Gravity in the 3+1-Split Formalism II: Self-Duality and the Emergence of the Gravitational Chern-Simons in the Boundary

TL;DR

This paper explores how self-dual solutions in 3+1 gravity with a cosmological constant relate to boundary Chern-Simons theory, revealing a holographic correspondence between bulk configurations and boundary topological field theories.

Contribution

It demonstrates that bulk self-dual gravitational configurations are holographically encoded by the three-dimensional gravitational Chern-Simons theory, connecting bulk solutions to boundary topological actions.

Findings

Lorentzian self-dual configurations are conformally flat

Boundary data are determined by classical solutions of 3D gravitational Chern-Simons

Bulk self-dual configurations are described by boundary gravitational Chern-Simons theory

Abstract

We study self-duality in the context of the 3+1-split formalism of gravity with non-zero cosmological constant. Lorentzian self-dual configurations are conformally flat spacetimes and have boundary data determined by classical solutions of the three-dimensional gravitational Chern-Simons. For Euclidean self-dual configurations, the relationship between their boundary initial positions and initial velocity is also determined by the three-dimensional gravitational Chern-Simons. Our results imply that bulk self-dual configurations are holographically described by the gravitational Chern-Simons theory which can either viewed as a boundary generating functional or as a boundary effective action.