Dirac operators with torsion, spectral Einstein functionals and the noncommutative residue

TL;DR

This paper extends spectral functionals to include torsion in Dirac operators, relating them to noncommutative residues on manifolds with boundary, advancing the understanding of spectral geometry with torsion.

Contribution

It introduces new spectral functionals incorporating torsion, extending previous work to the noncommutative setting and connecting them with noncommutative residues.

Findings

New spectral functionals with torsion are constructed.

The relationship between these functionals and noncommutative residues is established.

The approach generalizes spectral functionals to manifolds with boundary.

Abstract

Recently Dabrowski etc. \cite{DL} obtained the metric and Einstein functionals by two vector fields and Laplace-type operators over vector bundles, giving an interesting example of the spinor connection and square of the Dirac operator. Pf$\ddot{a}$ffle and Stephan \cite{PS1} considered orthogonal connections with arbitrary torsion on compact Riemannian manifolds and computed the spectral action. Motivated by the spectral functionals and Dirac operators with torsion, we give some new spectral functionals which is the extension of spectral functionals to the noncommutative realm with torsion, and we relate them to the noncommutative residue for manifolds with boundary. Our method of producing these spectral functionals is the noncommutative residue and Dirac operators with torsion.