The operator $\sqrt{-1}\widehat{c}(V)(d+\delta)$ and the Kastler-Kalau-Walze type theorems

TL;DR

This paper derives Lichnerowicz formulas and proves Kastler-Kalau-Walze type theorems for specific differential operators on 3- and 4-dimensional manifolds, extending geometric analysis in this context.

Contribution

It introduces new Lichnerowicz formulas and establishes Kastler-Kalau-Walze theorems for these operators on manifolds with and without boundary.

Findings

Lichnerowicz type formulas for the operators

Kastler-Kalau-Walze theorems on 3- and 4-dimensional manifolds

Results applicable to manifolds with boundary

Abstract

In this paper, we obtain two Lichnerowicz type formulas for the operators $\sqrt{-1}\widehat{c}(V)(d+\delta)$ and $-\sqrt{-1}(d+\delta)\widehat{c}(V)$. And we give the proof of Kastler-Kalau-Walze type theorems for the operators $\sqrt{-1}\widehat{c}(V)(d+\delta)$ and $-\sqrt{-1}(d+\delta)\widehat{c}(V)$ on 3,4-dimensional oriented compact manifolds with (resp.without) boundary.