$sl(2,\mathds{C})\times D$ symmetry and conformal primary basis for massless fields

TL;DR

This paper introduces a group theoretic approach to construct conformal primary bases for massless fields with arbitrary helicity, revealing enhanced symmetry and explicit constraints, and deriving new conformal primary wavefunctions.

Contribution

It provides a novel group theoretic framework using $sl(2, ext{C}) imes D$ symmetry to explicitly construct conformal primary wavefunctions for massless fields of arbitrary helicity.

Findings

Enhanced $sl(2, ext{C}) imes D$ symmetry for massless fields.

Explicit constraints on bulk dilatation and Casimirs.

New set of independent conformal primary wavefunctions for helicity |s| ≥ 1.

Abstract

Alternative to the embedding formalism, we provide a group theoretic approach to the conformal primary basis for the massless field with arbitrary helicity. To this end, we first point out that $sl(2,\mathds{C})$ isometry gets enhanced to $sl(2,\mathds{C})\times D$ symmetry for the solution space of the massless field with arbitrary helicity. Then associated with $sl(2,\mathds{C})\times D$ symmetry, we introduce the novel quadratic Casimirs and relevant tensor/spinor fields to derive 2 explicit constraints on the bulk dilatation and $sl(2,\mathds{C})$ Casimirs. With this, we further argue that the candidate conformal primary basis can be constructed out of the infinite tower of the descendants of the left and right highest (lowest) conformal primary wavefunction of $sl(2,\mathds{C})$ Lie algebra, and the corresponding celestial conformal weights are determined by the bulk scaling dimension through solving out the exact on-shell conformal primary wavefunctions, where on top of the two kinds of familiar-looking on-shell conformal primary wavefunctions, we also obtain another set of independent on-shell conformal primary wavefunctions for the massless field with helicity $|s|\ge 1$. In passing, we also develop the relationship between the 4D Lorentz Lie algebra and 2D conformal Lie algebra from scratch, and present an explicit derivation for the two important properties associated with the conformal primary wavefunctions.