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.

Using a component formulation, we construct the supersymmetric action for a superparticle in a threedimensional Newton-Cartan supergravity background and clarify its symmetries. Our construction proceeds by first constructing the superparticle in a flat background. Next, by boosting up the background symmetries, we replace in a first step the flat background by a Galilean supergravity background. In a second step the Galilean supergravity background is replaced by a Newton-Cartan supergravity background. We extend our results by adding a supersymmetric cosmological constant and compare the nonrelativistic superparticle with the relativistic κ-symmetric three-dimensional superparticle. 1 The torsion occurs due to the fact that the boundary geometry is invariant under an anisotropic scaling symmetry. PHYSICAL REVIEW D 90, 065006 (2014) 1550-7998=2014=90(6)=065006 (16) 065006-1

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We construct a supersymmetric formulation of three-dimensional linearized New Massive Gravity without introducing higher derivatives. Instead, we introduce supersymmetrically a set of bosonic and fermionic auxiliary fields which, upon elimination by their equations of motion, introduce fourth-order derivative terms for the metric and third-order derivative terms for the gravitino. Our construction requires an off-shell formulation of the three-dimensional supersymmetric massive Fierz-Pauli theory. We discuss the non-linear extension of our results.