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

Understand

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.