Gravity in the 3+1-split formalism: II. Self-duality and the emergence of the gravitational Chern–Simons in the boundary

We study self-duality in the context of the 3+1-split formalism of gravity with a 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 be viewed as a boundary generating functional or as a boundary effective action.