Monopoles, Dirac operator and index theory for fuzzy $SU(3)\mathrm{/}(U(1)\times U(1))$

The intersection of the 10-dimensional fuzzy conifold $Y_F^{10}$ with $S^5_F \times S^5_F$isthecompact8-dimensionalfuzzyspace$X_F^8$\mathrm{.}Weshowthat$X_F^8$is(theanalogueof)aprincipal$U(1)\times U(1)$bundleoverfuzzy${SU(3)}/({U(1) \times U(1)}) \left(\equiv$\mathcal{M}$^6_F\right)$\mathrm{.}Weconstruct$$\mathcal{M}$_F^6$ using the Gell-Mann matrices by adapting Schwinger’s construction. The space ${\mathcal{M}}_F^6$ is of relevance in higher dimensional quantum Hall effect and matrix models of $D$-branes. Further we show that the sections of the monopole bundle can be expressed in the basis of $SU(3)$ eigenvectors. We construct the Dirac operator on ${\mathcal{M}}_F^6$ from the Ginsparg-Wilson algebra on this space. Finally, we show that the index of the Dirac operator correctly reproduces the known results in the continuum.