A projective Dirac operator on CP^2 within fuzzy geometry

Abstract

We propose an ansatz for the commutative canonical spin_c Dirac operator on CP^2 in a global geometric approach using the right invariant (left action-) induced vector fields from SU(3). This ansatz is suitable for noncommutative generalisation within the framework of fuzzy geometry. Along the way we identify the physical spinors and construct the canonical spin_c bundle in this formulation. The chirality operator is also given in two equivalent forms. Finally, using representation theory we obtain the eigenspinors and calculate the full spectrum. We use an argument from the fuzzy complex projective space CP^2_F based on the fuzzy analogue of the unprojected spin_c bundle to show that our commutative projected spin_c bundle has the correct SU(3)-representation content.

reduced to 27 pages, minor corrections, minor improvements, typos corrected

References in corpus (6)

• A universal Dirac operator and noncommutative spin bundles over fuzzy complex projective spaces
• Fuzzy Toric Geometries
• Noncommutative vector bundles over fuzzy CP^N and their covariant derivatives
• Spin $j$ Dirac Operators on the Fuzzy 2-Sphere
• Dirac operator on the q-deformed Fuzzy sphere and Its spectrum
• 'Schwinger Model' on the Fuzzy Sphere

Cited by in corpus (2)

• Scalar and Spinor Field Actions on Fuzzy $S^4$: fuzzy $C{P}^3$ as a $S_F^2$ bundle over $S_F^4$
• Dirac operators for matrix algebras converging to coadjoint orbits