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A projective Dirac operator on $$\mathbb{CP}^2$$ within fuzzy geometry. (English) Zbl 1294.81065
Summary: We propose an ansatz for the commutative canonical spin$${}_{c}$$ Dirac operator on $$\mathbb{CP}^2$$ 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 $$\mathbb{CP}_F^2$$ 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.

##### MSC:
 81R60 Noncommutative geometry in quantum theory 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 14N05 Projective techniques in algebraic geometry 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) 81R05 Finite-dimensional groups and algebras motivated by physics and their representations 20C35 Applications of group representations to physics and other areas of science
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