Deformed Carroll particle from 2+1 gravity

Understand

We consider a point particle coupled to 2+1 gravity, with de Sitter gauge group SO(3,1).

• We observe that there are two contraction limits of the gauge group: one resulting in the Poincare group, and the second with the gauge group having the form AN(2) \ltimes \an(2)^*.
• The former case was thoroughly discussed in the literature, while the latter leads to the deformed particle action with de Sitter momentum space, like in the case of kappa-Poincare particle.
• However, the construction forces the mass shell constraint to have the form p_0^2 = m^2, so that the effective particle action describes the deformed Carroll particle.