A projective Dirac operator on CP^2 within fuzzy geometry

TL;DR

This paper constructs a Dirac operator on CP^2 suitable for fuzzy geometry, identifies spinors and spectrum, and verifies the correct SU(3) representation content, advancing noncommutative geometric models.

Contribution

It introduces a global geometric ansatz for the Dirac operator on CP^2 compatible with fuzzy geometry, including spinor and spectrum analysis.

Findings

Constructed a Dirac operator on CP^2 compatible with fuzzy geometry.

Identified physical spinors and computed the eigenspectrum.

Confirmed the correct SU(3) representation content of the spinor bundle.

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.