On the conformal higher spin unfolded equation for a three-dimensional self-interacting scalar field

Bengt E.W Nilsson

doi:10.1007/jhep08(2016)142

Abstract

We propose field equations for the conformal higher spin system in three dimensions coupled to a conformal scalar field with a sixth order potential. Both the higher spin equation and the unfolded equation for the scalar field have source terms and are based on a conformal higher spin algebra which we treat as an expansion in multi-commutators. Explicit expressions for the source terms are suggested and subjected to some simple tests. We also discuss a cascading relation between the Chern-Simons action for the higher spin gauge theory and an action containing a term for each spin that generalizes the spin 2 Chern-Simons action in terms of the spin connection expressed in terms of the frame field. This cascading property is demonstrated in the free theory for spin 3 but should work also in the complete higher spin theory.

Highlights

Note, that these equations are taken from the topological gauging of three dimensional CF T s with eight supersymmetries [2, 3] where all coupling constants are determined in terms of the gravitational one g
We propose field equations for the conformal higher spin system in three dimensions coupled to a conformal scalar field with a sixth order potential
There it was emphasized that the unfolded equation DΦ|0 q = 0 for the higher spin algebra based on SO(3, 2), the conformal group in three dimensional space-time, realized in terms of two hermitian spinor operators qα, pα satisfying [qα, pβ] = iδβα, produces the correct Klein-Gordon equation for a conformal scalar coupled to the spin 2 metric and its generalization to spin 3.10

Summary

The conformal interacting higher spin equations

The two basic field equations for the SO(3, 2) conformal higher spin (HS) theory coupled to a scalar field with fifth order self-interactions that we propose and study here are the unfolded equation. Where the first term defines the usual scalar field φ(x) that will appear conformally coupled to spin 2 and all higher spin frame fields coming from A and with its own fifth order selfinteraction in the Klein-Gordon equation. As will be elaborated upon elsewhere [19] some of the constraint equations of the spin 3 system along other generators than Kab will contain the spin 2 Cotton tensor and will be affected by the introduction of source terms This description of the structure of F implies that F can not be made to act directly on the vacuum like in the unfolded equation (2.1) since we would loose information

Field equations with source terms

A cascading Lagrangian

Findings

Conclusions