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Poisson Structure of the Boundary Gravitons in 3D Gravity with Negative $Λ$

Cedric Troessaert

TL;DR

This work develops a unified Hamiltonian framework for 3D gravity with negative Λ, detailing general asymptotic fall-off conditions and the Poisson structure of boundary gravitons. It identifies a boundary phase-space encoded by $(P,J,M,Q)$ and derives the Dirac bracket and the differentiable generators that act on it, showing that admissible boundary conditions for the Lagrange multipliers correspond to boundary functionals $k_H(M,J,P,Q)$. The authors demonstrate how classic boundary conditions (Conformal, Brown–Henneaux, Chiral, and Constrained Chiral) arise as special cases within this framework, yielding familiar Virasoro and current algebras with explicit central charges and levels. This provides a systematic, model-independent method to study AdS$_3$ boundary dynamics and their holographic implications, while clarifying how different boundary choices translate into distinct asymptotic symmetries and boundary theories.

Abstract

We use the hamiltonian formalism to study the asymptotic structure of 3 dimensional gravity with a negative cosmological constant. We start by defining very general fall-off conditions for the canonical variables and study the implied poisson structure of the boundary gravitons. From the allowed differentiable gauge transformations, we can extract all the possible boundary conditions on the lagrange multipliers and the associated boundary hamiltonians. In the last section, we use this general framework to describe some of the previoussly known boundary conditions.

Poisson Structure of the Boundary Gravitons in 3D Gravity with Negative $Λ$

TL;DR

This work develops a unified Hamiltonian framework for 3D gravity with negative Λ, detailing general asymptotic fall-off conditions and the Poisson structure of boundary gravitons. It identifies a boundary phase-space encoded by and derives the Dirac bracket and the differentiable generators that act on it, showing that admissible boundary conditions for the Lagrange multipliers correspond to boundary functionals . The authors demonstrate how classic boundary conditions (Conformal, Brown–Henneaux, Chiral, and Constrained Chiral) arise as special cases within this framework, yielding familiar Virasoro and current algebras with explicit central charges and levels. This provides a systematic, model-independent method to study AdS boundary dynamics and their holographic implications, while clarifying how different boundary choices translate into distinct asymptotic symmetries and boundary theories.

Abstract

We use the hamiltonian formalism to study the asymptotic structure of 3 dimensional gravity with a negative cosmological constant. We start by defining very general fall-off conditions for the canonical variables and study the implied poisson structure of the boundary gravitons. From the allowed differentiable gauge transformations, we can extract all the possible boundary conditions on the lagrange multipliers and the associated boundary hamiltonians. In the last section, we use this general framework to describe some of the previoussly known boundary conditions.

Paper Structure

This paper contains 14 sections, 4 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Asymptotic structure
  3. Differentiable gauge transformations
  4. Boundary gravitons
  5. Dirac bracket for the boundary gravitons
  6. Boundary hamiltonian
  7. Some examples of boundary conditions
  8. Conformal
  9. Brown-Henneaux
  10. Chiral
  11. Constrained Chiral
  12. Radial decomposition
  13. Gauge transformations
  14. Fefferman-Graham gauge fixing

Key Result

Theorem 2.1

The set of gauge-like transformations preserving the asymptotic conditions eq:assympg-eq:assympR is given by: where the function $j$ is given in equation eq:fuckingj and the four functions $f, \chi, \psi$ and $Y$ are independent of $r$.

Theorems & Definitions (4)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1