Resonant superalgebras and $\mathcal{N}=1$ supergravity theories in three spacetime dimensions

Abstract: We explore $\mathcal{N}=1$ supersymmetric extensions of algebras going beyond the Poincar\'e and AdS ones in three spacetime dimensions. Besides reproducing two known examples, we present new superalgebras, which all correspond to supersymmetric extensions with one fermionic charge $Q_{\alpha}$ concerning the so-called resonant algebras being characterized by the presence of an additional bosonic generator $Z_{a}$, besides the Lorentz $J_{a}$ and translation $P_{a}$ generators. Obtained eight supersymmetric $JPZ+Q$ schemes result directly from obeying super-Jacobi identities. We point out particular requirements that superalgebras have to satisfy to be successfully incorporated within valid supergravity actions. The presented algebraic and Lagrangian framework organizes and helps us better understand relations between the various supergravity and supersymmetric Chern-Simons actions invariant under diverse resonant superalgebras.