Geometric Dirac operator on the fuzzy sphere

Abstract

We construct a Connes spectral triple or `Dirac operator' on the non-reduced fuzzy sphere $C_{\lambda }[S^2]$ as realised using quantum Riemannian geometry with a central quantum metric $g$ of Euclidean signature and its associated quantum Levi-Civita connection. The Dirac operator is characterised uniquely up to unitary equivalence within our quantum Riemannian geometric setting and an assumption that the spinor bundle is trivial and rank 2 with a central basis. The spectral triple has KO dimension 3 and in the case of the round metric, essentially recovers a previous proposal motivated by rotational symmetry.

16 pages AMSLATEX no figures; minor revision, added sample computation of a Connes spectral distance in the concluding remarks

References in corpus (4)

• A Lorentzian version of the non-commutative geometry of the standard model of particle physics
• Equivariant Lorentzian Spectral Triples
• Spin $j$ Dirac Operators on the Fuzzy 2-Sphere
• Fuzzy and discrete black hole models

Cited by in corpus (3)

• Dirac gauge theory for topological spinors in 3+1 dimensional networks
• Quantum geodesic flows and curvature
• From Noncommutative Geometry to Random Matrix Theory