Noncommutative Residue and sub-Dirac Operators for Foliations

TL;DR

This paper introduces lower dimensional volumes linked to sub-Dirac operators in foliations, proves Kastler-Kalau-Walze type theorems, and connects these results to gravitational actions in specific cosmological models.

Contribution

It defines new geometric invariants for foliations and extends fundamental theorems to this setting, providing insights into gravitational actions in Robertson-Walker spaces.

Findings

Computed lower dimensional volumes for certain foliations

Proved Kastler-Kalau-Walze type theorems for foliations with/without boundary

Connected geometric invariants to gravitational actions in cosmology

Abstract

In this paper, we define lower dimensional volumes associated to sub-Dirac operators for foliations. In some cases, we compute these lower dimensional volumes. We also prove the Kastler-Kalau-Walze type theorems for foliations with or without boundary. As a corollary, we give an explanation of the gravitational action for the Robertson-Walker space $[a, b]\times_{f}M^{3}$.