Spectral geometry of functional metrics on noncommutative tori

TL;DR

This paper introduces and studies a new class of metrics called functional metrics on noncommutative tori, analyzing their spectral properties, scalar curvature, and a noncommutative Gauss-Bonnet theorem.

Contribution

It defines functional metrics on noncommutative tori, derives spectral invariants, and proves a Gauss-Bonnet type theorem for these metrics.

Findings

Explicit formulas for scalar curvature density in various dimensions.

Heat trace asymptotics for the new class of metrics.

A noncommutative Gauss-Bonnet theorem for two tori.

Abstract

We introduce a new family of metrics, called functional metrics, on noncommutative tori and study their spectral geometry. We define a class of Laplace type operators for these metrics and study their spectral invariants obtained from the heat trace asymptotics. A formula for the second density of the heat trace is obtained. In particular, the scalar curvature density and the total scalar curvature of functional metrics are explicitly computed in all dimensions for certain classes of metrics including conformally flat metrics and twisted product of flat metrics. Finally a Gauss-Bonnet type theorem for a noncommutative two torus equipped with a general functional metric is proved.