Gravity in the 3+1-Split Formalism II Self-Duality and the Emergence of the Gravitational C.pdf

a r X i v : 0 8 0 8 .1 2 1 3 v 1 [ h e p - t h ] 8 A u g 2 0 0 8 Gravity in the 3+1-Split Formalism II: Self-Duality and the Emergence of the Gravitational Chern-Simons in the Boundary D. S. Mansi1 and A. C. Petkou2 Department of Physics, University of Crete, GR-71003 Heraklion. G. Tagliabue3 Dipartimento di Fisica dell’Universita? di Milano, Via Celoria 16, I-20133 Milano and INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano. 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 gravita- tional 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 holo- graphically described by the gravitational Chern-Simons theory which can either viewed as a boundary generating functional or as a boundary effective action. 1dmansi@physics.uoc.gr 2petkou@physics.uoc.gr 3giovanni.tagliabue@mi.infn.it Contents 1 Introduction 1 2 Resume? of the 3+1-split formalism in Lorentzian signature 3 2.1 Self-duality with non-vanishing cosmological constant . . . . . . . . . . . . . . . . . 5 3 4D Weyl Vacua vs 3D gravitational Chern-Simons 6 3.1 Weyl structures in the boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Euclidean signature and self-dual solutions 10 4.1 Simple self-dual solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Self-duality with non-vanishing cosmological constant . . . . . . . . . . . . . . . . . 12 5 Holography of self-dual configurations and the gravitational Chern-Simons 14 6 Conclusions 16 A Weyl’s Conformal Tensor 17 B Details of computations for non-vanishing e0i 19 C Euclidean Yan