Generalized Fuzzy Torus and its Modular Properties

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.

v2: discussion and references added; v3: published version

References in corpus (3)

• Emergent Geometry and Gravity from Matrix Models: an Introduction
• Dynamical generation of fuzzy extra dimensions, dimensional reduction and symmetry breaking
• Discrete curvature and the Gauss-Bonnet theorem

Cited by in corpus (9)

• On multimatrix models motivated by random Noncommutative Geometry I: the Functional Renormalization Group as a flow in the free algebra
• Scaling behaviour in random non-commutative geometries
• Finite spectral triples for the fuzzy torus
• On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model
• From Noncommutative Geometry to Random Matrix Theory
• The matrix regularization for Riemann surfaces with magnetic fluxes
• Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models
• Continuity of the Spectrum of Dirac Operators of Spectral Triples for the Spectral Propinquity
• Convergence of Spectral Triples on Fuzzy Tori to Spectral Triples on Quantum Tori