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) >= s(2) >= s(3) >= 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 nontrivial 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 two-dimensional conformal field theory. We also derive both parity-even and parity-odd vertices with Chern-Simons fields and comment on the analogous classification in two dimensions.