Superconformal Symmetry in Six-dimensions and Its Reduction to Four

TL;DR

The paper develops a superspace formulation of six-dimensional superconformal symmetry, deriving the Killing equations and proving that the full group is $OSp(2,6|N)$ for $(N,0)$ with $R$-symmetry $Sp(N)$. It provides explicit finite transformations via a coset construction, introduces superinversion and spatial reflection, and builds covariant two- and three-point correlators in ${f R}^{6|8N}$; these structures reduce consistently to ${f R}^{4|4N}$ under dimensional reduction, yielding four-dimensional superconformal invariants and correlators with adjusted $R$-symmetry (to $SU(N) imes U(1)$ or $SU(4)$ for $N=4$). The framework also outlines covariant differential operators and the algebraic closure of the $(N,0)$ superconformal algebra, establishing a bridge between higher-dimensional superconformal theories and their four-dimensional counterparts.

Abstract

Superconformal symmetry in six-dimensions is analyzed in terms of coordinate transformations on superspace. A superconformal Killing equation is derived and its solutions are identified in terms of supertranslations, dilations, Lorentz transformations, R-symmetry transformations and special superconformal transformations. The full superconformal symmetry, which is shown to form the group OSp(2,6|N), is possible only if the supersymmetry algebra has N spinorial generators of the same chirality, corresponding to (N,0) supersymmetry. The R-symmetry group is then Sp(N) and the corresponding superspace is R^{6|8N}. We define superinversion as a map to the associated superspace of opposite chirality. General formulae for two-point and three-point correlation functions of quasi-primary superfields are exhibited. The superconformal group in six-dimensions is reduced to a corresponding extended superconformal group in four-dimensions. Superconformally covariant differential operators are also discussed.