Modular Curvature for Noncommutative Two-Tori

By Alain Connes, H. Moscovici. 2011

140 citations recorded by the source corpus.

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 in the first derivatives of the factor. Further outcomes of the paper include a variational proof of the Gauss-Bonnet theorem for noncommutative 2-tori, the modular analogue of Polyakov's conformal anomaly formula for regularized determinants of Laplacians, a conceptual understanding of the modular curvature as gradient of the Ray-Singer analytic torsion, and the proof using operator positivity that the scale invariant version of the latter assumes its extreme value only at the flat metric.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2b835f8c4e16ed5c17a8110105450d4a2697cbcf

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

1. Javier Parcet
2. Marius Junge
3. Masoud Khalkhali
4. A. Gonz'alez-P'erez
5. Farzad Fathizadeh
6. M. Lesch
7. H. Moscovici
8. Quanhua Xu

In Statistical and Nonlinear Physics

1. M. Lesch
2. Masoud Khalkhali
3. Farzad Fathizadeh
4. H. Moscovici
5. Marius Junge
6. Javier Parcet
7. Stephan Rosswog

Citations in the corpus

• Modular Curvature and Morita Equivalence (2015, methodology): “uivalently to the computation of the total curvature for such a metric, verifying the validity of the Gauss-Bonnet formula. The calculation of the full, not just the total, curvature was completed in [CoMo14] and also in [FaKh13], with the partial aid of different computer algebra systems. The resul”
• Singular integrals in quantum Euclidean spaces (2017, background): “ori 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]. Unfortunately, the work of Connes and his collaborators does not include Lp estimates for p”
• Scalar Curvature for Noncommutative Four-Tori (2013, methodology): “formation that are encoded in heat coecients, we refer the reader to [17] and the references therein. A crucial point that is used to dene and compute the scalar curvature for noncommutative spaces [9, 11, 14] is that, up to a universal factor, the restriction of u 1 to the diagonal gives the scalar”