The Symmetries of the Carroll Superparticle

TL;DR

This paper explores the geometry and symmetries of Carroll space and superparticles, revealing infinite-dimensional symmetries in flat cases and differences between flat and curved spaces, with implications for supersymmetry and BPS states.

Contribution

It uncovers the symmetry structure of Carroll particles in flat and curved spaces, including supersymmetric extensions and their relation to known algebras, and clarifies differences with Galilei particles.

Findings

Carroll particles have infinite-dimensional symmetries in flat space.

The duality between Bargmann and Carroll algebra does not hold in curved space.

Flat N=2 superparticles are equivalent to N=1 superparticles and are non-BPS.

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.