On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for $C^*$-Dynamical Systems

TL;DR

This paper extends the Chern-Gauss-Bonnet theorem to $C^*$-dynamical systems by constructing conformally perturbed spectral triples and establishing a Hodge decomposition, linking topological invariants to spectral properties.

Contribution

It introduces a conformal perturbation framework for spectral triples in noncommutative geometry and proves a Hodge decomposition theorem in this setting.

Findings

Established a Hodge decomposition theorem for conformally perturbed metrics.

Constructed spectral triples and twisted spectral triples with preserved summability.

Demonstrated conformal invariance of the Euler characteristic in noncommutative geometry.

Abstract

The analog of the Chern-Gauss-Bonnet theorem is studied for a $C^*$-dynamical system consisting of a $C^*$-algebra $A$ equipped with an ergodic action of a compact Lie group $G$. The structure of the Lie algebra $\mathfrak{g}$ of $G$ is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalgebra $\mathcal{A} \subset A$ as noncommutative differential forms on the dynamical system. We conformally perturb the standard metric, which is associated with the unique $G$-invariant state on $A$, by means of a Weyl conformal factor given by a positive invertible element of the algebra, and consider the Hermitian structure that it induces on the complex. A Hodge decomposition theorem is proved, which allows us to relate the Euler characteristic of the complex to the index properties of a Hodge-de Rham operator for the perturbed metric. This operator, which is shown to be selfadjoint, is a key ingredient in our construction of a spectral triple on $\mathcal{A}$ and a twisted spectral triple on its opposite algebra. The conformal invariance of the Euler characteristic is interpreted as an indication of the Chern-Gauss-Bonnet theorem in this setting. The spectral triples encoding the conformally perturbed metrics are shown to enjoy the same spectral summability properties as the unperturbed case.