On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C∗-Dynamical Systems
Research output: Contribution to journal › Journal article › Research › peer-review
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 coef ficients in the smooth subalgebra A ⊂ A as noncommutative dif ferential
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.
Original language | English |
---|---|
Article number | 016 |
Journal | Symmetry, Integrability and Geometry: Methods and Applications |
Volume | 12 |
Number of pages | 21 |
ISSN | 1815-0659 |
DOIs | |
Publication status | Published - 2016 |
- Faculty of Science - C*-dynamical systems, ergodic action, invariant state, conformal factor, Hodge-de Rham operator, noncommutative de Rham complex, Euler characteristic, Chern-Gauss-Bonnet theorem, spectral triple, spectral dimension
Research areas
ID: 155425012