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Enhanced asymptotic symmetry algebra of AdS3

Cedric Troessaert

TL;DR

This work generalizes Brown-Henneaux boundary conditions for AdS$_3$ gravity by promoting a dynamical boundary conformal factor, yielding new boundary degrees of freedom and an enhanced asymptotic symmetry algebra. The resulting algebra is the semi-direct product of two Virasoro algebras with two $U(1)$ currents, and the charges acquire a central extension with three nonzero constants: the two Brown-Henneaux central charges $c^{\pm}$ and a new central term $k$. Through a basis change tied to the exact AdS$_3$ Killing vectors, the standard Brown-Henneaux central charges are recovered while preserving the new $k$, illustrating a richer symmetry structure. The analysis combines boundary dynamics, surface charges, and central extensions to illuminate additional degrees of freedom and potential avenues for understanding 3D gravity and BTZ entropy within a broader symmetry framework.

Abstract

A generalization of the Brown-Henneaux boundary conditions is introduced for pure gravity with negative cosmological constant in 3 dimensions. This leads to new degrees of freedom and to an enhancement of the symmetry algebra. Up to the zero modes, it consists of two copies of the semi-direct product of a Virasoro algebra with a U(1) current algebra. The associated surface charge algebra now contains three non-zero central charges: the two usual Brown-Henneaux central charges and one new quantity.

Enhanced asymptotic symmetry algebra of AdS3

TL;DR

This work generalizes Brown-Henneaux boundary conditions for AdS gravity by promoting a dynamical boundary conformal factor, yielding new boundary degrees of freedom and an enhanced asymptotic symmetry algebra. The resulting algebra is the semi-direct product of two Virasoro algebras with two currents, and the charges acquire a central extension with three nonzero constants: the two Brown-Henneaux central charges and a new central term . Through a basis change tied to the exact AdS Killing vectors, the standard Brown-Henneaux central charges are recovered while preserving the new , illustrating a richer symmetry structure. The analysis combines boundary dynamics, surface charges, and central extensions to illuminate additional degrees of freedom and potential avenues for understanding 3D gravity and BTZ entropy within a broader symmetry framework.

Abstract

A generalization of the Brown-Henneaux boundary conditions is introduced for pure gravity with negative cosmological constant in 3 dimensions. This leads to new degrees of freedom and to an enhancement of the symmetry algebra. Up to the zero modes, it consists of two copies of the semi-direct product of a Virasoro algebra with a U(1) current algebra. The associated surface charge algebra now contains three non-zero central charges: the two usual Brown-Henneaux central charges and one new quantity.

Paper Structure

This paper contains 9 sections, 1 theorem, 56 equations.

Table of Contents

  1. Introduction
  2. New asymptotic conditions
  3. Asymptotic symmetries
  4. Asymptotic solutions to the EOM
  5. Surface charges
  6. Centrally extended algebra
  7. Brown-Henneaux central charges recovered
  8. Conclusions
  9. Central Extension

Key Result

Theorem 4.1

Any solution of Einstein's equations with negative cosmological constant satisfying to the usual Brown-Henneaux boundary conditions with $\bar{\gamma}_{AB}dx^Adx^B = -d\tau^2 + d\phi^2$ will also satisfy to the generalized boundary conditions eq:condasymp1-eq:condasymp4.

Theorems & Definitions (1)

  • Theorem 4.1