Parity singlets and doublets of massive spin-3 particles in $D=2+1$ via Nother gauge embedding

Here we demonstrate that the sixth order (in derivatives) spin-3 self-dual model can be obtained from the fifth order self-dual model via a Noether Gauge Embedding (NGE) of longitudinal Weyl transformations $\eta_{(\mu\nu}\partial_{\alpha)}\Phi$. In the case of doublet models we can show that the massive spin-3 Singh-Hagen theory is dual to a fourth and to a sixth order theory, via a double round of the NGE procedure by imposing traceless longitudinal (reparametrization-like) symmetries $\partial_{(\mu}\tilde{\xi}_{\nu\alpha)}$ in the first round and transverse Weyl transformations $\eta_{(\mu\nu}\psi^T_{\alpha)}$ in the second one. Our procedure automatically furnishes the dual maps between the corresponding fields.