On short and semi-short representations for four-dimensional superconformal symmetry

Possible short and semi-short representations for N = 2 and N = 4 superconformal symmetry in four dimensions are discussed. For N = 4 the well known short supermultiplets whose lowest dimension conformal primary operators correspond to 1/2-BPS or 1/4-BPS states and are scalar fields belonging to the SU(4) R-symmetry representations [0, p, 0] and [q, p, q] and having scale dimension Delta = p and Delta = 2q + p, respectively, are recovered. The representation content of semi-short multiplets, which arise at the unitarity threshold for long multiplets, is discussed. It is shown how, at the unitarity threshold, a long multiplet can be decomposed into four semi-short multiplets. If the conformal primary state is spinless one of these becomes a short multiplet. For N = 4 a 1/4-BPS multiplet need not have a protected dimension unless the primary state belongs to a [1, p, 1] representation. (C) 2003 Elsevier Science (USA). All rights reserved.