Multiplets of Superconformal Symmetry in Diverse Dimensions

TL;DR

This paper develops an efficient algorithm to classify superconformal multiplets in dimensions greater than three, revealing constraints on the existence of certain superconformal field theories and their multiplets.

Contribution

It introduces a conjectural but consistent algorithm for generating superconformal multiplets and classifies those containing conserved currents or free fields.

Findings

Classified superconformal multiplets with conserved currents and free fields.

Showed SCFTs with >16 supercharges cannot exist in d ≥ 4.

Identified that such theories in d=3 are necessarily free.

Abstract

We systematically analyze the operator content of unitary superconformal multiplets in $d > 3$ spacetime dimensions. We present a simple, general, and efficient algorithm that generates all of these multiplets by correctly eliminating possible null states. The algorithm is conjectural, but passes a vast web of consistency checks. We apply it to tabulate a large variety of superconformal multiplets. In particular, we classify and construct all multiplets that contain conserved currents or free fields, which play an important role in superconformal field theories (SCFTs). Some currents that are allowed in conformal field theories cannot be embedded in superconformal multiplets, and hence they are absent in SCFTs. We use the structure of superconformal stress tensor multiplets to show that SCFTs with more than 16 Poincar\'e supercharges cannot arise in $d \geq 4$, even when the corresponding superconformal algebras exist. We also show that such theories do arise in $d = 3$, but are necessarily free.