Curvature of the Determinant Line Bundle for the Noncommutative Two Torus

Fathi, Ali; Ghorbanpour, Asghar; Khalkhali, Masoud

doi:10.1007/s11040-016-9234-9

Cited by 15 publications

(34 citation statements)

References 25 publications

(53 reference statements)

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“…So far we have seen that these analytic techniques, suitably modified and enhanced, has been quite successful in dealing with scalar and Ricci curvature. Along this idea, in [25] the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus is computed. Following Quillen's original construction for Riemann surfaces [53] and using zeta regularized determinant of Laplacians, the determinant line bundle is endowed with a natural Hermitian metric.…”

Section: Curvature Of the Determinant Line Bundlementioning

confidence: 99%

“…In this section we shall recall and comment on results obtained in [25] on the curvature of the determinant line bundle on a noncommutative torus. 11.1.…”

Section: Curvature Of the Determinant Line Bundlementioning

confidence: 99%

“…In [53] Quillen shows that for families of Cauchy-Riemann operators on a Riemann surface one can correct the Hilbert space metric by multiplying it by zeta regularized determinant and in this way one obtains a smooth Hermitian metric on the induced determinant line bundle. In [25] a similar construction for the noncommutative two torus is given as we explain later in this section. 11.2.…”

Section: Curvature Of the Determinant Line Bundlementioning

confidence: 99%

“…The canonical trace and noncommutative residue. To carry the calculations, an analogue of the canonical trace of [45] for the noncommutative torus is constructed in [25]. First we need to extend our original algebra of pseudodifferential operatprs to classical pseudodifferential operators.…”

Section: Curvature Of the Determinant Line Bundlementioning

confidence: 99%

“…We provide the explicit formulas for a few of the functions appearing in (25), and we refer the reader to [17] for the remaining functions, most of which have lengthy expressions. We have for example: Figure 6.…”

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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: Curvature Of the Determinant Line Bundlementioning

confidence: 99%

“…In this section we shall recall and comment on results obtained in [25] on the curvature of the determinant line bundle on a noncommutative torus. 11.1.…”

Section: Curvature Of the Determinant Line Bundlementioning

confidence: 99%

Section: Curvature Of the Determinant Line Bundlementioning

confidence: 99%

Section: Curvature Of the Determinant Line Bundlementioning

confidence: 99%

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Curvature in noncommutative geometry

Fathizadeh

Khalkhali

2019

Advances in Noncommutative Geometry

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On the scalar curvature for the noncommutative four torus

Fathizadeh

2015

Journal of Mathematical Physics

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The scalar curvature for noncommutative four tori T 4 Θ , where their flat geometries are conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3-sphere. This method is more convenient since it does not require the rearrangement lemma and it is advantageous as it explains the simplicity of the final functions of one and two variables, which describe the curvature with the help of a modular automorphism. In particular, it readily allows to write the function of two variables as the sum of a finite difference and a finite product of the one variable function. The curvature formula is simplified for dilatons of the form sp, where s is a real parameter and p ∈ C ∞ (T 4 Θ ) is an arbitrary projection, and it is observed that, in contrast to the two dimensional case studied by Connes and Moscovici, J. Am. Math. Soc. 27(3), 639-684 (2014), unbounded functions of the parameter s appear in the final formula. An explicit formula for the gradient of the analog of the Einstein-Hilbert action is also calculated. C 2015 AIP Publishing LLC. [http://dx

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