For N -extended superconformal field theories in three spacetime dimensions (3D), with 1 ≤ N ≤ 3, we compute the two-and three-point correlation functions of the supercurrent and the flavour current multiplets. We demonstrate that supersymmetry imposes additional restrictions on the correlators of conserved currents as compared with the non-supersymmetric case studied by Osborn and Petkou in hep-th/9307010. It is shown that the three-point function of the supercurrent is determined by a single functional form consistent with the conservation equation and all the symmetry properties. Similarly, the three-point function of the flavour current multiplets is also determined by a single functional form in the N = 1 and N = 3 cases. The specific feature of the N = 2 case is that two independent structures are allowed for the three-point function of flavour current multiplets, but only one of them contributes to the three-point function of the conserved currents contained in these multiplets. Since the supergravity and super-Yang-Mills Ward identities are expected to relate the coefficients of the two-and three-point functions under consideration, the results obtained for 3D superconformal field theory are analogous to those in 2D conformal field theory.In addition, we present a new supertwistor construction for compactified Minkowski superspace. It is suitable for developing superconformal field theory on 3D spacetimes other than Minkowski space, such as S 1 × S 2 and its universal covering space R × S 2 .

We construct the classical action of the Aharony-Bergman-Jafferis-Maldacena (ABJM) model in the N =3, d=3 harmonic superspace. In such a formulation three out of six supersymmetries are realized off shell while the other three mix the superfields and close on shell. The superfield action involves two hypermultiplet superfields in the bifundamental representation of the gauge group and two Chern-Simons gauge superfields corresponding to the left and right gauge groups. The N =3 superconformal invariance allows only for a minimal gauge interaction of the hypermultiplets. Amazingly, the correct sextic scalar potential of ABJM emerges after the elimination of auxiliary fields. Besides the original U(N ) × U(N ) ABJM model, we also construct N =3 superfield formulations of some generalizations. For the SU(2) × SU(2) case we give a simple superfield proof of its enhanced N =8 supersymmetry and SO(8) R-symmetry.

We reveal non-manifest gauge and SO(1, 5) Lorentz symmetries in the Lagrangian description of a six-dimensional free chiral field derived from the Bagger-Lambert-Gustavsson model in arXiv:0804.3629 and make this formulation covariant with the use of a triplet of auxiliary scalar fields. We consider the coupling of this self-dual construction to gravity and its supersymmetrization. In the case of the non-linear model of arXiv:0805.2898 we solve the equations of motion of the gauge field, prove that its non-linear field strength is self-dual and find a gauge-covariant form of the non-linear action. Issues of the relation of this model to the known formulations of the M5-brane worldvolume theory are discussed.