Topological aspects of $\mathbb{Z}/2\mathbb{Z}$ eigenfunctions for the Laplacian on $S^2$

TL;DR

This paper investigates how eigenfunctions and eigenvalues of the Laplacian on a sphere, acting on sections of a real line bundle with punctures, behave as functions on configuration spaces, revealing topological influences.

Contribution

It introduces a topological framework for understanding eigenfunctions of the Laplacian on punctured spheres, focusing on the effects of the bundle's structure and point configurations.

Findings

Eigenvalues vary with point configurations on the sphere.

Eigenfunctions exhibit topological dependence on the line bundle structure.

Configuration space analysis reveals new spectral properties.

Abstract

This paper concerns the behavior of the eigenfunctions and eigenvalues of the round sphere's Laplacian acting on the space of sections of a real line bundle which is defined on the complement of an even numbers of points in $S^2$. Of particular interest is how these eigenvalues and eigenvectors change when viewed as functions on the configuration spaces of points.