scholarly journals Noncommutative geometry and conformal geometry, II. Connes–Chern character and the local equivariant index theorem

2016 ◽  
Vol 10 (1) ◽  
pp. 303-374 ◽  
Author(s):  
Raphaël Ponge ◽  
Hang Wang
Keyword(s):  
Noncommutative Geometry ◽  
Conformal Geometry ◽  
Chern Character ◽  
Index Theorem ◽  
Equivariant Index

The Chern character of a transversally elliptic symbol and the equivariant index

1996 ◽  
Vol 124 (1-3) ◽  
pp. 11-49 ◽  
Author(s):  
Nicole Berline ◽  
Michèle Vergne
Keyword(s):  
Chern Character ◽  
Equivariant Index

scholarly journals The Noncommutative Infinitesimal Equivariant Index Formula

2014 ◽  
Vol 14 (1) ◽  
pp. 73-102 ◽  
Author(s):  
Yong Wang
Keyword(s):  
Heat Kernel ◽  
Explicit Formula ◽  
Noncommutative Geometry ◽  
Index Formula ◽  
Equivariant Index ◽  
Heat Kernel Asymptotics

AbstractIn this paper, we establish an infinitesimal equivariant index formula in the noncommutative geometry framework using Greiner's approach to heat kernel asymptotics. An infinitesimal equivariant index formula for odd dimensional manifolds is also given. We define infinitesimal equivariant eta cochains, prove their regularity and give an explicit formula for them. We also establish an infinitesimal equivariant family index formula and introduce the infinitesimal equivariant eta forms as well as compare them with the equivariant eta forms.

scholarly journals The equivariant index theorem in entire cyclic cohomology

2008 ◽  
Vol 3 (2) ◽  
pp. 261-307 ◽  
Author(s):  
Denis Perrot
Keyword(s):  
Homology Class ◽  
Chern Character ◽  
Index Theorem ◽  
Complete Riemannian Manifold ◽  
Cyclic Cohomology ◽  
Convolution Algebra ◽  
Cup Product ◽  
Compactly Supported Functions ◽  
Assembly Map ◽  
Banach Completion

AbstractLet G be a locally compact group acting smoothly and properly by isometries on a complete Riemannian manifold M, with compact quotient G\M. There is an assembly map which associates to any G-equivariant K-homology class on M, an element of the topological K-theory of a suitable Banach completion of the convolution algebra of continuous compactly supported functions on G. The aim of this paper is to calculate the composition of the assembly map with the Chern character in entire cyclic homology . We prove an index theorem reducing this computation to a cup-product in bivariant entire cyclic cohomology. As a consequence we obtain an explicit localization formula which includes, as particular cases, the equivariant Atiyah-Segal-Singer index theorem when G is compact, and the Connes-Moscovici index theorem for G-coverings when G is discrete. The proof is based on the bivariant Chern character introduced in previous papers.

scholarly journals TWISTED ENTIRE CYCLIC COHOMOLOGY, J-L-O COCYCLES AND EQUIVARIANT SPECTRAL TRIPLES

2004 ◽  
Vol 16 (05) ◽  
pp. 583-602 ◽  
Author(s):  
DEBASHISH GOSWAMI
Keyword(s):  
Spectral Data ◽  
Noncommutative Geometry ◽  
Positive Operator ◽  
Chern Character ◽  
Cyclic Cohomology ◽  
Spectral Triple ◽  
Usual Sense ◽  
Spectral Triples ◽  
Definition Of ◽  
Made In

We study the "quantized calculus" corresponding to the algebraic ideas related to "twisted cyclic cohomology" introduced in [12]. With very similar definitions and techniques as those used in [9], we define and study "twisted entire cyclic cohomology" and the "twisted Chern character" associated with an appropriate operator theoretic data called "twisted spectral data", which consists of a spectral triple in the conventional sense of noncommutative geometry [1] and an additional positive operator having some specified properties. Furthermore, it is shown that given a spectral triple (in the conventional sense) which is equivariant under the (co-) action of a compact matrix pseudogroup, it is possible to obtain a canonical twisted spectral data and hence the corresponding (twisted) Chern character, which will be invariant (in the usual sense) under the (co-)action of the pseudogroup, in contrast to the fact that the Chern character coming from the conventional noncommutative geometry need not to be invariant. In the last section, we also try to detail out some remarks made in [3], in the context of a new definition of invariance satisfied by the conventional (untwisted) cyclic cocycles when lifted to an appropriate larger algebra.

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