Connes-Hida calculus in index theory. (English) Zbl 1106.58012

Léandre, Rémi

Zambrini, Jean-Claude (ed.), XIVth international congress on mathematical physics (ICMP 2003), Lisbon, Portugal, 28 July – 2 August 2003. Selected papers based on the presentation at the conference. Hackensack, NJ: World Scientific (ISBN 981-256-201-X/hbk). 493-497 (2005).

From the introduction: There are two ways to see the relations between the index theorem and the algebraic properties of the complex auxiliary bundle associated to a Dirac operator:
– The first one uses the Bismut-Chern character over the free loop space, associated to the equivariant cohomology of the free loop space.
– The second one uses the cyclic complex associated to the algebra of complex valued functions on a manifold. It enters in the heart of noncommutative differential geometry of Connes.
The goal of this communication is to apply tools of Hida calculus in order to understand the role of the auxiliary bundle in index theory in these two aspects.
For the entire collection see [Zbl 1089.81005].

MSC:

58J20	Index theory and related fixed-point theorems on manifolds
46L52	Noncommutative function spaces
46L87	Noncommutative differential geometry
58J42	Noncommutative global analysis, noncommutative residues
58J65	Diffusion processes and stochastic analysis on manifolds
60H40	White noise theory