A Kastler-Kalau-Walze type theorem and the spectral action for perturbations of Dirac operators on manifolds with boundary

TL;DR

This paper proves a Kastler-Kalau-Walze type theorem for perturbed Dirac operators on manifolds with boundary, providing operator-theoretic insights into gravitational action and computing spectral actions for specific perturbations.

Contribution

It establishes a new theorem for perturbed Dirac operators on manifolds with boundary and analyzes the spectral action for two-form perturbations.

Findings

Proved a Kastler-Kalau-Walze type theorem for perturbed Dirac operators.

Provided operator-theoretic explanations for gravitational action on boundary.

Computed spectral action for Dirac operators with two-form perturbations on 4D manifolds.

Abstract

In this paper, we prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-form perturbations on $4$-dimensional compact manifolds.