The Ricci Flow on Noncommutative Two-Tori

Bhuyain, Tanvir Ahamed; Marcolli, Matilde

doi:10.1007/s11005-012-0550-0

Cited by 25 publications

(57 citation statements)

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“…First, the final one and two variable functions in the curvature formulas are significantly simple, which indicates an enormous amount of cancellations in the algebraically lengthy formulas that involve hundreds of terms with numerous functions from the rearrangement lemma involved [10,9,1,17]. Second, the function of two variables for the curvature of T 2 θ can be recovered from the one variable function [9] by finite differences.…”

Section: Discussionmentioning

confidence: 99%

“…Also, the explicit computation of the gradient of the analog of the Einstein-Hilbert action for T 4 Θ prepares the ground for further studies of the natural associated geometric flows in this context, cf. [1,9].…”

Section: Discussionmentioning

confidence: 99%

“…Following these works, the local differential geometry of noncommutative tori equipped with curved metrics has received considerable attention in recent years [1,16,11,15,12]. See also [29,13].…”

mentioning

confidence: 99%

“…This is overcome by the rearrangement lemma [10,9] (cf. [1,17,23]), which uses the modular automorphism of the state implementing the conformal perturbation and delicate Fourier analysis to reorder the integrands and computes the integrals explicitly. The integrals are then expressed as somewhat complicated functions of the modular automorphism acting on relevant elements of the C * -algebra.…”

mentioning

confidence: 99%

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Scalar curvature for the noncommutative two torus

Fathizadeh¹,

Khalkhali²

2013

J. Noncommut. Geom.

139

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Abstract. The scalar curvature for the noncommutative four torus T 4 Θ , where its flat geometry is 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 A. Connes and H. Moscovici, 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.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Scalar curvature for the noncommutative two torus

Fathizadeh¹,

Khalkhali²

2013

J. Noncommut. Geom.

139

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“…More recently, it appears in the study of the contribution of black holes in Euclidean quantum gravity [16,15]. In [1], the author describes an appropriate analog of Hamilton's Ricci flow for the noncommutative two tori, which is the prototype example of noncommutative manifolds. It is still an interesting question to find more ways to define the Ricci flow in noncommutative geometry.…”

Section: Introductionmentioning

confidence: 99%

Heat equation in a model matrix geometry

2015

Comptes Rendus. Mathématique

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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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The Ricci Flow on Noncommutative Two-Tori

Abstract: Abstract. In this paper we construct a version of Ricci flow for noncommutative 2-tori, based on a spectral formulation in terms of the eigenvalues and eigenfunction of the Laplacian and recent results on the Gauss-Bonnet theorem for noncommutative tori.

Cited by 25 publications

References 44 publications

Scalar curvature for the noncommutative two torus

Scalar curvature for the noncommutative two torus

Heat equation in a model matrix geometry

Curvature in noncommutative geometry

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