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

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 ≥ s 2 ≥ s 3 ≥ 2 of massless bosons if the triangle inequalities are satisfied (s 1 < s 2 þ s 3 ) and none otherwise. These vertices involve two (three) derivatives for odd (even) values of the sum s 1 þ s 2 þ s 3 . A nontrivial relation between parityeven 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.

The uniqueness of hypergravity

“…Therefore, up to an overall coupling constant, the graviton cubic self coupling is the same that appears in General Relativity. Moreover, this is the off-shell form of the only parity-preserving vertex that one would expect in D = 3 from the classification of cubic vertices of bosonic gauge fields [24,25].…”

“…One possible way to generalize our analysis is to relax the assumption of parity. After all, parity-odd cubic vertices do exist in D = 3, at least for bosons [25]. This would give rise to an additional graviton self coupling at the cubic level.…”

“…Because we did not assume parity invariance of T µν , it follows as a byproduct of our analysis that the parity non-invariant 3-derivative cubic self coupling of gravitons, reported in[25], must be non-Abelian.…”

Cubic interaction vertices for N=1 arbitrary spin massless supermultiplets in flat space

“…[49]- [55] and references therein). We think that the light-cone gauge approach will simplify considerably the whole analysis of higherspin massive and conformal fields in three dimensions.…”

Cubic interaction for higher spins in AdS+1 space in the explicit covariant form

“…It is worth to mention also that all these activities supplemented with the parallel development of Vasiliev's frame like formalism to cubic interaction in AdS space [38]- [40]. It is interesting also in this aspects that covariant classification of cubic vertices was done for parity even dimensions d ≥ 4 in [33] but classification including parity odd vertices for four and three dimensions was completed only recently in [41,42,43]. The last point we want to mention here is that although cubic interaction in AdS space has formulation developed in ambient space some years ago [44]- [48] the direct formulation on the language of AdS d+1 covariant derivatives is still unknown and realized before in [48] for some simplest part of interaction only.…”