On Short and Semi-Short Representations for Four Dimensional Superconformal Symmetry

Abstract

Possible short and semi-short representations for $\mathbb{N}=2$ and $\mathbb{N}=4$ superconformal symmetry in four dimensions are discussed. For $\mathbb{N}=4$ the well known short supermultiplets whose lowest dimension conformal primary operators correspond to $\half$-BPS or $\frac{1}{4}$-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 $\mathbb{N}=4$ a $\frac{1}{4}$-BPS multiplet need not have a protected dimension unless the primary state belongs to a $[1,p,1]$ representation.

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