Actions, Charges and Off-Shell Fields in the Unfolded Dynamics Approach

TL;DR

This paper develops an action-centered formulation of unfolded dynamics for massless fields of all spins in flat Minkowski space, recasting actions and conserved charges as elements of the $Q$-cohomology of the underlying $L_\infty$-algebra. It introduces a minimal off-shell, flat-space HS system with 1-forms $A(p,y|x)$ and 0-forms $F(p,y|x)$ valued in a Weyl star-product algebra, obeying zero-curvature and covariant-constancy equations that admit a consistent off-shell dynamics and, upon linearization, reproduce the known free unfolded descriptions. The work then shows how nonlinear off-shell constraints close in closed forms for Yang–Mills and gravity within this unfolded framework, clarifies the role of the $sp(2)$ generators, and discusses contractions to Poisson/commutative algebras to yield finite spin towers. It also explains how the independent $Q$-cohomology classes encode invariant actions and charges, offering a promising route to action principles for higher-spin theories and potential extensions to AdS backgrounds and superstring contexts. Overall, the paper provides a coherent, algebraic approach to actions, charges, and off-shell HS fields in Minkowski space and sets the stage for further development of nonlinear unfolded dynamics in broader settings.

Abstract

Within unfolded dynamics approach, we represent actions and conserved charges as elements of cohomology of the $L_\infty$ algebra underlying the unfolded formulation of a given dynamical system. The unfolded off-shell constraints for symmetric fields of all spins in Minkowski space are shown to have the form of zero curvature and covariant constancy conditions for 1-forms and 0-forms taking values in an appropriate star product algebra. Unfolded formulation of Yang-Mills and Einstein equations is presented in a closed form.