Spectral action for scalar perturbations of Dirac operators

TL;DR

This paper analyzes the spectral action for odd-dimensional spin manifolds with scalar perturbations of the Dirac operator, computing key coefficients and exploring specific cases including spheres, torsion, and quantum groups.

Contribution

It provides explicit calculations of spectral action coefficients for scalar-perturbed Dirac operators and extends analysis to noncommutative geometries like SU_q(2).

Findings

Computed first two Gilkey-de Witt coefficients for scalar perturbations.

Explicitly analyzed the case of n-spheres with symmetric Dirac operators.

Studied the noncommutative case of SU_q(2) with torsion perturbations.

Abstract

We investigate the leading terms of the spectral action for odd-dimensional Riemannian spin manifolds with the Dirac operator perturbed by a scalar function. We calculate first two Gilkey-de Witt coefficients and make explicit calculations for the case of $n$-spheres with a completely symmetric Dirac. In the special case of dimension 3, when such perturbation corresponds to the completely antisymmetric torsion we carry out the noncommutative calculation following Chamseddine and Connes and study the case of $SU_q(2)$.