A projective Dirac operator on mathbb{C}Pⁿ and extended SUSY

We construct a universal spin_c Dirac operator on mathbb{C}Pⁿ built by projecting su(n+1) left actions and prove its equivalence to the standard right action Dirac operator on mathbb{C}Pⁿ. The eigenvalue problem is solved and the spinor space constructed thereof, showing that the proposed Dirac operator is universal, changing only its domain for different spin_c structures. Explicit expressions for the chirality and the eigenspinors are also found and consistency with the index theorem is established. Also, the extended mathcal{N} =2 supersymmetry algebra is realised through the Dirac operator and its companion supercharge, and an expression for the superpotential of any spin_c connection on mathbb{C}Pⁿ is found and generalised to any spin coset manifold G/H with G,H compact, connected, and G semisimple. The R-symmetry of this superalgebra is found to be equivalent to the U(1) holonomy of the spin_c connection.