFT stress-energy tensor via holographic renormalization. In four dimensions, self-duality provides explicit, computable relations between the renormalized action, boundary invariants, and the stress-energy, with several explicit self-dual examples and a near-uniqueness result for analytic boundary data. The discussion extends to de Sitter space via a Wick-rotation correspondence, including global existence results for small de Sitter perturbations and constructions of self-similar spacetimes, highlighting both the potential and limitations of a fully gravitational semi-classical partition function in AdS/CFT.

Abstract

We discuss classical gravitational aspects of the AdS/CFT correspondence, with the aim of obtaining a rigorous (mathematical) understanding of the semi-classical limit of the gravitational partition function. The paper surveys recent progress in the area, together with a selection of new results and open problems.

Geometric aspects of the AdS/CFT correspondence

TL;DR

The paper develops a rigorous mathematical framework for the Euclidean AdS/CFT correspondence through conformally compact (AH) Einstein metrics, focusing on the Fefferman–Graham expansion and the Dirichlet–Neumann data that connect boundary conformal structure to bulk geometry. It analyzes global existence and uniqueness of AH fillings, the role of positive boundary scalar curvature in preventing cusp Degenerations, and the renormalized action that parallels the CFT stress-energy tensor via holographic renormalization. In four dimensions, self-duality provides explicit, computable relations between the renormalized action, boundary invariants, and the stress-energy, with several explicit self-dual examples and a near-uniqueness result for analytic boundary data. The discussion extends to de Sitter space via a Wick-rotation correspondence, including global existence results for small de Sitter perturbations and constructions of self-similar spacetimes, highlighting both the potential and limitations of a fully gravitational semi-classical partition function in AdS/CFT.

Abstract

We discuss classical gravitational aspects of the AdS/CFT correspondence, with the aim of obtaining a rigorous (mathematical) understanding of the semi-classical limit of the gravitational partition function. The paper surveys recent progress in the area, together with a selection of new results and open problems.

Paper Structure

This paper contains 7 sections, 9 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Conformally compact Einstein metrics
  3. Uniqueness Issue.
  4. Existence Issue.
  5. Role of $R \geq 0$.
  6. Self-Duality.
  7. Continuation to de Sitter and self-similar vacuum space-times.

Key Result

Theorem 1.1

[5,6] Let $M$ be a compact, oriented $(n+1)$-manifold with boundary $\partial M.$ If ${\cal E}$ is non-empty, then ${\cal E}$ is a smooth infinite dimensional manifold. Further, the boundary map is a $C^{\infty}$ smooth Fredholm map of index 0. Thus the derivative $D\Pi$ has finite dimensional kernel and cokernel, has closed range, and

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark 2.3
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • Remark 3.4
  • Theorem 4.1
  • Remark 4.2
  • Theorem 5.1
  • Remark 5.2
  • ...and 5 more