Superconformal Symmetry in Three-dimensions

TL;DR

This work develops a comprehensive three-dimensional ${\cal N}$-extended superconformal framework on superspace by deriving the superconformal Killing equation and solving it to reveal the generator structure, including translations, dilations, Lorentz rotations, ${\rm SO}({\cal N})$ R-symmetry, and special superconformal transformations. It identifies the full group with the supermatrix group ${\rm OSp}({\cal N}|2,\mathbb{R})$ and provides explicit coset constructions and finite transformations, constructing covariant two-, three-, and general $n$-point building blocks and invariants for superconformal correlators. The paper presents explicit forms for two-point and three-point functions, establishes the general $n$-point reduction to an $(n-2)$-point core, and develops the associated superconformal invariants and covariant operators. These results extend the 4D/6D constructions to 3D and provide a robust framework for off-shell superfield formulations and invariant actions in 3D superconformal theories.

Abstract

Three-dimensional N-extended superconformal symmetry is studied within the superspace formalism. A superconformal Killing equation is derived and its solutions are classified in terms of supertranslations, dilations, Lorentz transformations, R-symmetry transformations and special superconformal transformations. Superconformal group is then identified with a supermatrix group, OSp(N|2,R), as expected from the analysis on simple Lie superalgebras. In general, due to the invariance under supertranslations and special superconformal transformations, superconformally invariant n-point functions reduce to one unspecified (n-2)-point function which must transform homogeneously under the remaining rigid transformations, i.e. dilations, Lorentz transformations and R-symmetry transformations. After constructing building blocks for superconformal correlators, we are able to identify all the superconformal invariants and obtain the general form of n-point functions. Superconformally covariant differential operators are also discussed.