Deformed Noncommutative Tori

TL;DR

This paper constructs deformed noncommutative tori from matrix geometries, showing their C*-algebra completions are isomorphic to standard noncommutative tori and exploring their limits and modules.

Contribution

It introduces a new deformation framework linking fuzzy geometries to noncommutative tori, enabling concrete study of deformations and their modules.

Findings

Deformed algebras can be completed into standard noncommutative tori.

Fuzzy sphere and torus are limits of finite-dimensional representations.

Projective modules with constant curvature connections are described.

Abstract

We recall a construction of non-commutative algebras related to a one-parameter family of (deformed) spheres and tori, and show that in the case of tori, the *-algebras can be completed into C*-algebras isomorphic to the standard non-commutative torus. As the former was constructed in the context of matrix (or fuzzy) geometries, it provides an important link to the framework of non-commutative geometry, and opens up for a concrete way to study deformations of non-commutative tori. Furthermore, we show how the well-known fuzzy sphere and fuzzy torus can be obtained as formal scaling limits of finite-dimensional representations of the deformed algebras, and their projective modules are described together with connections of constant curvature.