The Gauss-Bonnet Theorem for the noncommutative two torus

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Closest papers

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   The Gauss-Bonnet Theorem for Noncommutative Two Tori With a General Conformal Structure

   Farzad Fathizadeh, M. Khalkhali · 2010 · arXiv: Operator Algebras · preprint, not peer reviewed · cited 84 times · Not summarized yet

   Abstract

   In this paper we give a proof of the Gauss-Bonnet theorem of Connes and Tretkoff for noncommutative two tori $\mathbb{T}_{\theta}^2$ equipped with an arbitrary translation invariant complex structure. More precisely, we show that for any complex number $\tau$ in the upper half plane, representing the conformal class of a metric on $\mathbb{T}_{\theta}^2$, and a Weyl factor given by a positive invertible element $k \in C^{\infty}(\mathbb{T}_{\theta}^2)$, the value at the origin, $\zeta (0)$, of the spectral zeta function of the Laplacian $\triangle'$ attached to $(\mathbb{T}_{\theta}^2, \tau, k)$ is independent of $\tau$ and $k$.

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   Curved noncommutative torus and Gauss–Bonnet

   L. Dąbrowski, A. Sitarz · 2012 · Journal of Mathematical Physics · cited 55 times · Not summarized yet

   Abstract

   We study perturbations of the flat geometry of the noncommutative two-dimensional torus Tθ2 (with irrational θ). They are described by spectral triples (Aθ,H,D), with the Dirac operator D, which is a differential operator with coefficients in the commutant of the (smooth) algebra Aθ of Tθ. We show, up to the second order in perturbation, that the ζ-function at 0 vanishes and so the Gauss-Bonnet theorem holds. We also calculate first two terms of the perturbative expansion of the corresponding local scalar curvature.

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   Weyl’s Law and Connes’ Trace Theorem for Noncommutative Two Tori

   Farzad Fathizadeh, M. Khalkhali · 2011 · Letters in Mathematical Physics · cited 25 times · Not summarized yet

   Abstract

   We prove the analogue of Weyl’s law for a noncommutative Riemannian manifold, namely the noncommutative two torus $${\mathbb{T}_{\theta}^{2}}$$ equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed Laplacian on $${\mathbb{T}_{\theta}^{2}}$$ . We also prove the analogue of Connes’ trace theorem by showing that the Dixmier trace and a noncommutative residue coincide on pseudodifferential operators of order −2 on $${\mathbb{T}_{\theta}^{2}}$$ .

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   Einstein-Hilbert Action and on the Gauss-Bonnet Theorem for Riemannian Noncommutative Tori

   Javad Golipour · 2017 · arXiv: Differential Geometry · preprint, not peer reviewed · Not summarized yet

   Abstract

   We show the non-positivity of the Einstein-Hilbert action for conformal flat Riemannian metrics. The action vanishes only when the metric is constant flat. This recovers an earlier result of Fathizadeh-Khalkhali in the setting of spectral triples on noncommutative four-torus. Furthermore, computations of the gradient flow and the scalar curvature of this space based on modular operator are given. We also show the Gauss-Bonnet theorem for a parametrized class of non-diagonal metrics on noncommutative two-torus.

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   Gauss-Bonnet for matrix conformally rescaled Dirac

   M. Khalkhali, A. Sitarz · 2017 · Journal of Mathematics and Physics · cited 10 times · Not summarized yet

   Abstract

   We derive an explicit formula for the scalar curvature over a two-torus with a Dirac operator conformally rescaled by a globally diagonalizable matrix. We show that the Gauss-Bonnet theorem holds and extend the result to all Riemann surfaces with Dirac operators modified in the same way.

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   A noncommutative Tauberian theorem and Weyl asymptotics in noncommutative geometry

   E. Mcdonald, F. Sukochev, D. Zanin · 2022 · Letters in Mathematical Physics · cited 9 times · Not summarized yet
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   On Certain Spectral Invariants of Dirac Operators on Noncommutative Tori

   Ali Fathi, M. Khalkhali · 2015 · arXiv: Quantum Algebra · preprint, not peer reviewed · cited 6 times · Not summarized yet

   Abstract

   The spectral eta function for certain families of Dirac operators on noncommutative $3$-torus is considered and the regularity at zero is proved. By using variational techniques, we show that $\eta_{D}(0)$ is a conformal invariant. By studying the Laurent expansion at zero of $\text{TR} (|D|^{-z})$, the conformal invariance of $\zeta'_{|D|}(0)$ for noncommutative $3$-torus is proved. Finally, for the coupled Dirac operator, a local formula for the variation $\partial_A\eta_{D+A}(0)$ is derived which is the analogue of the so called induced Chern-Simons term in quantum field theory literature.

