Higher spin dynamics in gravity and $w_{1 + \infty}$ celestial symmetries

TL;DR

This work uncovers an infinite tower of higher-spin charges in four-dimensional gravity obtained from the large-$r$ expansion of vacuum Einstein equations. It develops a canonical action of these charges on the gravity phase space, shows that quadratic truncations reproduce an infinite set of soft theorems, and demonstrates a canonical realization of the loop algebra $Lw_{1+\infty}$ via their brackets. By linking to celestial holography, it matches the charge action with celestial OPEs and with $w$-currents, providing a unified spacetime interpretation of higher-spin celestial symmetries. The results bridge bulk asymptotic gravity and celestial CFT structures, suggesting deep constraints on gravitational dynamics from infinite-dimensional symmetry algebras.

Abstract

In this paper we extract from a large-$r$ expansion of the vacuum Einstein's equations a dynamical system governing the time evolution of an infinity of higher-spin charges. Upon integration, we evaluate the canonical action of these charges on the gravity phase space. The truncation of this action to quadratic order and the associated charge conservation laws yield an infinite tower of soft theorems. We show that the canonical action of the higher spin charges on gravitons in a conformal primary basis, as well as conformally soft gravitons reproduces the higher spin celestial symmetries derived from the operator product expansion. Finally, we give direct evidence that these charges form a canonical representation of a $w_{1+\infty}$ loop algebra on the gravitational phase space.

Figures (3)

• Figure 1: Spin-$s$ diamond associated with negative-helicity soft gravitons and $s =-1, 0, 1, 2$. Operators connected by long edges have weights $h=(\Delta+J)/2, \bar{h}=(\Delta-J)/2$ related by $(h, \bar{h}) \leftrightarrow (1 - h, \bar{h})$. Operators connected by short edges have $(h, \bar{h}) \leftrightarrow (h, 1 - \bar{h})$. Diagonally opposite corners are related by $(h, \bar{h}) \leftrightarrow (1 - h, 1 - \bar{h})$.
• Figure 2: Diamond associated with a negative helicity soft graviton of dimension $\Delta = 1 - s$ for $s \geq 3$. The weigth labels are $(\Delta,J)$. Operators connected by long edges have weights related by $(h, \bar{h}) \leftrightarrow (1 - h, \bar{h})$. Operators connected by short edges have $(h, \bar{h}) \leftrightarrow (h, 1 - \bar{h})$. Diagonally opposite corners are related by $(h, \bar{h}) \leftrightarrow (1 - h, 1 - \bar{h})$.
• Figure 3: There are two maps from a soft graviton $N_s$ with $(\Delta, J) = (1 - s, -2)$ to an operator of $(\Delta, J) = (3, s):$ the light transform defined in \ref{['LT']} and the action of $s + 2$ derivatives $\partial_{z_1}^{2 + s}$. The resulting operators have the same OPE with massless celestial operators upon trading $2\pi\delta^{(2)}(z)$ for $1/(z{\bar{z}})$.