On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C&lt;sup&gt;*&lt;/sup&gt;-Dynamical Systems

Fathizadeh, Farzad; Gabriel, Olivier

doi:10.3842/sigma.2016.016

Cited by 13 publications

(21 citation statements)

References 57 publications

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“…Therefore, for simplicity, we use these unitary maps to transfer the operator d + d * to an unbounded operator D acting on the Hilbert space H that is the direct sum of all H k,0 . We can now state the following result from [29].…”

Section: 3mentioning

confidence: 92%

“…Conformally twisted spectral triples for C * -dynamical systems. The example in §7.3 inspired the construction of twisted spectral triples for general ergodic C * -dynamical systems in [29]. The Dirac operator used in this work, following more closely the geometric approach taken originally in [9], is the analog of the de Rham operator.…”

Section: 3mentioning

confidence: 99%

“…The Dirac operator used in this work, following more closely the geometric approach taken originally in [9], is the analog of the de Rham operator. An important reason for this choice is that an important goal in [29] was to confirm the validity of the analog of the Chern-Gauss-Bonnet theorem in the vast setting of ergodic C * -dynamical systems.…”

Section: 3mentioning

confidence: 99%

“…Relevant to our discussion is indeed the Fredholm index of the de Rham operator d + d * , where d is the de Rham exterior derivative and d * is its adjoint with respect to the metric on the differential forms induced by the Riemannian metric. The Fredholm index of d + d * should be calculated when the operator is viewed as a map from the direct sum of all even differential forms to the direct sum of all odd differential forms on M : In [29], this spectral approach is taken to show that the analog of the Chern-Gauss-Bonnet theorem can be established for ergodic C * -dynamical systems. Let us consider the set up and the constructions presented in §7.4 for a C * -algebra A with an ergodic action of a compact Lie group G of dimension n. Then, one of the main results proved in [29] is the following statement.…”

Section: 3mentioning

confidence: 99%

“…The Fredholm index of d + d * should be calculated when the operator is viewed as a map from the direct sum of all even differential forms to the direct sum of all odd differential forms on M : In [29], this spectral approach is taken to show that the analog of the Chern-Gauss-Bonnet theorem can be established for ergodic C * -dynamical systems. Let us consider the set up and the constructions presented in §7.4 for a C * -algebra A with an ergodic action of a compact Lie group G of dimension n. Then, one of the main results proved in [29] is the following statement. Here, d is given by (40), h = h * ∈ A ∞ is the element that was used to implement with e h a conformal perturbation of the metric, and the Hilbert space H k,h is the completion of the k-differential forms Ω k (A, G) with respect to the perturbed metric.…”

Section: 3mentioning

confidence: 99%

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Curvature in noncommutative geometry

Fathizadeh

Khalkhali

2019

Advances in Noncommutative Geometry

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Dedicated to Alain Connes with admiration, affection, and much appreciation Contents 1. Introduction 2. Curvature in noncommutative geometry 2.1. A brief history of curvature 2.2. Laplace type operators and Gilkey's theorem 2.3. Noncommutative Chern-Weil theory 2.4. From spectral geometry to spectral triples 3. Pseudodifferential calculus and heat expansion 3.1. Classical pseudodifferential calculus 3.2. Small-time heat kernel expansion 3.3. Pseudodifferential calculus and heat kernel expansion for noncommutative tori 4. Gauss-Bonnet theorem and curvature for noncommutative 2-tori 4.1. Scalar curvature and Gauss-Bonnet theorem for T 2 θ 4.2. The Laplacian on (1, 0)-forms on T 2θ with curved metric 5. Noncommutative residues for noncommutative tori and curvature of noncommutative 4-tori 5.1. Noncommutative residues 5.2. Scalar curvature of the noncommutative 4-torus 6.The Riemann curvature tensor and the term a 4 for noncommutative tori 6.1. Functional relations 6.2. A partial differential system associated with the functional relations 6.3. Action of cyclic groups in the differential system, invariant expressions and simple flow of the system 6.4. Gradient calculations leading to functional relations 6.5. The term a 4 for non-conformally flat metrics on noncommutative four tori 7. Twisted spectral triples and Chern-Gauss-Bonnet theorem for ergodic C * -dynamical systems 7.1. Twisted spectral triples 7.2. Conformal perturbation of a spectral triple 7.3. Conformal perturbation of the flat metric on T 2 θ 7.4. Conformally twisted spectral triples for C * -dynamical systems 7.5. The Chern-Gauss-Bonnet theorem for C * -dynamical systems

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Section: 3mentioning

confidence: 92%

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

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Curvature in noncommutative geometry

Fathizadeh

Khalkhali

2019

Advances in Noncommutative Geometry

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Induced C*-Complexes in Metaplectic Geometry

Krýsl

2018

Commun. Math. Phys.

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For a symplectic manifold admitting a metaplectic structure and for a Kuiper map, we construct a complex of differential operators acting on exterior differential forms with values in the dual of Kostant's symplectic spinor bundle. Defining a Hilbert C * -structure on this bundle for a suitable C * -algebra, we obtain an elliptic C * -complex in the sense of Mishchenko-Fomenko. Its cohomology groups appear to be finitely generated projective Hilbert C * -modules. The paper can serve as a guide for handling of differential complexes and PDEs on Hilbert bundles. * E-mail address: Svatopluk.Krysl@mff.cuni.cz † The author thanks for financial supports from the founding No. 17-01171S granted by the Czech Science Foundation. We thank to the anonymous reviewer for his comments and suggestions.[14]. In [66] and [61], C * -compact perturbations of the complexes' differentials are allowed. For the infinite rank bundles, one cannot expect that, in general, the space of harmonic elements represents the cohomology groups homeomorphically (quotient topology) and linearly (quotient projections). Therefore, it makes sense to find conditions when this happens. This was investigated by the author in the past years (see [35], [36] or an overview in [37]).In the connection to sheaf cohomology, Banach and Fréchet bundles are studied in the papers of Illusie [25] and Röhrl [57]. To present a sample of the broad context (foremost connected to C * -algebras, K-theory and homological algebra) in which such bundles are considered, let us mention the papers of Maeda, Rosenberg [43], Freed, Lott [16], Larraín-Hubach [39], the author [36] and Fathizadeh, Gabriel [12]. There are also works in which holomorphic Banach bundles and their sheaf cohomology groups are treated. See Lempert [40] and Kim [27]. A further reason for an investigation of these complexes might originate in ), which we do not touch here explicitly, although we give a modest interpretation of the introduced structures in the realm of quantum theory.The aim of our paper is to study elliptic complexes for the case of infinite rank Hilbert C * -bundles which are induced by Lie group representations, and to show as well how the theory of connections and appropriately generalized elliptic complexes apply in this situation. In order to make the considered situation not too general, we choose a specific Lie group representation (the Segal-Shale-Weil representation of the metaplectic group). We hope that the reader may follow the text easier. Our further aim is to find examples for the theory of Mishchenko and Fomenko ([14]) that would not be trivial or covered by the theory developed around the classical Atiyah-Singer index theorem, and to show how the Hodge theory for certain C * -bundles (derived in [35] and [36]) make us able to describe the cohomology of elliptic complexes on Hilbert bundles easily.Let (V, ω 0 ) be a symplectic vector space and L be a Lagrangian subspace of it. The Segal-Shale-Weil representation is a non-trivial representation of the connected double cover G...

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