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   A Riemann–Roch theorem for the noncommutative two torus

   M. Khalkhali, Ali Moatadelro · 2013 · Journal of Geometry and Physics · cited 17 times · Not summarized yet

   Abstract

   Abstract We prove the analogue of the Riemann–Roch formula for the noncommutative two torus A θ = C ( T θ 2 ) equipped with an arbitrary translation invariant complex structure and a Weyl factor represented by a positive element k ∈ C ∞ ( T θ 2 ) . We consider a topologically trivial line bundle equipped with a general holomorphic structure and the corresponding twisted Dolbeault Laplacians. We define a spectral triple ( A θ , H , D ) that encodes the twisted Dolbeault complex of A θ and whose index gives the left hand side of the Riemann–Roch formula. Using Connes’ pseudodifferential calculus and heat equation techniques, we explicitly compute the b 2 terms of the asymptotic expansion of Tr…

Prior work it builds on

1. Close* Not relevant

   On the Non-commutative Torus of Real Dimension Two

   Paula B. Cohen · 1990 · cited 3 times · Not summarized yet
2. Related* Not relevant

   Noncommutative Geometry

   A. Lesniewski · 1997 · cited 5,429 times · Not summarized yet
3. Related Not relevant

   Type III and spectral triples

   A. Connes, H. Moscovici · 2006 · arXiv: Operator Algebras · preprint, not peer reviewed · cited 138 times · Not summarized yet

   Abstract

   We explain how a simple twisting of the notion of spectral triple allows to incorporate type III examples, such as those arising from the transverse geometry of codimension one foliations. Since the twisting of the commutators turns the usual hypertrace constructed out of the Dixmier trace into a twisted trace on the coordinate algebra, one would be tempted to interpret that as a manifestation of twisting at the level of cyclic cohomology, akin to that introduced by the authors in the context of Hopf cyclic cohomology. The main point of this note, besides giving simple natural examples of the general notion and developing the first basic steps of the theory, is to show that contrary to the i…

4. Related Not relevant

   The Spectral Action Principle

   A. Chamseddine, A. Connes · 1996 · Communications in Mathematical Physics · cited 796 times · Not summarized yet

   Abstract

   Abstract:We propose a new action principle to be associated with a noncommutative space . The universal formula for the spectral action is where is a spinor on the Hilbert space, is a scale and a positive function. When this principle is applied to the noncommutative space defined by the spectrum of the standard model one obtains the standard model action coupled to Einstein plus Weyl gravity. There are relations between the gauge coupling constants identical to those of SU(5) as well as the Higgs self-coupling, to be taken at a fixed high energy scale.

5. Related* Not relevant

   Quasi homomorphismes, cohomologie cyclique et positivité

   A. Connes, J. Cuntz · 1988 · Communications in Mathematical Physics · cited 62 times · Not summarized yet

Papers that cite it

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   Modular Curvature for Noncommutative Two-Tori

   A. Connes, H. Moscovici · 2011 · arXiv: Quantum Algebra · preprint, not peer reviewed · cited 141 times · Not summarized yet

   Abstract

   In this paper we investigate the curvature of conformal deformations by noncommutative Weyl factors of a flat metric on a noncommutative 2-torus, by analyzing in the framework of spectral triples functionals associated to perturbed Dolbeault operators. The analogue of Gaussian curvature turns out to be a sum of two functions in the modular operator corresponding to the non-tracial weight defined by the conformal factor, applied to expressions involving derivatives of the same factor. The first is a generating function for the Bernoulli numbers and is applied to the noncommutative Laplacian of the conformal factor, while the second is a two-variable function and is applied to a quadratic form…

2. Close Uses this paper in depth Not relevant

   Scalar Curvature for Noncommutative Four-Tori

   Farzad Fathizadeh, M. Khalkhali · 2013 · arXiv: Quantum Algebra · preprint, not peer reviewed · cited 64 times · Not summarized yet

   Abstract

   In this paper we study the curved geometry of noncommutative 4-tori $\mathbb{T}_\theta^4$. We use a Weyl conformal factor to perturb the standard volume form and obtain the Laplacian that encodes the local geometric information. We use Connes' pseudodifferential calculus to explicitly compute the terms in the small time heat kernel expansion of the perturbed Laplacian which correspond to the volume and scalar curvature of $\mathbb{T}_\theta^4$. We establish the analogue of Weyl's law, define a noncommutative residue, prove the analogue of Connes' trace theorem, and find explicit formulas for the local functions that describe the scalar curvature of $\mathbb{T}_\theta^4$. We also study the an…

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   An Asymmetric Noncommutative Torus

   L. Dąbrowski, A. Sitarz · 2014 · Symmetry Integrability and Geometry-methods and Applications · cited 55 times · Not summarized yet

   Abstract

   We introduce a family of spectral triples that describe the curved noncommuta- tive two-torus. The relevant family of new Dirac operators is given by rescaling one of two terms in the flat Dirac operator. We compute the dressed scalar curvature and show that the Gauss{Bonnet theorem holds (which is not covered by the general result of Connes and Moscovici).

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   Wodzicki residue and minimal operators on a noncommutative 4-dimensional torus

   A. Sitarz · 2013 · Journal of Pseudo-Differential Operators and Applications · cited 19 times · Not summarized yet

   Abstract

   We compute the Wodzicki residue of the inverse of a conformally rescaled Laplace operator over a 4-dimensional noncommutative torus. We show that the straightforward generalisation of the Laplace-Beltrami operator to the noncommutative case is not the minimal operator.

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   Spectral geometry of functional metrics on noncommutative tori

   Asghar Ghorbanpour, M. Khalkhali · 2018 · Communications in Number Theory and Physics · cited 4 times · Not summarized yet

   Abstract

   We introduce a new family of metrics, called functional metrics, on noncommutative tori and study their spectral geometry. We define a class of Laplace type operators for these metrics and study their spectral invariants obtained from the heat trace asymptotics. A formula for the second density of the heat trace is obtained. In particular, the scalar curvature density and the total scalar curvature of functional metrics are explicitly computed in all dimensions for certain classes of metrics including conformally flat metrics and twisted product of flat metrics. Finally a Gauss-Bonnet type theorem for a noncommutative two torus equipped with a general functional metric is proved.