Star-Product and Massless Free Field Dynamics in $AdS_4$

TL;DR

This work formulates massless free fields of arbitrary spin in $AdS_4$ within a star-product framework based on auxiliary spinor variables, recasting the dynamics as covariant-constancy and zero-curvature conditions. It constructs an explicit gauge function that reproduces the $AdS_4$ geometry in the star-product language and uses it to derive plane-wave-like solutions for the matter and Weyl fields, expressed through Gaussian integrals over spinor variables. The authors also provide an explicit, though intricate, expression for higher-spin gauge potentials $w_1$ that solves the corresponding equation and encodes the full tower of higher-spin dynamics. The results demonstrate that star-product techniques can yield concrete, locally valid solutions and suggest broad applicability to other backgrounds, dimensions, and possible supersymmetric extensions.

Abstract

Generic solution of free equations for massless fields of an arbitrary spin in $AdS_4$ is built in terms of the star-product algebra with spinor generating elements. A class of "plane wave" solutions is described explicitly.