Symmetries at Null Boundaries: Two and Three Dimensional Gravity Cases

TL;DR

This work develops a general framework for symmetry and charge analysis near generic null boundaries in 2d dilaton gravity and 3d gravity, without imposing boundary conditions. It shows that maximal boundary degrees of freedom exist and that integrable charges can be achieved in infinitely many bases, with a fundamental algebra of Heisenberg ⊕ Diff$(d-2)$ capturing the null boundary dynamics in the absence of Bondi news. In 2d, the integrable basis yields a Heisenberg structure, while in 3d it yields a Heisenberg ⊕ Witt (Diff) structure; the authors also construct a fundamental 3d basis and identify two additional integrable algebras (twisted-Sugawara and BMS3 ⊕ Witt) arising from specific basis changes. The paper further develops a general change-of-basis formalism for integrable charges, discusses the relation to boundary conditions and variational principles, and outlines higher-dimensional extensions and connections to the membrane paradigm and soft hair ideas.

Abstract

We carry out in full generality and without fixing specific boundary conditions, the symmetry and charge analysis near a generic null surface for two and three dimensional (2d and 3d) gravity theories. In 2d and 3d there are respectively two and three charges which are generic functions over the codimension one null surface. The integrability of charges and their algebra depend on the state-dependence of symmetry generators which is a priori not specified. We establish the existence of infinitely many choices that render the surface charges integrable. We show that there is a choice, the "fundamental basis", where the null boundary symmetry algebra is the Heisenberg+Diff(d-2) algebra. We expect this result to be true for d>3 when there is no Bondi news through the null surface.