Irregular higher-spin generating equations and chiral perturbation theory

TL;DR

This work develops irregular higher-spin generating equations as a complementary framework to Vasiliev's equations to achieve minimally nonlocal HS vertices in four dimensions. By extending holomorphic/antiholomorphic generating systems to include the mixed sector via generalized projector identities and an irregular large star product, the authors construct a chiral perturbation theory that organizes interactions by a degree $k$ of holomorphic–antiholomorphic entanglement. They demonstrate the existence of consistent vertices up to degree $k=2$ (i.e., quadratic and cubic order) with arbitrary holomorphic order $N$, and show that holomorphic/holo-mixed vertices reproduce known Vasiliev structures in the appropriate limits, while revealing structure dualities between sectors. The formalism yields a maximally local holomorphic sector and constrains the mixed sector, offering a plausible route to addressing higher-spin locality and enabling systematic exploration of quartic and higher-order vertices with potential links to dualities with Vasiliev’s theory. These results provide new tools for probing HS locality and point toward broader applications, including possible extensions to Coxeter higher-spin models. The approach also identifies a clear hierarchy of consistency conditions (e.g., the $b$-constraint) that must be satisfied for a unitary completion of the full HS theory.

Abstract

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.