Supersymmetric Poisson and Poisson-supersymmetric sigma models

We revisit and construct new examples of supersymmetric 2D topological sigma models whose target space is a Poisson supermanifold. Inspired by the AKSZ construction of topological field theories, we follow a graded-geometric approach and identify two commuting homological vector fields compatible with the graded symplectic structure, which control the gauge symmetries and the supersymmetries of the sigma models. Exemplifying the general structure, we first revisit the dilaton supergravity in 2 dimensions and then we show that two distinguished cases exhibiting global supersymmetry exist, one being the differential Poisson sigma model constructed before by Arias, Boulanger, Sundell and Torres-Gomez and the other a contravariant differential Poisson sigma model. The new model features nonlinear global supersymmetry transformations that are generated by the Poisson structure on the body of the target supermanifold, giving rise to a Poisson supersymmetry. Further examples are characterised by supersymmetry transformations controlled by the anchor map of a Lie algebroid, when this map is invertible, in which case we determine the geometric conditions for invariance under supersymmetry and closure of the supersymmetry algebra. Moreover, we show that the common thread through this type of models is that their supersymmetry-generating vector field is the coadjoint representation up to homotopy of a Lie algebroid.