4D Higher Spin Gravity with Dynamical Two-Form as a Frobenius--Chern--Simons Gauge Theory

TL;DR

This paper develops an off-shell four-dimensional higher spin gravity framework using a covariant Hamiltonian action on a nine-dimensional Poisson manifold, incorporating a dynamical two-form and extending Vasiliev's equations.

Contribution

It introduces a novel off-shell formulation of higher spin gravity with a dynamical two-form, expanding the algebraic and geometric structure of the theory.

Findings

Reproduces Vasiliev's extended higher spin equations in a specific background

Provides a covariant Hamiltonian action on a nine-dimensional Poisson manifold

Incorporates a dynamical two-form and duality extension in the gauge theory

Abstract

We provide an off-shell formulation of four-dimensional higher spin gravity based on a covariant Hamiltonian action on an open nine-dimensional Poisson manifold whose boundary consists of the direct product of spacetime and a noncommutative twistor space of S^2 x S^2 topology. The fundamental field is a superconnection consisting of even and odd differential forms valued in the odd and even sectors of a 3-graded associative algebra given by the direct product of an eight-dimensional Frobenius algebra and a higher spin algebra extended by inner Klein operators. The superconnection consists of two one-forms gauging the one-sided actions of the higher spin algebra, two bi-fundamental real forms given by the Weyl zero-form and a new dynamical two-form, an additional set of forms providing a maximal duality extension, and, finally, the Lagrange multipliers required for the covariant Hamiltonian action. In a particular two-form background, the model yields Vasiliev's recently proposed extended higher spin gravity equations, whose interaction terms are accounted for by de Rham closed globally defined forms arising in the dynamical two-form.