Superconformal symmetry in three dimensions

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. A 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.