Metric- and frame-like higher-spin gauge theories in three dimensions

We study the relation between the frame-like and metric-like formulation of higher-spin gauge theories in three space-time dimensions. We concentrate on the theory that is described by an SL(3) x SL(3) Chern-Simons theory in the frame-like formulation. The metric-like theory is obtained by eliminating the generalised spin connection by its equation of motion, and by expressing everything in terms of the metric and a spin-3 Fronsdal field. We give an exact map between fields and gauge parameters in both formulations. To work out the gauge transformations explicitly in terms of metric-like variables, we have to make a perturbative expansion in the spin-3 field. We describe an algorithm how to do this systematically, and we work out the gauge transformations to cubic order in the spin-3 field. We use these results to determine the gauge algebra to this order, and explain why the commutator of two spin-3 transformations only closes on-shell.