Quantum double and $\kappa$-Poincar\'e symmetries in (2+1)-gravity and Chern-Simons theory

TL;DR

This paper reviews the role of Drinfeld doubles and kappa-Poincare symmetries in quantized (2+1)-gravity and Chern-Simons theory, clarifying their compatibility and physical implications.

Contribution

It explicitly determines the conditions for Hopf algebra symmetries to be compatible with (2+1)-gravity and clarifies the relationship between kappa-Poincare symmetries and Chern-Simons theory.

Findings

Usual kappa-Poincare symmetries with timelike deformation are not associated with (2+1)-gravity.

Kappa-Poincare symmetries are linked to Chern-Simons theory only in the de Sitter case.

The relevant Chern-Simons theory in this context is physically inequivalent to (2+1)-gravity.

Abstract

We review the role of Drinfeld doubles and kappa-Poincare symmetries in quantised (2+1)-gravity and Chern-Simons theory. We discuss the conditions under which a given Hopf algebra symmetry is compatible with a Chern-Simons theory and determine this compatibility explicitly for the Drinfeld doubles and kappa-Poincare symmetries associated with the isometry groups of (2+1)-gravity. In particular, we explain that the usual kappa-Poincare symmetries with a timelike deformation are not directly associated with (2+1)-gravity. These kappa-Poincare symmetries are linked to Chern-Simons theory only in the de Sitter case, and the relevant Chern-Simons theory is physically inequivalent to (2+1)-gravity.