The non-linear coupled spin 2 - spin 3 Cotton equation in three dimensions

TL;DR

This paper derives the full non-linear spin 3 Cotton equation coupled to spin 2 in three-dimensional conformal higher spin theory using a frame field approach, solving gauge theory equations and verifying symmetry properties.

Contribution

It provides the first explicit derivation of the non-linear spin 3 Cotton equation in 3D, connecting gauge theory and metric formulations, with computationally obtained results.

Findings

Derived the non-linear spin 3 Cotton equation in 3D

Verified symmetry properties of the equation

Connected gauge and metric formulations of the theory

Abstract

In the context of three-dimensional conformal higher spin theory we derive, in the frame field formulation, the full non-linear spin 3 Cotton equation coupled to spin 2. This is done by solving the corresponding Chern-Simons gauge theory system of equations, that is, using $F=0$ to eliminate all auxiliary fields and thus expressing the Cotton equation in terms of just the spin 3 frame field and spin 2 covariant derivatives and tensors (Schouten). In this derivation we neglect the spin 4 and higher spin sectors and approximate the star product commutator by a Poisson bracket. The resulting spin 3 Cotton equation is complicated but can be related to linearized versions in the metric formulation obtained previously by other authors. The expected symmetry (spin 3 "translation", "Lorentz" and "dilatation") properties are verified for Cotton and other relevant tensors but some perhaps unexpected features emerge in the process, in particular in relation to the non-linear equations. We discuss the structure of this non-linear spin 3 Cotton equation but its explicit form is only presented here, in an exact but not completely refined version, in appended files obtained by computer algebra methods. Both the frame field and metric formulations are provided.