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
DOI:
https://doi.org/10.1090/S0894-0347-2014-00793-1
Published electronically:
April 8, 2014
MathSciNet review:
3194491
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Abstract | References | Similar Articles | Additional Information
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
MR Author ID:
51015
Email:
alain@connes.org
Henri Moscovici
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Email:
henri@math.ohio-state.edu
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.