Multiplets of Superconformal Symmetry in Diverse Dimensions

TL;DR

This work delivers a comprehensive framework for classifying unitary superconformal multiplets in dimensions $\ge3$, introducing a conjectural yet consistency-checked algorithm to generate all multiplets by eliminating null states. It unifies the treatment of long and short multiplets via Racah–Speiser methods and careful null-state analysis, enabling explicit tabulations of multiplets, currents, and stress-tensor multiplets across dimensions. A central outcome is the strong constraint on maximal supersymmetry: SCFTs with more than 16 Poincaré supercharges are forbidden in $d\ge4$, while such theories may exist in $d=3$ only if free. The work also clarifies recombination structures, absolutely protected multiplets, and provides tools for computing superconformal indices and studying allowed deformations.

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 Poincaré 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.

Figures (1)

• Figure 1: Hierarchy of unitary long and short multiplets. Long multiplets exist when $\Delta > \Delta_A$. Their continuum extends down to short $A$-type multiplets with $\Delta = \Delta_A$. Isolated short multiplets exist when $\Delta = \Delta_{B,C,D}$, but not if $\Delta$ lies in a gap between $\Delta_A>\Delta_B >\Delta_C>\Delta_D$.