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Irregular higher-spin generating equations and chiral perturbation theory

We present a complementary approach to the standard Vasiliev framework for nonlinear higher-spin interactions in four dimensions, aimed at identifying their minimally nonlocal form. Our proposal introduces a generating system for higher-spin vertices at the level of classical equations, which we refer to as irregular, in contrast to the regular case described by Vasiliev. This system extends the recently proposed equations for (anti)holomorphic interactions by incorporating the mixed sector. Its perturbative series encompasses the entire (anti)holomorphic sector in the leading order, with vertices related to powers of the complex parity-breaking parameter $η$ or $\barη$. The subsequent corrections facilitate the mixing of the two sectors, with vertices carrying mixed powers of $η$ and $\barη$. The consistency relies on the nonlinear algebraic constraint, which is shown to be satisfied at least in the quadratic and cubic approximations. As a result, the previously discussed (anti)holomorphic interactions in the literature can be systematically extended to generate vertices of the form $η^N \barη^k$ and their conjugate, at least for $k \leq 2$ and any $N$. As a byproduct of our analysis, we also identify the new higher-spin structure dualities.