A projective Dirac operator on $\mathbb{C}P^n$ and extended SUSY

Abstract

We construct a universal spin c Dirac operator on CP n built by projecting su(n + 1) left actions and prove its equivalence to the standard right action Dirac operator on CP n . 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 N = 2 supersymmetry algebra is realised through the Dirac operator and its companion supercharge, an expression for the superpotential of any spin c connection on CP n is found and generalised to any 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.