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We provide an off-shell formulation of four-dimensional higher spin gravitybased on a covariant Hamiltonian action on an open nine-dimensional Poissonmanifold whose boundary consists of the direct product of spacetime and anoncommutative twistor space of S^2 x S^2 topology. The fundamental field is asuperconnection consisting of even and odd differential forms valued in the oddand even sectors of a 3-graded associative algebra given by the direct productof an eight-dimensional Frobenius algebra and a higher spin algebra extended byinner Klein operators. The superconnection consists of two one-forms gaugingthe one-sided actions of the higher spin algebra, two bi-fundamental real formsgiven by the Weyl zero-form and a new dynamical two-form, an additional set offorms providing a maximal duality extension, and, finally, the Lagrangemultipliers required for the covariant Hamiltonian action. In a particulartwo-form background, the model yields Vasiliev's recently proposed extendedhigher spin gravity equations, whose interaction terms are accounted for by deRham closed globally defined forms arising in the dynamical two-form.

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