Cubic interactions of massless bosonic fields in three dimensions II: Parity-odd and Chern-Simons vertices

This work completes the classification of the cubic vertices for arbitrary spin massless bosons in three dimensions started in a previous companion paper by constructing parity-odd vertices. Similarly to the parity-even case, there is a unique parity-odd vertex for any given triple $s_1\geq s_2\geq s_3\geq 2$ of massless bosons if the triangle inequalities are satisfied ($s_1<s_2+s_3$) and none otherwise. These vertices involve two (three) derivatives for odd (even) values of the sum $s_1+s_2+s_3$. A non-trivial relation between parity-even and parity-odd vertices is found. Similarly to the parity-even case, the scalar and Maxwell matter can couple to higher spins through current couplings with higher derivatives. We comment on possible lessons for 2d CFT. We also derive both parity-even and parity-odd vertices with Chern-Simons fields and comment on the analogous classification in two dimensions.Comment: 29 page

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.Comment: 26 pages, no figure