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Morita Classes of Microdifferential Algebroids

By Andrea D&apos and Pietro Polesello


Following Kashiwara, any complex contact manifold X can be canonically quantized. This means that X is endowed with a canonical microdifferential algebroid – a linear stack locally equivalent to an algebra of microdifferential operators. In this paper, we prove that Morita (resp. equivalence) classes of microdifferential algebroids on X are classified by H 2 (Y,C ×), for Y the symplectification of X. We also show that any stack locally equivalent to a stack of microdifferential modules is globally equivalent to the stack of modules over a microdifferential algebroid. To obtain these results we use techniques of microlocal calculus, non commutative cohomology and Morita theory for linear stacks

Year: 2011
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