An action principle for Vasiliev's four-dimensional higher-spin gravity

TL;DR

This work establishes a generalized Hamiltonian action principle for Vasiliev's four-dimensional higher-spin gravity by extending the original system to include higher-degree differential forms. The duality-extended formulation introduces two distinct interaction freedoms, one in the Q-structure and one in a generalized Poisson (Lagrange-multiplier) sector, with gauge invariance requiring at least one to be linear in the master fields. The authors derive on-shell duality-extended equations that embed Vasiliev's original equations as a consistent sub-sector, and they discuss spectral flows, boundary conditions, and consistent truncations to Type A and B models. They further develop a graded cyclic chiral trace and bulk actions for odd and even-dimensional bases, proving Cartan integrability in bilinear cases and outlining global formulations and potential paths toward AKSZ-BV quantization. Overall, the paper provides a rigorous, background-independent action framework for unfolded higher-spin gravity, opening avenues for quantum formulations and holographic interpretations while leaving key questions about nontrivial couplings and twistor-space boundaries for future work.

Abstract

We provide Vasiliev's fully nonlinear equations of motion for bosonic gauge fields in four spacetime dimensions with an action principle. We first extend Vasiliev's original system with differential forms in degrees higher than one. We then derive the resulting duality-extended equations of motion from a variational principle based on a generalized Hamiltonian sigma-model action. The generalized Hamiltonian contains two types of interaction freedoms: One set of functions that appears in the Q-structure of the generalized curvatures of the odd forms in the duality-extended system; and another set depending on the Lagrange multipliers, encoding a generalized Poisson structure, i.e. a set of polyvector fields of ranks two or higher in target space. We find that at least one of the two sets of interaction-freedom functions must be linear in order to ensure gauge invariance. We discuss consistent truncations to the minimal Type A and B models (with only even spins), spectral flows on-shell and provide boundary conditions on fields and gauge parameters that are compatible with the variational principle and that make the duality-extended system equivalent, on shell, to Vasiliev's original system.