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We develop the superspace geometry of N -extended conformal supergravity in three space-time dimensions. General off-shell supergravity-matter couplings are constructed in the cases N ≤ 4.

This paper presents a projective superspace formulation for 4D N = 2 mattercoupled supergravity. We first describe a variant superspace realization for the N = 2 Weyl multiplet. It differs from that proposed by Howe in 1982 by the choice of the structure group SO(3, 1) × SU(2) versus SO(3, 1) × U(2) , which implies that the super-Weyl transformations are generated by a covariantly chiral parameter instead of a real unconstrained one. We introduce various off-shell supermultiplets which are curved superspace analogues of the superconformal projective multiplets in global supersymmetry and which describe matter fields coupled to supergravity. A manifestly locally supersymmetric and super-Weyl invariant action principle is given. Off-shell locally supersymmetric nonlinear sigma models are presented in this new superspace.

We review self-duality of nonlinear electrodynamics and its extension to several Abelian gauge fields coupled to scalars. We then describe self-duality in supersymmetric models, both N = 1 and N = 2. The self-duality equations, which have to be satisfied by the action of any self-dual system, are found and solutions are discussed. One important example is the Born-Infeld action. We explain why the N = 2 supersymmetric actions proposed so far are not the correct world-volume actions for D3 branes in d = 6.