Modular Curvature for Noncommutative Two-Tori

TL;DR

This paper explores the concept of curvature in noncommutative geometry, specifically on noncommutative 2-tori, by analyzing spectral triples and conformal deformations, leading to new insights into geometric invariants and the Gauss-Bonnet theorem.

Contribution

It introduces a novel formulation of modular curvature for noncommutative 2-tori, including a variational proof of the Gauss-Bonnet theorem and a modular analogue of Polyakov's anomaly formula.

Findings

Derived a noncommutative analogue of Gaussian curvature as a sum of two functions.

Provided a variational proof of the Gauss-Bonnet theorem for noncommutative 2-tori.

Established a modular version of Polyakov's conformal anomaly formula.

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 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.