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

• Опубліковано в: : Symmetry, Integrability and Geometry: Methods and Applications
• Дата: 2016
• Автори: Fathizadeh, F., Gabriel, O.
• Формат: Стаття
• Мова: English
• Опубліковано: Інститут математики НАН України 2016
• Онлайн доступ: https://nasplib.isofts.kiev.ua/handle/123456789/147426
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C*-Dynamical Systems / F. Fathizadeh, O. Fathizadeh // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 60 назв. — англ.

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.