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 propose a new off-shell formulation for N -extended conformal supergravity in three spacetime dimensions. Our construction is based on the gauging of the N -extended superconformal algebra in superspace. Covariant constraints are imposed such that the algebra of covariant derivatives is given in terms of a single curvature superfield which turns out to be the super Cotton tensor. An immediate corollary of this construction is that the curved superspace is conformally flat if and only if the super Cotton tensor vanishes. Upon degauging of certain local symmetries, our formulation is shown to reduce to the conventional one with the local structure group SL(2, R) × SO(N ).1 In the N = 1 case, the superconformal tensor calculus was independently developed in [7]. Early superspace approaches to N = 1 and N = 2 supergravity theories were given in [8,9,10,11]. 2 We are grateful to Jim Gates for bringing Ref.[6] to our attention. 3 This construction is a natural generalization of Howe's superspace formulation for 4D Nextended conformal supergravity in four dimensions [13]. 4 The special cases of N = 8 and N = 16 conformal supergravity theories were independently worked out in [15,16] and [17] respectively. 5 If a prepotential formulation is available, the conformal supergravity action may be written as a superspace integral in terms of the prepotentials.

Long ago, Achúcarro and Townsend discovered that in three dimensions (3D) Nextended anti-de Sitter (AdS) supergravity exists in several incarnations, which were called the (p, q) AdS supergravity theories with non-negative integers p ≥ q such that N = p+q. Using the superspace approach to 3D N -extended supergravity developed in arXiv:1101.4013, we present three superfield formulations for N = 2 supergravity that allow for well defined cosmological terms and supersymmetric AdS solutions. The conformal compensators corresponding to these theories are respectively: (i) a chiral scalar multiplet; (ii) a vector multiplet; and (iii) an improved complex linear multiplet. The theories corresponding to (i) and (iii) are shown to provide two dually equivalent realizations of the (1,1) AdS supergravity, while (ii) describes the (2,0) AdS supergravity. We associate with each supergravity formulation, with and without a cosmological term, a consistent supercurrent multiplet. The supercurrents in the (1,1) and (2,0) AdS backgrounds are derived for the first time. We elaborate on rigid supersymmetric theories in (1,1) and (2,0) AdS superspaces.As in four dimensions, the Type I minimal and the w = −1 (or n = −1) non-minimal formulations can be used to describe AdS supergravity by modifying the supergravity action (in the Type I case) or deforming the complex linear constraint (in the non-minimal case). The two realizations turn out to be dually equivalent and lead to the same (1, 1) AdS supergravity. Unlike the situation in four dimensions, the Type II theory can also 3 The off-shell supergravity versions are traditionally labelled by the real parameter n introduced by Gates and Siegel [28].

The projective superspace formulation for four-dimensional N = 2 mattercoupled supergravity presented in arXiv:0805.4683 makes use of the variant superspace realization for the N = 2 Weyl multiplet in which the structure group is SL(2, C) × SU(2) and the super-Weyl transformations are generated by a covariantly chiral parameter. An extension to Howe's realization of N = 2 conformal supergravity in which the tangent space group is SL(2, C) × U(2) and the super-Weyl transformations are generated by a real unconstrained parameter was briefly sketched. Here we give the explicit details of the extension.