Scalar field in an arbitrary dimension from the standpoint of higher-spin gauge theory

We formulate the equations of motion of a free scalar field in the flat and AdS spaces of arbitrary dimension in the form of “higher-spin” covariant constancy conditions. The Klein–Gordon equation describes a nontrivial cohomology of a certain "$\sigma_-$-complex". The action principle for a scalar field is formulated in terms of the “higher-spin” covariant derivatives for an arbitrary mass in AdS$_d$ and for a nonzero mass in the flat space. The free-field part of the constructed action coincides with the standard first-order Klein–Gordon action, but the interaction part is different because of the presence of an infinite set of auxiliary fields, which do not contribute at the free level. We consider the example of Yang–Mills current interaction and show how the proposed action generates the pseudolocally exact form of the matter currents in AdS$_d$.