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

The intersection of the 10-dimensional fuzzy conifold $Y_F^{10}$ with $S^5_F \times S^5_F$ is the compact 8-dimensional fuzzy space $X_F^8$. We show that $X_F^8$ is (the analogue of) a principal $U(1)\times U(1)$ bundle over fuzzy ${SU(3)}/({U(1) \times U(1)}) \left(\equiv\mathcal{M}^6_F\right)$. We construct $\mathcal{M}_F^6$ 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.