Generalised Chern–Simons actions for 3d gravity and κ-Poincaré symmetry

Abstract

We consider Chern-Simons theories for the Poincaré, de Sitter and anti-de Sitter groups in three dimensions which generalise the Chern-Simons formulation of 3d gravity. We determine conditions under which κ-Poincaré 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 κ-Poincaré 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 κ-de Sitter symmetry, and carry out the first steps in that construction.

Noncommutative Relativistic Spacetimes and Worldlines from 2 + 1 Quantum (Anti-)de Sitter Groups

“…It is worth stressing that the connection between the Poisson-Lie group approach presented here and the role that classical -matrices and Drinfel' d-doubles play in the context of 2 + 1 quantum gravity [76][77][78][79][80][81][82][83][84] has been studied in detail in the works [85][86][87][88][89]. Also, the deformed Casimir operators (57) (or (64)) can be used to provide modified dispersion relations, which should be related to those appearing in several phenomenological approaches to quantum gravity (see [90][91][92][93]).…”

On κ-deformation and triangular quasibialgebra structure

“…Moreover it is usually supposed [43][44][45] -though cf. [37] -that the role of quantum field theory with κ-deformed Poincaré symmetry, if any, will be that of an effective description in a regime intermediate between standard QFT and full quantum gravity.…”

“…1) to match the power of the dimensionful deformation parameter κ in the expansion (3.3). The 1+2 dimensional case is very special because there does exist such an invariant, obtained by splitting the Casimir ǫ µνρ M µν P ρ = −2N k P k +ǫ ij M ij E symmetrically over the tensor product: ǫ µνρ M µν ∨ P ρ [37,38].…”

Basic quantizations of D = 4 Euclidean, Lorentz, Kleinian and quaternionic $$ {\mathfrak{o}}^{\star }(4) $$ symmetries

“…Following Fock-Rosly construction [77,78], in such framework we describe gravitational degrees of freedom as parameterizing the Poisson-Lie group manifold. If we search for quantum deformations of Fock-Rosly construction, it appears that only classical r-matrices obtained from Drinfeld double (DD) structures [10,11] are allowed [79][80][81]. The DD structures and corresponding classical r-matrices for o(3, 1) and o(2, 2) algebras were recently constructed and classified [74][75][76].…”