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

TL;DR

This paper constructs a universal Dirac operator on complex projective space, solves its eigenvalue problem, and explores its connection to extended supersymmetry, providing explicit formulas and generalizations to other spin coset manifolds.

Contribution

It introduces a universal spin$_c$ Dirac operator on $ ext{CP}^n$, linking it to supersymmetry and generalizing to other spin coset spaces.

Findings

Eigenvalues and eigenspinors explicitly computed.

Established equivalence with standard Dirac operator.

Connected R-symmetry to U(1) holonomy.

Abstract

We construct a universal spin$_c$ Dirac operator on $\mathbb{C}P^n$ built by projecting $su(n+1)$ left actions and prove its equivalence to the standard right action Dirac operator on $\mathbb{C}P^n$. 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 $\mathcal{N} =2$ supersymmetry algebra is realised through the Dirac operator and its companion supercharge, and an expression for the superpotential of any spin$_c$ connection on $\mathbb{C}P^n$ 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.