A universal Dirac operator and noncommutative spin bundles over fuzzy complex projective spaces

Abstract

We present a universal Dirac operator for noncommutative spin and spin^c bundles over fuzzy complex projective spaces. We give an explicit construction of these bundles, which are described in terms of finite dimensional matrices, calculate the spectrum and explicitly exhibit the Dirac eigenspinors. To our knowledge the spin^c spectrum for CP^n with n>=3 is new.

25 pages, typeset in LaTeX, uses youngtab package, v3 updates to published version and includes a minor correction in the conclusions

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• Dirac Operators on Quantum Projective Spaces
• Dimensional Reduction, Monopoles and Dynamical Symmetry Breaking
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• Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
• Dimensional Reduction and Vacuum Structure of Quiver Gauge Theory
• Instantons, Twistors, and Emergent Gravity
• On multimatrix models motivated by random Noncommutative Geometry I: the Functional Renormalization Group as a flow in the free algebra
• Higher-Dimensional Unification with continuous and fuzzy coset spaces as extra dimensions
• Scalar and Spinor Field Actions on Fuzzy $S^4$: fuzzy $C{P}^3$ as a $S_F^2$ bundle over $S_F^4$
• On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model
• Dirac operators for matrix algebras converging to coadjoint orbits
• Monopole harmonics on ${\mathbb{CP}}^{n-1}$
• Computing the spectral action for fuzzy geometries: from random noncommutative geometry to bi-tracial multimatrix models
• 'Schwinger Model' on the Fuzzy Sphere
• Noncommutative Spherically Symmetric Spaces
• Berezin Quantization From Ergodic Actions of Compact Quantum Groups, and Quantum Gromov-Hausdorff Distance
• A projective Dirac operator on CP^2 within fuzzy geometry
• Oscillator Calculus on Coadjoint Orbits and Index Theorems
• A projective Dirac operator on $\mathbb{C}{P}^n$ and extended SUSY
• The twisted Spin${}^c$ Dirac operator on Kähler submanifolds of the complex projective space
• Gravity Amplitudes from a Gaussian Matrix Model