  The $\cal{N}=4$ supersymmetric spinning particle admits several consistent
quantizations, related to the gauging of different subgroups of the $SO(4)$
$R$-symmetry on the worldline. We construct the background independent BRST
quantization for all of these choices which are shown to reproduce either the
massless NS-NS spectrum of the string, or Einstein theory with or without the
antisymmetric tensor field and/or dilaton corresponding to different
restrictions. Quantum consistency of the worldline implies equations of motion
for the background which, in addition to the admissible string backgrounds,
admit Einstein manifolds with or whithout a cosmological constant. The vertex
operators for the Kalb-Ramond, graviton and dilaton fields are obtained from
the linear variations of the BRST charge. They produce the physical states by
action on the diffeomorphism ghost states.
