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We study self-duality in the context of the 3+1-split formalism of gravitywith non-zero cosmological constant. Lorentzian self-dual configurations areconformally flat spacetimes and have boundary data determined by classicalsolutions of the three-dimensional gravitational Chern-Simons. For Euclideanself-dual configurations, the relationship between their boundary initialpositions and initial velocity is also determined by the three-dimensionalgravitational Chern-Simons. Our results imply that bulk self-dualconfigurations are holographically described by the gravitational Chern-Simonstheory which can either viewed as a boundary generating functional or as aboundary effective action.

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