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

TL;DR

This paper formulates an action principle for Vasiliev's four-dimensional higher-spin gravity, extending the original system with higher-degree forms and deriving duality-extended equations from a generalized Hamiltonian variational principle.

Contribution

It introduces a new action principle for Vasiliev's higher-spin equations, incorporating duality extensions and generalized Hamiltonian structures, ensuring gauge invariance and consistency.

Findings

Derived duality-extended equations of motion from a variational principle.

Identified conditions for interaction functions to maintain gauge invariance.

Established boundary conditions compatible with the variational principle.

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.