On the index theorem for symplectic orbifolds

Boris Fedosov; Bert-Wolfang Schulze; Nikolai Tarkhanov

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CONTENTSI. Introduction............................................................................................................................................... 5II. Preliminary notions................................................................................................................................ 7III. Geometric quantization.........................................................................................................................12 A. Elements of...

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A Liouville form on a symplectic manifold $(X, ω)$ is by definition a potential $β$ of the symplectic form $- d β = ω$ . Its center $M$ is given by $β^{- 1} (0)$ . A normal form for certain Liouville forms in a neighborhood of its center is given.

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A geometric quantization of a Kähler manifold, viewed as a symplectic manifold, depends on the complex structure compatible with the symplectic form. The quantizations form a vector bundle over the space of such complex structures. Having a canonical quantization would amount to finding a natural (projectively) flat connection on this vector bundle. We prove that for a broad class of manifolds, including symplectic homogeneous spaces (e.g., the sphere), such connection does not exist....

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The notion of a local line bundle on a manifold, classified by 2-cohomology with real coefficients, is introduced. The twisting of pseudodifferential operators by such a line bundle leads to an algebroid with elliptic elements with real-valued index, given by a twisted variant of the Atiyah-Singer index formula. Using ideas of Boutet de Monvel and Guillemin the corresponding twisted Toeplitz algebroid on any compact symplectic manifold is shown to yield the star...