Geroch group for Einstein spaces and holographic integrability

TL;DR

This paper explores the extension of Geroch's reduction method to Einstein spaces, revealing a holographic perspective on integrability and solution-generating symmetries in asymptotically AdS spacetimes.

Contribution

It demonstrates how holography provides an alternative approach to integrability, introducing a holographic U-duality group that extends Geroch's solution-generating symmetries.

Findings

Geroch's reduction extends to Einstein spacetimes with a solution-generating subgroup.

Holography links boundary data to exact Einstein geometries via integrability conditions.

Holographic U-duality enables solution mixing, such as mass and nut charge.

Abstract

We review how Geroch's reduction method is extended from Ricci-flat to Einstein spacetimes. The Ehlers-Geroch SL(2,R) group is still present in the three-dimensional sigma-model that captures the dynamics, but only a subgroup of it is solution-generating. Holography provides an alternative three-dimensional perspective to integrability properties of Einstein's equations in asymptotically anti-de Sitter spacetimes. These properties emerge as conditions on the boundary data (metric and energy-momentum tensor) ensuring that the hydrodynamic derivative expansion be resummed into an exact four-dimensional Einstein geometry. The conditions at hand are invariant under a set of transformations dubbed holographic U-duality group. The latter fills the gap left by the Ehlers-Geroch group in Einstein spaces, and allows for solution-generating maps mixing e.g. the mass and the nut charge.