A Star Product for Differential Forms on Symplectic Manifolds

We present a star product between differential forms to second order in the deformation parameter $\mathrm{\hslash }$. The star product obtained is consistent with a graded differential Poisson algebra structure on a symplectic manifold. The form of the graded differential Poisson algebra requires the introduction of a connection with torsion on the manifold, and places various constraints upon it. The star product is given to second order in ${\mathrm{\hslash }}^2$.