Symmetries of Curved Superspace

We review the superspace technique to determine supersymmetric spacetimes in the framework of off-shell formulations for supergravity in diverse dimensions using the case of 3D N = 2 supergravity theories as an illustrative example.This geometric formalism has several advantages over other approaches advocated in the last four years.Firstly, the infinitesimal isometry transformations of a given curved superspace form, by construction, a finite-dimensional Lie superalgebra, with its odd part corresponding to the rigid supersymmetry transformations.Secondly, the generalised Killing spinor equation, which must be obeyed by the supersymmetry parameters, is a consequence of the more fundamental superfield Killing equation.Thirdly, general rigid supersymmetric theories on a curved spacetime are readily constructed in superspace by making use of the known off-shell supergravity-matter couplings and restricting them to the background chosen.It is the superspace techniques which make it possible to generate arbitrary off-shell supergravitymatter couplings.Fourthly, all maximally supersymmetric Lorentzian spaces correspond to those off-shell supergravity backgrounds for which the Grassmann-odd components of the superspace torsion and curvature tensors vanish, while the Grassmann-even components of these tensors are annihilated by the spinor derivatives.