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

TL;DR

This paper demonstrates that self-dual gravity in AdS$_4$ can be formulated using a scalar field with a cubic interaction, revealing a new symmetry algebra and establishing a double copy relation with self-dual Yang-Mills theory, with implications for celestial holography.

Contribution

It introduces a novel scalar field formulation of self-dual gravity in AdS$_4$, uncovers a new kinematic algebra, and connects it to the double copy framework and celestial holography.

Findings

Self-dual gravity in AdS$_4$ described by a scalar field with cubic interaction.

Discovery of an AdS$_4$ kinematic algebra as a deformation of $w_{1+ abla}$ algebra.

Establishment of a double copy relation between self-dual gravity and Yang-Mills 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.