The Gauss-Bonnet Theorem for Noncommutative Two Tori With a General Conformal Structure
By Farzad Fathizadeh, Masoud Khalkhali. 2010
84 citations recorded by the source corpus.
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 $k$.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/b86692dd0015fc093a3cc7f9b8d0a73347067450
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In Geometry and Topology
- Masoud Khalkhali
- Farzad Fathizadeh
- Alain Connes
- H. Moscovici
- Marius Junge
- Javier Parcet
- V. Ginzburg
- B. Rangipour
In Mathematical Physics
- Javier Parcet
- Masoud Khalkhali
- Marius Junge
- Farzad Fathizadeh
- A. Gonz'alez-P'erez
- H. Moscovici
- Alain Connes
- Quanhua Xu
Citations in the corpus
- Singular integrals in quantum Euclidean spaces (2017, background): “Other applications in the context of quantum tori include a well-established elliptic operator theory [17], the Gauss-Bonnet theorem for 2D quantum tori [23, 28] and recent results on the local differential geometry of non-flat noncommutative tori [6, 22].”
- Modular Curvature for Noncommutative Two-Tori (2011, background): “The obtained terms are, except for this nuance, similar to those of [14] and [15], but for the operator (0,1) φ there is a nontrivial term T0,1 which is given by T0,1 = ir (2)b0kδ1(k)b0kδ2(k) + ir (2)b0kδ1(k)b0δ2(k)k − ir(4)b0kδ1(k)k(2)b(2)0kδ2(k) − ir(4)b0kδ1(k)k(2)b(2)0δ2(k)k − ir(2)b0kδ2(k)b0kδ1(”
- Scalar Curvature for Noncommutative Four-Tori (2013, methodology): “One can use Connes’ pseudodifferential calculus [5] to apply similar heat kernel techniques in order to compute geometric invariants of noncommutative spaces [12, 11, 13, 14, 15].”