Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 016, 21 pages      arXiv:1506.07913      https://doi.org/10.3842/SIGMA.2016.016

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

Farzad Fathizadeh a and Olivier Gabriel b
a) Department of Mathematics, Mail Code 253-37, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA
b) University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark

Received October 26, 2015, in final form February 04, 2016; Published online February 10, 2016

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.

Key words: $C^*$-dynamical systems; ergodic action; invariant state; conformal factor; Hodge-de Rham operator; noncommutative de Rham complex; Euler characteristic; Chern-Gauss-Bonnet theorem; ordinary and twisted spectral triples; unbounded selfadjoint operators; spectral dimension.

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