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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Modular curvature for noncommutative two-tori


Authors: Alain Connes and Henri Moscovici
Journal: J. Amer. Math. Soc. 27 (2014), 639-684
MSC (2010): Primary 46L87, 58B34, 81R60
Published electronically: April 8, 2014
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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 the 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.


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Additional Information

Alain Connes
Affiliation: Collége de France, 3, rue d’Ulm, Paris, F-75005 France – and – IHES, 91440 Bures-Sur-Yvette, France – and – The Ohio State University, Columbus, Ohio 43210
Email: alain@connes.org

Henri Moscovici
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Email: henri@math.ohio-state.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-2014-00793-1
PII: S 0894-0347(2014)00793-1
Received by editor(s): December 6, 2011
Received by editor(s) in revised form: October 22, 2013
Published electronically: April 8, 2014
Additional Notes: The work of the first author was partially supported by the National Science Foundation award no. DMS-0652164
The work of the second author was partially supported by the National Science Foundation award no. DMS-0969672
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.