Generalised Chern-Simons actions for 3d gravity and kappa-Poincare symmetry

TL;DR

This paper explores generalized Chern-Simons theories for 3d gravity and their relation to kappa-Poincare symmetry, concluding that such symmetry is not compatible with classical or quantum 3d gravity, but can be realized in multi-particle models.

Contribution

It identifies conditions for associating kappa-Poincare symmetry with 3d Chern-Simons theories and constructs a multi-particle model invariant under kappa-de Sitter symmetry.

Findings

Kappa-Poincare symmetry cannot be associated with 3d gravity.

The de Sitter case allows for a symmetry association, but the resulting theory differs from gravity.

A multi-particle model invariant under kappa-de Sitter symmetry is constructed.

Abstract

We consider Chern-Simons theories for the Poincare, de Sitter and anti-de Sitter groups in three dimensions which generalise the Chern-Simons formulation of 3d gravity. We determine conditions under which kappa-Poincare symmetry and its de Sitter and anti-de Sitter analogues can be associated to these theories as quantised symmetries. Assuming the usual form of those symmetries, with a timelike vector as deformation parameter, we find that such an association is possible only in the de Sitter case, and that the associated Chern-Simons action is not the gravitational one. Although the resulting theory and 3d gravity have the same equations of motion for the gauge field, they are not equivalent, even classically, since they differ in their symplectic structure and the coupling to matter. We deduce that kappa-Poincare symmetry is not associated to either classical or quantum gravity in three dimensions. Starting from the (non-gravitational) Chern-Simons action we explain how to construct a multi-particle model which is invariant under the classical analogue of kappa-de Sitter symmetry, and carry out the first steps in that construction.