Morita classes of microdifferential algebroids

Details Morita classes of microdifferential algebroids Morita classes of microdifferential algebroids

2011-12-21

arxiv.org/abs/1112.5005v1

Mathematics

Algebraic Geometry

Scientific paper

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 $H^2(Y,\C^\times)$, 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.

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