Remarks on 2+1 self-dual Chern-Simons gravity

We study 2+1 Chern-Simons gravity at the classical action level. In particular we rederive the linear combinations of the “standard” and “exotic” Einstein actions, from the (anti-)self-duality of the “internal” Lorentzian indices. The relation to a genuine four-dimensional (anti-)self-dual topological theory greatly facilitates the analysis and its relation to hyperbolic three-dimensional geometry. Finally a non-Abelian vector field “dual” action is also obtained.