Supergravity as a special case of local nilpotent fermionic symmetry

Abstract

We start with a four-dimensional (4D) system only with local nilpotent fermionic symmetry, and show that massive $N=1$ supergravity is realized as a special case. Our field content in 4D is $(e_{\ensuremath{\mu}}{}^{m},{\ensuremath{\psi}}_{\ensuremath{\mu}},\ensuremath{\omega}_{\ensuremath{\mu}}{}^{rs},\ensuremath{\chi})$, where ${\ensuremath{\psi}}_{\ensuremath{\mu}}$ is a vector-spinor in the Majorana representation in four dimenisons, while $\ensuremath{\chi}$ is a compensator Majorana spinor, and $\ensuremath{\omega}_{\ensuremath{\mu}}{}^{rs}$ is the Lorentz connection in the first-order formalism. Applying a similar method to ten dimensions, we start with the field content $(e_{\ensuremath{\mu}}{}^{m},{\ensuremath{\psi}}_{\ensuremath{\mu}},\ensuremath{\omega}_{\ensuremath{\mu}}{}^{rs},{A}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}},{B}_{\ensuremath{\mu}\ensuremath{\nu}},\ensuremath{\lambda},\ensuremath{\varphi},\ensuremath{\chi})$ with nilpotent fermionic symmetry, and show that the conventional massive type-IIA supergravity comes out as a special case of our system. These explicit results indicate that the most known massive supergravity theories are just special cases of more fundamental systems with nilpotent fermionic symmetry. Our nilpotent fermionic charge ${N}_{\ensuremath{\alpha}}$ satisfying ${{N}_{\ensuremath{\alpha}},{N}_{\ensuremath{\beta}}}=0$ resembles the Becchi-Rouet-Stora-Tyutin charge ${Q}_{\mathrm{B}}$ in topological field theory with the ``twisting of supersymmetry.'' If we interpret our charge ${N}_{\ensuremath{\alpha}}$ as twisted supersymmetry, it becomes clear how our system evades the Haag-Lopusza\ifmmode \acute{n}\else \'{n}\fi{}ski-Sohnius theorem for the uniqueness of supergravity.