Quasiclassical approximation for magnetic monopoles

TL;DR

This paper develops a quasiclassical approximation using the WKB method to analyze magnetic Laplacian eigenvalues on manifolds with non-exact magnetic fields, demonstrated on the Dirac monopole.

Contribution

It introduces a novel quasiclassical approach employing the Maslov canonical operator for magnetic Laplacians with non-exact fields on manifolds.

Findings

Approximate eigenvalues for magnetic Laplacian on manifolds with non-exact fields.

Application of the method to the Dirac magnetic monopole on a sphere.

Validation of the approximation in a specific geometric setting.

Abstract

A quasiclassical approximation is constructed to describe the eigenvalues of the magnetic Laplacian on a compact Riemannian manifold in the case when the magnetic field is not given by an exact 2-form. For this, the multidimensional WKB method in the form of Maslov canonical operator is applied. In this case, the canonical operator takes values in sections of a nontrivial line bundle. The constructed approximation is demonstrated for the Dirac magnetic monopole on the two-dimensional sphere.