Remarks on 2+1 Self-dual Chern-Simons Gravity

TL;DR

This paper explores 2+1 dimensional Chern-Simons gravity, deriving new action combinations from self-duality conditions, linking to 4D topological theories and hyperbolic geometry, and introducing a dual vector field action.

Contribution

It rederives standard and exotic Einstein actions from self-duality, connecting 3D gravity to 4D topological theories and hyperbolic geometry, and introduces a dual vector field action.

Findings

Derived new action combinations from self-duality.

Linked 3D gravity to 4D topological theories.

Introduced a non-abelian dual vector field action.

Abstract

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.