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The $Λ$-BMS$_4$ Charge Algebra

Geoffrey Compère, Adrien Fiorucci, Romain Ruzziconi

TL;DR

The paper derives the Λ-BMS$_4$ charge algebra for asymptotically locally (A)dS$_4$ spacetimes without imposing boundary conditions, showing Weyl charges vanish and boundary diffeomorphism charges form a centerless algebra under an adjusted bracket. By implementing a Dirichlet boundary gauge and a careful holographic renormalization that includes corner terms, it constructs a Λ-BMS$_4$ algebra and demonstrates its smooth flat limit to the generalized BMS$_4$ algebra, including a nontrivial corner-induced 2-cocycle that vanishes at null-infinity corners for nonradiative configurations. A precise dictionary between Fefferman–Graham and Bondi variables is developed to carry the flat limit into Bondi gauge, where the symplectic structure and charges remain well-defined. The results establish a covariant-phase-space framework for infrared symmetries in AdS/dS spacetimes and their flat limit, clarifying how corner terms and foliation data encode the BMS$_4$ structure without central extensions. This has implications for infrared aspects of gravity, holography, and potential quantum-gravity applications at null infinity.

Abstract

The surface charge algebra of generic asymptotically locally (A)dS$_4$ spacetimes without matter is derived without assuming any boundary conditions. Surface charges associated with Weyl rescalings are vanishing while the boundary diffeomorphism charge algebra is non-trivially represented without central extension. The $Λ$-BMS$_4$ charge algebra is obtained after specifying a boundary foliation and a boundary measure. The existence of the flat limit requires the addition of corner terms in the action and symplectic structure that are defined from the boundary foliation and measure. The flat limit then reproduces the BMS$_4$ charge algebra of supertranslations and super-Lorentz transformations acting on asymptotically locally flat spacetimes. The BMS$_4$ surface charges represent the BMS$_4$ algebra without central extension at the corners of null infinity under the standard Dirac bracket, which implies that the BMS$_4$ flux algebra admits no non-trivial central extension.

The $Λ$-BMS$_4$ Charge Algebra

TL;DR

The paper derives the Λ-BMS charge algebra for asymptotically locally (A)dS spacetimes without imposing boundary conditions, showing Weyl charges vanish and boundary diffeomorphism charges form a centerless algebra under an adjusted bracket. By implementing a Dirichlet boundary gauge and a careful holographic renormalization that includes corner terms, it constructs a Λ-BMS algebra and demonstrates its smooth flat limit to the generalized BMS algebra, including a nontrivial corner-induced 2-cocycle that vanishes at null-infinity corners for nonradiative configurations. A precise dictionary between Fefferman–Graham and Bondi variables is developed to carry the flat limit into Bondi gauge, where the symplectic structure and charges remain well-defined. The results establish a covariant-phase-space framework for infrared symmetries in AdS/dS spacetimes and their flat limit, clarifying how corner terms and foliation data encode the BMS structure without central extensions. This has implications for infrared aspects of gravity, holography, and potential quantum-gravity applications at null infinity.

Abstract

The surface charge algebra of generic asymptotically locally (A)dS spacetimes without matter is derived without assuming any boundary conditions. Surface charges associated with Weyl rescalings are vanishing while the boundary diffeomorphism charge algebra is non-trivially represented without central extension. The -BMS charge algebra is obtained after specifying a boundary foliation and a boundary measure. The existence of the flat limit requires the addition of corner terms in the action and symplectic structure that are defined from the boundary foliation and measure. The flat limit then reproduces the BMS charge algebra of supertranslations and super-Lorentz transformations acting on asymptotically locally flat spacetimes. The BMS surface charges represent the BMS algebra without central extension at the corners of null infinity under the standard Dirac bracket, which implies that the BMS flux algebra admits no non-trivial central extension.

Paper Structure

This paper contains 30 sections, 151 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Charge algebra of asymptotically locally (A)dS$_{\mathbf 4}$ spacetimes
  3. Holographically renormalized variational principle
  4. Holographically renormalized symplectic structure
  5. Canonical surface charges
  6. Charge algebra under the adjusted Dirac bracket
  7. $\Lambda$-BMS$_{4}$ charge algebra
  8. Boundary gauge fixing and $\Lambda$-BMS$_{4}$ asymptotic symmetry algebra
  9. Symplectic structure, Hamiltonians and $\Lambda$-BMS$_4$ charge algebra
  10. Flat limit of the $\Lambda$-BMS$_4$ phase space
  11. Dictionary between Starobinsky/Fefferman-Graham and Bondi
  12. Flat limit of the action and corner terms
  13. BMS$_4$ charge algebra
  14. Discussion
  15. Notations and conventions
  16. ...and 15 more sections

Figures (1)

  • Figure 1: Geometry of Al(A)dS$_4$ boundaries with background structure.