Isolated Surfaces and Symmetries of Gravity

TL;DR

This paper explores the algebraic structure of conserved charges on isolated surfaces in gravity, revealing a maximal symmetry algebra independent of bulk dynamics and demonstrating its representation through Noether charges in Einstein-Hilbert gravity.

Contribution

It provides an off-shell algebraic derivation and geometric description of the maximal corner symmetry algebra for arbitrary embedded surfaces in gravity.

Findings

Derived the algebra ${ m Diff}(S) times GL(k,b R) times b R^k$ for isolated surfaces.

Showed the algebra is independent of bulk dynamics or geometry.

Demonstrated the algebra's representation via Noether charges in Einstein-Hilbert 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.