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Inspired by the results on symmetries of the symplectic Dirac operator, we realize symplectic spinor fields and the symplectic Dirac operator in the framework of (the double cover of) homogeneous projective structure in two real dimensions. The symmetry group of the homogeneous model of the double cover of projective geometry in two real dimensions is $\widetilde{\mathrm{SL}}(3,{\mathbb R})$.

Projective structure, $\widetilde{\operatorname{SL}}(3,\mathbb{R})$ and the symplectic Dirac operator