Cyclic cocycles on deformation quantizations and higher index theorems

2008/05/09 by M. Pflaum, Pflaum, M., H. Posthuma +3
Mathematics · #FOS: Mathematics #K-Theory and Homology (math.KT) #Quantum Algebra (math.QA) #math.KT #math.QA

paper · pdf · doi:10.48550/arxiv.0805.1411

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

arxiv created 2009/08/13 · arxiv updated 2009/12/01

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.

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