On Perrot's index cocycles

TL;DR

This paper simplifies Perrot's algebraic construction of index cocycles, deriving the Todd class from a cyclic cocycle without using traditional analytic methods, introducing new algebraic tools and traces.

Contribution

It presents a simplified, algebraic version of Perrot's index cocycle construction, replacing analytic components with algebraic series and a novel trace, offering a new approach to index theory.

Findings

Recovering the Todd class from a cyclic cocycle algebraically

Replacing heat kernel with series expansion in the construction

Introducing a new trace that replaces the operator trace

Abstract

We shall present a simplified version of a construction due to Denis Perrot that recovers the Todd class of the complexified tangent bundle from a JLO-type cyclic cocycle. The construction takes place within an algebraic framework, rather than the customary functional-analytic framework for the JLO theory. The series expansion for the exponential function is used in place of the heat kernel from the functional-analytic theory; the Dirac operator chosen is far from elliptic; and a remarkable new trace discovered by Perrot replaces the operator trace. In its full form Perrot's theory constitutes a wholly new approach to index theory. The account presented here covers most but not all of this approach.