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

Abstract

The intersection of the 10-dimensional fuzzy conifold $Y_F^{10}$ with $S_F^5\times S_F^5$ 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 $SU(3)/(U(1)\times U(1))\left(\equiv {\mathcal{M}}_F^6\right)$. 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.

arXiv comment14 pages

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Cited by in corpus (3)

• Matricial bridges for "Matrix algebras converge to the sphere"
• Vector bundles for "Matrix algebras converge to the sphere"
• Dirac operators for matrix algebras converging to coadjoint orbits