The Connes-Chamseddine cycle and the noncommutative integral

TL;DR

This paper extends the study of the Connes-Chamseddine cycle from 4-dimensional to 6-dimensional manifolds using noncommutative integrals, providing new computations and insights in noncommutative geometry.

Contribution

It introduces a novel approach to analyze the Connes-Chamseddine cycle on higher-dimensional manifolds via noncommutative integrals, expanding the framework beyond previous 4D cases.

Findings

Computed noncommutative integrals on n-dimensional manifolds.

Derived the Connes-Chamseddine cycle for 6-dimensional manifolds.

Established a normal coordinated method for integral calculations.

Abstract

In [5], Connes and Chamseddine defined a cycle in the general framework of noncommutative geometry. They computed this cycle for the Dirac operator on 4-dimensioanl manifolds. We propose a way to study the Connes-Chamseddine cycle from the viewpoint of the noncommutative integral on 6-dimensional manifolds in this paper. Furthermore, we compute several interesting noncommutative integral defined in [8] by the normal coodinated way on n-dimensional manifolds. As a corollary, the Connes-Chamseddine cycle on 6-dimensional manifolds is obtained.