Geometric Dirac operator on the fuzzy sphere

TL;DR

This paper constructs a Dirac operator on the fuzzy sphere within quantum Riemannian geometry, providing a spectral triple with KO dimension 3 that generalizes previous models.

Contribution

It introduces a new spectral triple on the fuzzy sphere using quantum Riemannian geometry, characterizing the Dirac operator uniquely under specific assumptions.

Findings

Spectral triple has KO dimension 3.

Recovers previous models in the round metric case.

Provides a quantum geometric framework for fuzzy spheres.

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.