Parity-violating vertices for spin-3 gauge fields

The problem of constructing consistent parity-violating interactions for spin-3 gauge fields is considered in Minkowski space. Under the assumptions of locality, Poincar\'e invariance, and parity noninvariance, we classify all the nontrivial perturbative deformations of the Abelian gauge algebra. In space-time dimensions $n=3$ and $n=5$, deformations of the free theory are obtained which make the gauge algebra non-Abelian and give rise to nontrivial cubic vertices in the Lagrangian, at first order in the deformation parameter g. At second order in g, consistency conditions are obtained which the five-dimensional vertex obeys, but which rule out the $n=3$ candidate. Moreover, in the five-dimensional first-order deformation case, the gauge transformations are modified by a new term which involves the second de Wit-Freedman connection in a simple and suggestive way.