Scalar Field in Any Dimension from the Higher Spin Gauge Theory Perspective
O. V. Shaynkman, M. A. Vasiliev
TL;DR
The paper reformulates scalar field dynamics in arbitrary dimensions using an unfolded, higher-spin-inspired framework in flat space and $AdS_d$, interpreting the Klein-Gordon equation as a nontrivial $\sigma_{-}$-cohomology class of the extended covariant derivative. It constructs a free action $S^{2}$ built from higher-spin covariant derivatives that is on-shell equivalent to the standard first-order Klein-Gordon action for arbitrary mass, while revealing an infinite tower of auxiliary fields that modify interactions; Yang-Mills couplings yield pseudolocally exact conserved currents in AdS via infinite derivative expansions. It further develops an off-mass-shell extension by enlarging the field content to $\tilde{C}$ and defining a generalized covariant derivative $\tilde{\mathcal{D}}$ with trivial first cohomology, offering a path toward nonlinear higher-spin interactions. The work links higher-spin symmetry, AdS geometry, and string-inspired derivative expansions, and has potential implications for AdS/CFT and the structure of consistent higher-spin couplings.
Abstract
We formulate the equations of motion of a free scalar field in the flat and $AdS$ space of an arbitrary dimension in the form of some "higher spin" covariant constancy conditions. Klein-Gordon equation is interpreted as a non-trivial cohomology of a certain "\sgm-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 non-zero mass in the flat space. The constructed action is shown to be equivalent to the standard first-order Klein-Gordon action at the quadratic level but becomes different at the interaction level because of the presence of an infinite set of auxiliary fields which do not contribute at the free level. The example of Yang-Mills current interaction is considered in some detail. It is shown in particular how the proposed action generates the pseudolocally exact form of the matter currents in $AdS_d$.