Generalized Fuzzy Torus and its Modular Properties

TL;DR

This paper introduces a generalized fuzzy torus with modular properties, analyzing its algebraic structure, spectral characteristics, and semi-classical limit to approximate classical tori with arbitrary modular parameters.

Contribution

It extends fuzzy torus models to include non-trivial modular parameters, providing new insights into their geometric and spectral properties.

Findings

The generalized fuzzy torus accurately reproduces classical torus geometry in the semi-classical limit.

The spectrum of the Laplacian and Dirac operator are computed explicitly.

The modular properties of the fuzzy torus are characterized and analyzed.

Abstract

We consider a generalization of the basic fuzzy torus to a fuzzy torus with non-trivial modular parameter, based on a finite matrix algebra. We discuss the modular properties of this fuzzy torus, and compute the matrix Laplacian for a scalar field. In the semi-classical limit, the generalized fuzzy torus can be used to approximate a generic commutative torus represented by two generic vectors in the complex plane, with generic modular parameter $\tau$. The effective classical geometry and the spectrum of the Laplacian are correctly reproduced in the limit. The spectrum of a matrix Dirac operator is also computed.