The equivariant noncommutative Atiyah–Patodi–Singer index theorem

Equivariant KK-theory and noncommutative index theory

Equivariant noncommutative index on braided sphere

To some Hecke symmetries (i.e. Yang-Baxter braidings of Hecke type) we associate ”noncommutative varieties” called braided spheres. An example of such a variety is the Podles’ nonstandard quantum sphere. On any braided sphere we introduce and compute an ”equivariant” analogue of Connes’ noncommutative index. In contrast with the Connes’ construction our version of equivariant NC index is based ...

Index Theorem in Finite Noncommutative Geometry

Noncommutative (NC) geometry is interesting since it is related to string theories and matrix models,1)–3) and it may capture some nature of quantum gravity. It is also a candidate for a new regularization of quantum field theory.4) Study about topological aspects of gauge theory on it is important since compactification of extra dimensions with nontrivial index in string theory can realize chi...

The equivariant index theorem in entire cyclic cohomology

Let G be a locally compact group acting smoothly and properly by isometries on a complete Riemannian manifoldM , with compact quotient GnM . There is an assembly map W K .M/ ! K .B/ which associates to any Gequivariant K-homology class onM , an element of the topological K-theory of a suitable Banach completion B of the convolution algebra of continuous compactly supported functions on G. The a...

Noncommutative topology and the world's simplest index theorem.

In this article we outline an approach to index theory on the basis of methods of noncommutative topology. We start with an explicit index theorem for second-order differential operators on 3-manifolds that are Fredholm but not elliptic. This low-brow index formula is expressed in terms of winding numbers. We then proceed to show how it is derived as a special case of an index theorem for hypoe...