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

TL;DR

This paper explores the structure of short and semi-short superconformal representations in four-dimensional N=2 and N=4 theories, detailing their properties, decompositions, and conditions for protected dimensions.

Contribution

It provides a comprehensive analysis of short and semi-short supermultiplets, including their classification, decomposition at unitarity thresholds, and conditions for protected conformal dimensions.

Findings

Recovered known short supermultiplets for N=4 superconformal symmetry.

Described the decomposition of long multiplets into semi-short multiplets at unitarity thresholds.

Identified conditions under which 1/4-BPS multiplets have protected dimensions.

Abstract

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 $\half$-BPS or ${1\over 4}$-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 ${1\over 4}$-BPS multiplet need not have a protected dimension unless the primary state belongs to a $[1,p,1]$ representation.