Phase Space Renormalization and Finite BMS Charges in Six Dimensions

TL;DR

This work establishes a complete treatment of the solution space for six-dimensional Einstein gravity near future null infinity, revealing that an infinite-dimensional Weyl-BMS (WBMS/GBMS) symmetry can act nontrivially on analytic solution spaces. A novel phase-space renormalization procedure using local covariant counterterms renders the large-r divergences finite while preserving the essential radiative data and the GBMS algebra in its integrable sector. The GBMS diffeomorphisms are shown to be non-canonical due to non-integrable contributions, yet a well-defined integrable charge algebra reproduces the expected symmetry structure and matches the leading and subleading soft-graviton theorems via Ward identities. Antipodal matching between I^+ and I^- sectors ties these asymptotic symmetries to observable soft theorems, strengthening the infrared triangle in higher even dimensions and providing a robust framework for flat-space holography in D=6. The results generalize to higher even dimensions and offer a principled way to handle IR structure in gravity beyond four dimensions, with potential implications for holographic correspondences and the universality of soft theorems.

Abstract

We perform a complete and systematic analysis of the solution space of six-dimensional Einstein gravity. We show that a particular subclass of solutions -- those that are analytic near $\mathcal{I}^+$ -- admit a non-trivial action of the generalised Bondi-Metzner-van der Burg-Sachs (GBMS) group which contains \emph{infinite-dimensional} supertranslations and superrotations. The latter consists of all smooth volume-preserving Diff$\times$Weyl transformations of the celestial $S^4$. Using the covariant phase space formalism and a new technique which we develop in this paper (phase space renormalization), we are able to renormalize the symplectic potential using counterterms which are \emph{local} and \emph{covariant}. The Hamiltonian charges corresponding to GBMS diffeomorphisms are non-integrable. We show that the integrable part of these charges faithfully represent the GBMS algebra and in doing so, settle a long-standing open question regarding the existence of infinite-dimensional asymptotic symmetries in higher even dimensional non-linear gravity. Finally, we show that the semi-classical Ward identities for supertranslations and superrotations are precisely the leading and subleading soft-graviton theorems respectively.

Figures (1)

• Figure 1: Penrose Diagram for Globally Asymptotically Flat Spacetimes (1) Timelike future and past infinity $i^+$ and $i^-$ are spacelike boundaries and timelike geodesics (red) start on $i^-$ and end on $i^+$. (2) Spatial infinity $i^0$ is a timelike boundary on which spacelike geodesics (orange) end. (3) Lightlike past and future infinity ${\cal I}^-$ and ${\cal I}^+$ are null boundaries and null geodesics (blue) start on ${\cal I}^-$ and end on ${\cal I}^+$. ${\cal I}^+$ (${\cal I}^-$) itself has future and past boundaries which denoted by ${\cal I}^+_+$ (${\cal I}^-_+$) and ${\cal I}^+_-$ (${\cal I}^-_-$) respectively.