Self-dual gravity and color/kinematics duality in AdS${}_4$

Abstract

We show that self-dual gravity in Euclidean four-dimensional Anti-de Sitter space (AdS${}_4$) can be described by a minimally coupled scalar field with a cubic interaction written in terms of a deformed Poisson bracket, providing a remarkably simple generalisation of the Plebanski action for self-dual gravity in flat space. This implies a novel symmetry algebra in self-dual gravity, notably an AdS${}_4$ version of the so-called kinematic algebra. We also obtain the 3-point interaction vertex of self-dual gravity in 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 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 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.

Updated to match published version. Minor clarifications added