Cyclic cocycles on deformation quantizations and higher index theorems

  • Pflaum M
  • Posthuma H
  • Tang X
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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.

Author-supplied keywords

  • Cyclic cocycles deformation quantizations higher index theorems

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Authors

  • M. J. Pflaum

  • H. Posthuma

  • X. Tang

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