The complete action for $\mathcal{N}=2$ de Sitter pure supergravity

TL;DR

This work resolves the existence and uniqueness of the simplest ${\cal N}=2$ pure supergravity in $dS_4$ by constructing a real, gauge-invariant Lagrangian to all orders, using antifield methods to classify cubic interactions and then completing to a full MacDowell–Mansouri–like action. It shows the graviphoton must carry a wrong-sign kinetic term and, contrary to prior beliefs, the gravitino sector also exhibits non-unitarity in Lorentzian signature, prompting a reconsideration of unitarity in the Euclidean setting. The resulting unique theory provides a robust Lorentzian baseline for exploring Euclidean de Sitter quantum gravity, including sphere-path-integral formulations and 1-loop analyses expressed via $so(4,1)$ characters, and motivates future work on higher-point fermionic correlators. The work thereby extends Pilch–van Nieuwenhuizen–Sohnius by explicitly including quartic fermion terms and clarifying the full structure of de Sitter ${\cal N}=2$ supergravity, with potential implications for holography and cosmological quantum gravity.

Abstract

Supergravity theories in de Sitter spacetime are known to be very constrained, and rather unnatural within String/M Theory. We revisit the seminal paper by Pilch, van Nieuwenhuizen and Sohnius, where the possible existence of a real Lagrangian for ${\cal N}=2$ pure supergravity in four-dimensional de Sitter spacetime was pointed out. We clarify several issues related to the non-unitarity of the theory and explicitly construct the unique, complete theory searched for long ago by the aforementioned authors. We argue that the lack of unitarity of the Lorentzian theory may be revisited in the Euclidean approach to de Sitter quantum gravity, where alternative definitions of unitarity can be introduced.