Cyclic cocycles on deformation quantizations and higher index theorems

Details Cyclic cocycles on deformation quantizations and higher index theorems Cyclic cocycles on deformation quantizations and higher index theorems

2008-05-09

arxiv.org/abs/0805.1411v3

Mathematics

K-Theory and Homology

59 pages, this is a major revision, orbifold analytic higher index is introduced

Scientific paper

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.

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