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Published 2016 | Published + Submitted
Journal Article Open

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


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 g of G is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalgebra A ⊂ 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 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.

Additional Information

The authors retain the copyright for their papers published in SIGMA under the terms of the Creative Commons Attribution-ShareAlike License. Received October 26, 2015, in final form February 04, 2016; Published online February 10, 2016. The authors thank the Hausdorff Research Institute for Mathematics (HIM) for their hospitality and support during the trimester program on Noncommutative Geometry and its Applications in 2014, where the present work was partially carried out. They also thank the anonymous referees for their constructive feedback. Parts of this article were obtained and written while the second author was working as a postdoc at the University of Glasgow. He would like to thank C. Voigt for enabling his stay in Scotland.

Attached Files

Published - sigma16-016.pdf

Submitted - 1506.07913v4.pdf


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August 22, 2023
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