Hint to Supersymmetry from GR Vacuum

The $S$-matrix formulation of gravity suggests that the $\theta$-vacuum structure must not be sustained by the theory. We point out that, when applied to the vacuum of general relativity, this criterion hints to supersymmetry. The topological susceptibility of gravitational vacuum induced by Eguchi-Hanson instantons can be eliminated neither by spin-$1/2$ fermions nor by an axion coupled via them since such fermions do not provide instanton zero modes. Instead, the job is done by a spin-$3/2$ fermion, hence realizing a local supersymmetry. This scenario also necessitates the spontaneous breaking of supersymmetry and predicts the existence of axion of $R$-symmetry which gets mass exclusively from the gravitational instantons. The $R$-axion can be a viable dark matter candidate. Matching between the index and the anomaly imposes a constraint that spin-$1/2$ fermions should not contribute to the chiral gravitational anomaly.