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

TL;DR

This paper develops conformal higher spin field equations in three dimensions coupled to a scalar with a sixth order potential, introducing source terms and exploring their algebraic structure and cascading relations.

Contribution

It presents new conformal higher spin equations with source terms and analyzes their algebraic and action-based structures in three dimensions.

Findings

Explicit source term expressions proposed and tested.

Demonstration of cascading relation in free spin 3 theory.

Extension of Chern-Simons action to higher spins.

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.