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

TL;DR

This paper explores the realization of symplectic spinor fields and the symplectic Dirac operator within the framework of homogeneous projective geometry in two real dimensions, highlighting the role of the symmetry group SL(3,R).

Contribution

It introduces a new geometric framework for symplectic spinors and Dirac operators using the double cover of projective structures in two dimensions.

Findings

Realization of symplectic spinor fields in projective geometry

Identification of SL(3,R) as the symmetry group

Connection between symmetries and Dirac operator properties

Abstract

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 $\widetilde{\mathrm{SL}}(3,{\mathbb R})$.