The generalized noncommutative residue and the Kastler-Kalau-Walze type theorem

TL;DR

This paper extends the concept of noncommutative residue for Dirac operators and proves Kastler-Kalau-Walze type theorems for 4D and 6D manifolds, with and without boundary.

Contribution

It introduces a generalized noncommutative residue for Dirac operators and establishes new Kastler-Kalau-Walze type theorems in higher dimensions.

Findings

Generalized noncommutative residue defined for Dirac operators

Proved Kastler-Kalau-Walze type theorems in 4D and 6D cases

Results applicable to manifolds with and without boundary

Abstract

In this paper, we define the generalized noncommutative residue of the Dirac operator. And we give the proof of Kastler-Kalau-Walze type theorems for the generalized noncommutative residue on 4-dimensional and 6-dimensional compact manifolds with (resp.without) boundary.