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.

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A simple systematic method for calculating derivative expansions of the one-loop effective action is presented. This method is based on using symbols of operators and well known deformation quantization theory. To demonstrate its advantages we present several examples of application for scalar theory, Yang-Mills theory, and scalar electrodynamics. The superspace formulation of the method is considered for Kählerian and non-Kählerian quantum corrections for Wess-Zumino and for Heisenberg-Euler lagrangians in super QED models. *

We study the effective actions in massive rank 2 and rank 3 antisymmetric tensor field models in curved space-time. These models are classically equivalent to massive vector field and massive scalar field with minimal coupling to gravity, respectively. We prove that the effective action for massive rank 2 antisymmetric tensor field is exactly equal to that for massive vector field and the effective action for massive rank 3 antisymmetric tensor field is exactly equal to that for massive scalar field. The proof is based on an identity for mass-dependent zeta functions associated with Laplacians acting on p forms.