We show that self-dual gravity in Euclidean four-dimensional anti–de Sitter space (${\mathrm{AdS}}_4$) can be described by a scalar field with a cubic interaction written in terms of a deformed Poisson bracket, providing a remarkably simple generalization of the Plebanski action for self-dual gravity in flat space. This implies a novel symmetry algebra in self-dual gravity, notably an ${\mathrm{AdS}}_4$ version of the so-called kinematic algebra. We also obtain the three-point interaction vertex of self-dual gravity in ${\mathrm{AdS}}_4$ from that of self-dual Yang-Mills by replacing the structure constants of the Lie group with the structure constants of the new kinematic algebra, implying that self-dual gravity in ${\mathrm{AdS}}_4$ can be derived from self-dual Yang-Mills in this background via a double copy. This provides a concrete starting point for defining the double copy for Einstein gravity in ${\mathrm{AdS}}_4$ by expanding around the self-dual sector. Moreover, we show that the new kinematic Lie algebra can be lifted to a deformed version of the $w_{1+\infty }$ algebra, which plays a prominent role in celestial holography.

• Received 15 May 2023
• Accepted 26 July 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.081501

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society