Interactions of massless fermionic fields in three dimensions

Abstract

All independent interaction vertices involving massless (Fang--Fronsdal) fermions in three dimensions are classified, completing the classification of interactions of massless fields of any spin. Similarly to the bosonic case, we get no independent vertices at quartic or higher order in the fields involving fields with spin $s\geq 3/2$, and cubic vertices only for spins satisfying triangle inequalities, apart from the cases involving (matter) fields with spin $s\leq 1$. Differently from the bosonic cases, we get only one vertex for each triple of spins with two Majorana fermions and one boson, which is parity even (odd) when the sum of the spins of all fields involved is odd (even). When the two Majorana fermions are identical, their coupling to an odd-spin boson is trivialized. We comment on the non-trivial holographic dictionary relating these vertices to $CFT$ correlators in two dimensions.