  We revisit the theory of background fields constructed on the BRST-algebra of
a spinning particle with $\mathcal{N}=4$ worldline supersymmetry, whose
spectrum contains the graviton but no other fields. On a generic background,
the closure of the BRST algebra implies the vacuum Einstein equations with a
cosmological constant that is undetermined. On the other hand, in the "vacuum"
background with no metric, the cohomology is given by a collection of free
scalar- and vector fields. Only certain combinations of linear excitations,
necessarily involving a vector field, can be extended beyond the linear level
with the vector field inducing an Einstein metric.
