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 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. 
Diego S. Mansi, Anastasios C. Petkou and Giovanni Tagliabue