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After a brief description of the $\mathbb{Z}$-graded differential Poissonalgebra, we introduce a covariant star product for exterior differential formsand give an explicit expression for it up to second order in the deformationparameter $\hbar$, in the case of symplectic manifolds. The graded differentialPoisson algebra endows the manifold with a connection, not necessarilytorsion-free, and places upon the connection various constraints.

Covariant Star Product for Exterior Differential Forms on Symplectic Manifolds