Superconformal structures on the three-sphere

TL;DR

This paper introduces superconformal structures on the three-sphere, defining superspheres with extended supersymmetry, and explores their realizations, symmetries, and geometric properties relevant for superconformal field theories.

Contribution

It constructs the supersphere S^{3|4n} as a homogeneous space of the superconformal group and analyzes its realizations and symmetries, especially for the n=1 case.

Findings

Defined superconformal superspheres S^{3|4n} as homogeneous spaces of OSp(2n|2,2)

Derived supertwistor and bi-supertwistor realizations of S^{3|4n}

Analyzed the unique properties of the n=1 case, including R-symmetry subgroup compactness

Abstract

With the motivation to develop superconformal field theory on S^3, we introduce a 2n-extended supersphere S^{3|4n}, with n=1,2,..., as a homogeneous space of the three-dimensional Euclidean superconformal group OSp(2n|2,2) such that its bosonic body is S^3. Supertwistor and bi-supertwistor realizations of S^{3|4n} are derived. We study in detail the n=1 case, which is unique in the sense that the R-symmetry subgroup SO^*(2n) of the superconformal group is compact only for n=1. In particular, we show that the OSp(2|2,2) transformations preserve the chiral subspace of S^{3|4}. Several supercoset realizations of S^{3|4n} are presented. Harmonic/projective extensions of the supersphere by auxiliary bosonic fibre directions are sketched.