The Symmetries of the Carroll Superparticle

TL;DR

This work characterizes Carroll space symmetries and particle dynamics in both flat and AdS backgrounds, including bosonic and ${\cal N}=1,2$ supersymmetric extensions. By contracting the AdS algebra, it constructs the AdS Carroll (AC) algebra, builds particle actions via non-linear realizations, and derives Killing equations that reveal an infinite-dimensional symmetry structure, with crucial differences between curved and flat cases. In the supersymmetric sector, the ${\cal N}=1$ AC superparticle exhibits an infinite-dimensional superalgebra in the flat limit, while the ${\cal N}=2$ Carroll superparticle is not BPS due to κ-symmetry, with distinct curved three-dimensional realizations. The results illuminate the symmetry geometry of ultra-relativistic systems and point toward possible couplings to (super) AdS gauge fields and Carroll gravity theories.

Abstract

Motivated by recent applications of Carroll symmetries we investigate the geometry of flat and curved (AdS) Carroll space and the symmetries of a particle moving in such a space both in the bosonic as well as in the supersymmetric case. In the bosonic case we find that the Carroll particle possesses an infinite-dimensional symmetry which only in the flat case includes dilatations. The duality between the Bargmann and Carroll algebra, relevant for the flat case, does not extend to the curved case. In the supersymmetric case we study the dynamics of the N=1 AdS Carroll superparticle. Only in the flat limit we find that the action is invariant under an infinite-dimensional symmetry that includes a supersymmetric extension of the Lifshitz Carroll algebra with dynamical exponent z=0. We also discuss in the flat case the extension to N=2 supersymmetry and show that the flat N=2 superparticle is equivalent to the (non-moving) N=1 superparticle and that therefore it is not BPS unlike its Galilei counterpart. This is due to the fact that in this case kappa-symmetry eliminates the linearized supersymmetry. In an appendix we discuss the N=2 curved case in three dimensions only and show that there are two N=2 theories that are physically different.