3d conformal fields with manifest $sl(2,\mathbb{C})$

TL;DR

This paper constructs all short representations of the conformal algebra $so(3,2)$ using $sl(2, ext{C})$ spinors, connecting to massless fields in AdS${}_4$, analyzing unitarity, and exploring their contraction to Poincare algebra.

Contribution

It provides a comprehensive construction of short $so(3,2)$ representations with manifest $sl(2, ext{C})$ symmetry, linking to spinor-helicity formalism and conformal field theory.

Findings

Identified all short $so(3,2)$ representations with $sl(2, ext{C})$ symmetry.

Analyzed unitarity and classified them as lowest-weight modules.

Compared properties under contraction to Poincare algebra.

Abstract

In the present paper we construct all short representation of $so(3,2)$ with the $sl(2,\mathbb{C})$ symmetry made manifest due to the use of $sl(2,\mathbb{C})$ spinors. This construction has a natural connection to the spinor-helicity formalism for massless fields in AdS${}_4$ suggested earlier. We then study unitarity of the resulting representations, identify them as the lowest-weight modules and as conformal fields in the three-dimensional Minkowski space. Finally, we compare these results with the existing literature and discuss the properties of these representations under contraction of $so(3,2)$ to the Poincare algebra.