Massive gravity from a first-quantized perspective

In this work, we investigate the BRST quantization of the massive $\mathcal{N}=4$ supersymmetric spinning particle, with a twofold purpose: exploring different approaches to give mass to spinning particle models and formulating a first-quantized theory for linearized massive gravity on both flat and curved spacetime. Our results suggest that achieving the nilpotency of the BRST charge requires an Einstein spacetime with vanishing cosmological constant as the only viable consistent background. In the course of the analysis, we take the $\mathcal{N}=2$ supersymmetric worldline as an exemplificative model, correctly producing the Proca theory on curved spacetime. Our analysis shows that the associated BRST system uniquely selects the minimal coupling to the background curvature.