Cyclic cocycles on deformation quantizations and higher index theorems

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Abstract

We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasi-isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We give a new proof of Nest-Tsygan's algebraic higher index theorem by computing the pairing between such cyclic cocycles and the K-theory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold. As an application, we obtain the analytic higher index theorem of Connes-Moscovici and its extension to orbifolds. © 2009 Elsevier Inc. All rights reserved.

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Pflaum, M. J., Posthuma, H., & Tang, X. (2010). Cyclic cocycles on deformation quantizations and higher index theorems. Advances in Mathematics, 223(6), 1958–2021. https://doi.org/10.1016/j.aim.2009.10.012

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