Abstract
We construct a generalisation of the three-dimensional Poincaré algebra that also includes a colour symmetry factor. This algebra can be used to define coloured Poincaré gravity in three space-time dimensions as well as to study generalisations of massive and massless free particle models. We present various such generalised particle models that differ in which orbits of the coloured Poincaré symmetry are described. Our approach can be seen as a stepping stone towards the description of particles interacting with a non-abelian background field or as a starting point for a worldline formulation of an associated quantum field theory.
Highlights
Space-time symmetries that are larger than those realised in conventional gravitational systems, including bosonic generators in non-trivial representations of isometry algebra are usually ruled out in field theories of finitely many interacting particles
This algebra can be used to define coloured Poincaré gravity in three space-time dimensions as well as to study generalisations of massive and massless free particle models. We present various such generalised particle models that differ in which orbits of the coloured Poincaré symmetry are described
The coloured Poincaré algebra in D = 3, cPoin3(N ), is defined to be the Lie algebra associated with the associative algebra given by the tensor product cPoin3(N ) = A ⊗ u(N )
Summary
Space-time symmetries that are larger than those realised in conventional gravitational systems, including bosonic generators in non-trivial representations of isometry algebra are usually ruled out in field theories of finitely many interacting particles. It is a generic problem to couple matter to the gravitational systems with extended symmetries such as the coloured (higher-spin) gravity given by a Chern-Simons action in three dimensions. In the case of constant electro-magnetic field the symmetry algebra is given by the Bacry-Combe-Richards algebra [28] that contains four spacetime translations that do not commute, two Lorentz boost transformations and two central charges Another possibility is to enlarge the Poincaré algebra with tensorial non-central charges, leading to the so-called Maxwell algebra [29].
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