Isolated Surfaces and Symmetries of Gravity

TL;DR

The paper identifies a maximal embedding algebra ${\cal A}_k$ of bulk diffeomorphisms that preserve an isolated codimension-k corner S, providing a metric-independent, off-shell description of corner geometry.It develops an intrinsic geometric framework based on an Ehresmann connection and a Randers-Papapetrou-like metric to encode the embedding data on the corner, separating kinematic corner structure from bulk dynamics.In the Einstein–Hilbert setting with codimension-2 corners, Noether charges realize a representation of ${\cal A}_2$, though dynamically only the subalgebra ${\tilde{\cal A}}_2 = Diff(S) \ltimes sl(2,R) \ltimes R^2$ is realized for finite-distance corners, with normal translations spontaneously broken by a fixed corner.The work connects to near-horizon BMS-like symmetries, showing that those are instances of the maximal embedding algebra, highlighting the role of corner degrees of freedom and edge modes for potential insights into quantum gravity.

Abstract

Conserved charges in theories with gauge symmetries are supported on codimension-2 surfaces in the bulk spacetime. It has recently been suggested that various classical formulations of gravity dynamics display different symmetries, and paying attention to the maximal such symmetry could have important consequences to further elucidate the quantization of gravity. After establishing an algebraic off-shell derivation of the maximal closed subalgebra of the full bulk diffeomorphisms in the presence of an isolated corner, we show how to geometrically describe the latter and its embedding in spacetime, without constraining the geometry away from the corner, such as by assuming a foliation. The analysis encompasses arbitrary embedded surfaces, of generic codimensions $k$. The resulting corner algebra ${\cal A}_k$, calling $S$ the embedded surface and $M$ the bulk, is that of the group $(Diff(S)\ltimes GL(k,\mathbb{R}))\ltimes \mathbb{R}^k$. This result is independent of any dynamics or pseudo-Riemannian structure in the bulk. We then evaluate the Noether charges of ${\cal A}_2$ for Einstein-Hilbert dynamics and show that the Noether charge algebra gives a representation of the algebra ${\cal A}_2$, for finite proper distance corners in the bulk, while all other charges associated with $Diff(M)$ vanish.