On Born-Infeld-type higher-spin interactions and supersymmetry

We develop a simple Lorentz-covariant construction of the Born–Infeld-type cubic interactions $(s,s_1,s_2)$ of massless higher-spin fields with $s+s_1+s_2$ derivatives in four-dimensiona flat space. For arbitrary spins $s\ge s_1\ge s_2$, the vertex is formulated in terms of a conserved higher-spin current and its trace. In contrast to the famous Berends–Burgers–van Dam currents, these currents are constructed from higher-spin Weyl tensors of a single chirality, schematically $J\sim \mathrm{\partial }C\mathrm{\partial }C$. Starting with a general ansatz, we derive recurrence relations fixing the current coefficients, identify the unique non-trivial conserved current after eliminating trivial (fake) vertices, and show that in the range $s\le s_1+s_2$ the interaction admits a manifestly gauge-invariant representation in terms of higher-spin Weyl tensors. The construction is then extended to $\mathcal{N}=1$ and $\mathcal{N}=2$ supersymmetry using the corresponding off-shell superfield formalisms. In the $\mathcal{N}=1$ case, we obtain the general conserved integer superspin $s$ supercurrent with manifest supersymmetry, which is bilinear in the Weyl supertensors of superspins $Y_1$ and $Y_2$. For identical superfields, $Y_1=Y_2$, this supercurrent vanishes for odd $s$ and is in agreement with earlier results by Gates and Koutrolikos. In the $\mathcal{N}=2$ case, using harmonic superspace, we construct the principal higher-spin supercurrent whose coefficients are uniquely fixed by the conservation equations. In contrast to the $\mathcal{N}=0,1$ cases, for identical $\mathcal{N}=2$ superfields, the corresponding vertex does not vanish. The results provide a systematic framework for Born–Infeld-type higher-spin interactions and their supersymmetric extensions.