# Monopole harmonics on $\mathbb{CP}^{n-1}$

## TL;DR

This paper computes the spectra and eigenfunctions of quantum models describing a charged particle on complex projective spaces with monopole fields, revealing simple wave functions and connections to orthogonal polynomials and Dirac operators.

## Contribution

It provides explicit spectra, eigenfunctions, and their algebraic structure for quantum particles on $ ext{CP}^{n-1}$ with monopole backgrounds, including supersymmetric cases.

## Key findings

- Eigenfunctions expressed via homogeneous coordinates
- Degenerate $SU(n)$ multiplets observed
- Connection to multidimensional orthogonal polynomials

## Abstract

We find the spectra and eigenfunctions of both ordinary and supersymmetric quantum-mechanical models describing the motion of a charged particle over the $\mathbb{CP}^{n-1}$ manifold in the presence of a background monopole-like gauge field. The states form degenerate $SU(n)$ multiplets and their wave functions acquire a very simple form being expressed via homogeneous coordinates. Their relationship to multidimensional orthogonal polynomials of a special kind is discussed. By the well-known isomorphism between the twisted Dolbeault and Dirac complexes, our construction also gives the eigenfunctions and eigenvalues of the Dirac operator on complex projective spaces in a monopole background.