Finite spectral triples for the fuzzy torus

Abstract

Finite real spectral triples are defined to characterise the non-commutative geometry of a fuzzy torus. The geometries are the non-commutative analogues of flat tori with moduli determined by integer parameters. Each of these geometries has four different Dirac operators, corresponding to the four unique spin structures on a torus. The spectrum of the Dirac operator is calculated. It is given by replacing integers with their quantum integer analogues in the spectrum of the corresponding commutative torus.

53 pages, 6 figures. v2: more detail on the square of the Dirac operator; some eigenvalue plots removed. Journal accepted MS

References in corpus (6)

• Noncommutative Gauge Theories on Fuzzy Sphere and Fuzzy Torus from Matrix Model
• From Large N Matrices to the Noncommutative Torus
• On Spin Structures and Dirac Operators on the Noncommutative Torus
• Generalized Fuzzy Torus and its Modular Properties
• Spin geometry of the rational noncommutative torus
• Dependence on the spin structure of the Dirac spectrum

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• On multimatrix models motivated by random Noncommutative Geometry I: the Functional Renormalization Group as a flow in the free algebra
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• Batalin-Vilkovisky quantization of fuzzy field theories
• Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models
• BV quantization of dynamical fuzzy spectral triples
• Reconstructing manifolds from truncated spectral triples
• Dirac operators on noncommutative hypersurfaces