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.
Highlights
The present paper is meant to complete the program set out in Ref. [1], concerning the classification of cubic interactions for massless bosons in three space-time dimensions
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
We complete the three-dimensional classification of cubic interactions deriving parity-odd vertices for massless bosonic fields as well as their couplings to Chern-Simons fields
Summary
The present paper is meant to complete the program set out in Ref. [1], concerning the classification of cubic interactions for massless bosons in three space-time dimensions. Cubic interactions of massless higher-spin fields in arbitrary space-time dimensions d ≥ 4 were studied extensively starting from the pioneering work [9], later extended to the complete light-cone gauge classification of vertices first in four dimensions [10] and in arbitrary dimensions d ≥ 4 [11]. [14,15,16,17,18,19,20,21] and references therein), resulting in the classification of parity-even cubic vertices in Minkowski space of any dimension d ≥ 4 [22], i.e., the covariant extension of Ref. We complete the three-dimensional classification of cubic interactions deriving parity-odd vertices for massless bosonic fields as well as their couplings to Chern-Simons fields. The Appendixes provide curious observations related to the parity-even vertices and classification of Fronsdal cubic vertices in two dimensions
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