Curved noncommutative torus and Gauss--Bonnet

TL;DR

This paper investigates the geometry of a noncommutative 2-torus under perturbations, demonstrating the Gauss-Bonnet theorem holds up to second order and computing the initial terms of scalar curvature expansion.

Contribution

It provides a perturbative analysis of the noncommutative torus's geometry, confirming the Gauss-Bonnet theorem and deriving curvature expansion terms.

Findings

Zeta function at 0 vanishes up to second order perturbation

Gauss-Bonnet theorem holds for the perturbed noncommutative torus

First two terms of scalar curvature expansion are calculated

Abstract

We study perturbations of the flat geometry of the noncommutative two-dimensional torus T^2_\theta (with irrational \theta). They are described by spectral triples (A_\theta, \H, D), with the Dirac operator D, which is a differential operator with coefficients in the commutant of the (smooth) algebra A_\theta of T_\theta. We show, up to the second order in perturbation, that the zeta-function at 0 vanishes and so the Gauss-Bonnet theorem holds. We also calculate first two terms of the perturbative expansion of the corresponding local scalar curvature.