Bilinear Fronsdal currents in the $Ad{S}_4$ higher-spin theory

We analyse higher-spin theory with general coupling constant $\eta$ at the second order, focusing on the gauge non-invariant vertices $\mathrm{{\rm Y}}(\omega ,\omega )$, $\mathrm{{\rm Y}}(\mathrm{\Omega },\omega ,C)$ and $\mathrm{{\rm Y}}(\omega ,C)$, that are shown to generate nontrivial currents in the Fronsdal equations. Explicit expressions for the currents are found in the frame-like formalism counterpart of the TT gauge worked out in the paper. The nonlinear higher-spin theory is shown to generate all types of Metsaev’s currents with the coupling constants manifestly expressed via the complex coupling constant $\eta$ of the higher-spin theory. It is shown that all currents in the higher-spin theory are conformal in the TT gauge except for those bilinear in the higher-spin gauge fields $\omega$.