Aspects of the noncommutative torus

Abstract

In this thesis a class of finite real spectral triples for the geometry on a fuzzy torus is introduced. The geometries are shown to be related via an action of a general integral matrix. Each geometry is shown to have four real spectral triples corresponding to the four unique spin structures found on the 2-torus. The spectrum of the Dirac operator on each geometry, and spin structure, is calculated and shown to be the quantum integer analogues of the spectrum of the Dirac operator on the corresponding commutative 2-torus. The spectrum of the noncommutative Dirac operator is then shown to converge to the spectrum of the commutative Dirac operator as the algebra becomes commutative. Finally, an outline for the proof of a fuzzy torus converging to a commutative torus, via the defined Dirac operator, is presented