On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C∗-Dynamical Systems
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On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C∗-Dynamical Systems. / Fathizadeh, Farzad ; Gabriel, Olivier.
I: Symmetry, Integrability and Geometry: Methods and Applications, Bind 12, 016, 2016.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C∗-Dynamical Systems
AU - Fathizadeh, Farzad
AU - Gabriel, Olivier
PY - 2016
Y1 - 2016
N2 - 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.
AB - 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.
KW - Faculty of Science
KW - C-dynamical systems
KW - ergodic action
KW - invariant state
KW - conformal factor
KW - Hodge-de Rham operator
KW - noncommutative de Rham complex
KW - Euler characteristic
KW - Chern-Gauss-Bonnet theorem
KW - spectral triple
KW - spectral dimension
U2 - 10.3842/SIGMA.2016.016
DO - 10.3842/SIGMA.2016.016
M3 - Journal article
VL - 12
JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
SN - 1815-0659
M1 - 016
ER -
ID: 155425012