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

TL;DR

This paper completes the classification of cubic vertices for massless bosonic fields in three dimensions by constructing parity-odd vertices, revealing unique vertices under certain conditions and exploring their relations to parity-even vertices and matter couplings.

Contribution

It introduces the complete set of parity-odd cubic vertices for massless bosons in three dimensions, extending previous work on parity-even vertices and analyzing their relations and matter interactions.

Findings

Unique parity-odd vertices exist for specific spin triples satisfying triangle inequalities.

Parity-odd vertices involve two or three derivatives depending on the sum of spins.

Relations between parity-even and parity-odd vertices are established.

Abstract

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.