Morita classes of microdifferential algebroids

Projective cotangent bundles of complex manifolds are the local models of complex contact manifolds. Such bundles are quantized by the algebra of microdifferential operators (a localization of the algebra of differential operators on the base manifold). Kashiwara proved that any complex contact manifold $X$ is quantized by a canonical microdifferential algebroid (a linear stack locally equivalent to an algebra of microdifferential operators). Besides the canonical one, there can be other microdifferential algebroids on $X$. Our aim is to classify them. More precisely, let $Y$ be the symplectification of $X$. We prove that Morita (resp. equivalence) classes of microdifferential algebroids on $X$ are described by $H^2(Y;{\mathbb C}^\times)$. We also show that any linear stack locally equivalent to a stack of microdifferential modules is in fact a stack of modules over a microdifferential algebroid. To obtain these results we use techniques of microlocal calculus, non-abelian cohomology and Morita theory for linear stacks.