Geometric Dirac operator on the fuzzy sphere

Abstract

We construct a Connes spectral triple or ‘Dirac operator’ on the non-reduced fuzzy sphere mathbb {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.