On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for $C^{\ast }$-Dynamical Systems

Abstract

The analog of the Chern-Gauss-Bonnet theorem is studied for a $C^{\ast }$-dynamical system consisting of a $C^{\ast }$-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.

References in corpus (8)

• Type III and spectral triples
• The Gauss-Bonnet Theorem for the noncommutative two torus
• Modular curvature and Morita equivalence
• Local index formula and twisted spectral triples
• Index theory for actions of compact Lie groups on C*-algebras
• Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras
• Semifinite spectral triples associated with graph C*-algebras
• Elliptic complexes over $C^{\ast }$-algebras of compact operators

Cited by in corpus (8)

• On twisting real spectral triples by algebra automorphisms
• The term a_4 in the heat kernel expansion of noncommutative tori
• Pseudodifferential calculus on noncommutative tori, I. Oscillating integrals
• A Scalar Curvature Formula For the Noncommutative 3-Torus
• Gauss-Bonnet for matrix conformally rescaled Dirac
• Pseudodifferential calculus on noncommutative tori, II. Main properties
• Induced $C^{\ast }$-complexes in metaplectic geometry
• A twisted local index formula for curved noncommutative two tori