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