In this paper we investigate the curvature of conformal deformations bynoncommutative Weyl factors of a flat metric on a noncommutative 2-torus, byanalyzing in the framework of spectral triples functionals associated toperturbed Dolbeault operators. The analogue of Gaussian curvature turns out tobe a sum of two functions in the modular operator corresponding to thenon-tracial weight defined by the conformal factor, applied to expressionsinvolving derivatives of the same factor. The first is a generating functionfor the Bernoulli numbers and is applied to the noncommutative Laplacian of theconformal factor, while the second is a two-variable function and is applied toa quadratic form in the first derivatives of the factor. Further outcomes ofthe paper include a variational proof of the Gauss-Bonnet theorem fornoncommutative 2-tori, the modular analogue of Polyakov's conformal anomalyformula for regularized determinants of Laplacians, a conceptual understandingof the modular curvature as gradient of the Ray-Singer analytic torsion, andthe proof using operator positivity that the scale invariant version of thelatter assumes its extreme value only at the flat metric.
展开▼
