The Moyal Sphere

TL;DR

This paper constructs a family of constant curvature metrics on the Moyal plane, including a noncommutative sphere, by conformal rescaling, and computes their Gauss-Bonnet terms, advancing noncommutative geometry understanding.

Contribution

It introduces a new family of metrics on the Moyal plane and explicitly computes their geometric invariants, linking noncommutative algebra to classical geometric structures.

Findings

Identification of a noncommutative sphere geometry

Explicit computation of Gauss-Bonnet terms for these metrics

Connection between conformal rescaling and noncommutative curvature

Abstract

We construct a family of constant curvature metrics on the Moyal plane and compute the Gauss-Bonnet term for each of them. They arise from the conformal rescaling of the metric in the orthonormal frame approach. We find a particular solution, which corresponds to the Fubini-Study metric and which equips the Moyal algebra with the geometry of a noncommutative sphere.