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2011

Modular Curvature for Noncommutative Two-Tori

Connes, Alain, Moscovici, Henri

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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.

Modular Curvature for Noncommutative Two-Tori · Around

Built on

D. B. Ray, I. M. Singer, R R -torsion and the Laplacian on Riemannian manifolds
1971
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D. B. Ray, I. M. Singer, Analytic torsion for complex manifolds
1973
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A. Connes, C ∗ C^{*} -algèbres et géométrie différentielle
1980
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M. A. Rieffel, C ∗ C^{\ast} -algebras associated with irrational rotations
1981
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P. Gilkey, Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem
1984
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T. P. Branson, B. Ørsted, Conformal indices of Riemannian manifolds
1986
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A. Connes, P. B. Tretkoff The Gauss-Bonnet Theorem for the noncommutative two torus
Original
2011
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