## Modular curvature for noncommutative two-tori

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- by Alain Connes and Henri Moscovici
- J. Amer. Math. Soc.
**27**(2014), 639-684 - DOI: https://doi.org/10.1090/S0894-0347-2014-00793-1
- 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.## References

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## Bibliographic 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 - © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3194491