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. However, we show that only certain combinations of linear excitations, involving a vector field, can be extended beyond the linear level in turn, an Einstein metric in space-time.
