N=1 Superconformal Symmetry in Four Dimensions

TL;DR

This paper develops a group-theoretical framework for N=1, four-dimensional superconformal symmetry by constructing finite transformations from the fundamental generators of the superconformal group. It shows that left-invariant derivatives and a class of superfields, including supercurrents, transform covariantly as quasi-primary fields, enabling closed-form determinations of two- and three-point functions through representation theory. The two-point function of the supercurrent is shown to be unique up to a constant, while the three-point function is determined up to two free parameters, with conservation further restricting the form in interacting theories to a two-parameter structure. These results extend familiar 2D CFT techniques to 4D SCFT and provide a foundation for exploring operator product expansions and correlation functions in four-dimensional supersymmetric conformal theories. The approach yields explicit expressions for scalar and vector superfield correlators and highlights the role of superinversion-type transformations in ensuring full symmetry constraints.

Abstract

N=1, d=4 superconformal group is studied and its representations are discussed. Under superconformal transformations, left invariant derivatives and some class of superfields, including supercurrents, are shown to follow these representations. In other words, these superfields are quasi-primary by analogy with two dimensional conformal field theory. Based on these results, we find the general forms of the two-point and the three-point correlation functions of the quasi-primary superfields in a group theoretical way. In particular, we show that the two-point function of the supercurrent is unique up to a constant and the general form of the three-point function of the supercurrent has two free parameters.