Star-Product Functions in Higher-Spin Theory and Locality

TL;DR

This work develops a cohomological framework for nonlinear higher-spin equations in $AdS_4$ by organizing star-product functionals on twistor-like variables $Z^A,Y^A$ into the HS field algebra $\mathcal{H}$ and a stricter local subalgebra $\mathcal{H}^{loc}$. It proposes a locality conjecture distinguishing local field redefinitions (in $\mathcal{H}^{loc}$) from gauge transformations (in $\mathcal{H}$), and shows that physical HS fields reside in $d_Z$-cohomology with perturbative corrections captured by a homotopy operator acting on the spaces $V_{k,l}$. The analysis reveals how interaction terms are cohomologically controlled and explains why certain central elements are excluded from the HS equations, thereby constraining possible extensions of the theory. The framework aims to rule out integrating-flow nonlocalities (pseudolocal transformations) and provides a structured path toward constructing locality-respecting HS theories with potential applications to black-hole and vectorial HS models.

Abstract

Properties of the functional classes of star-product elements associated with higher-spin gauge fields and gauge parameters are elaborated. Cohomological interpretation of the nonlinear higher-spin equations is given. An algebra ${\mathcal H}$, where solutions of the nonlinear higher-spin equations are valued, is found. A conjecture on the classes of star-product functions underlying (non)local maps and gauge transformations in the nonlinear higher-spin theory is proposed.