An $L^2$-index formula of monopoles with Dirac-type singularities

TL;DR

This paper establishes the Fredholm property and computes the $L^2$-indices of Dirac operators associated with monopoles having Dirac-type singularities on complete 3-dimensional Riemannian manifolds, advancing understanding of their analytical properties.

Contribution

It provides the first proof of Fredholmness and explicit $L^2$-index formulas for monopole Dirac operators with singularities on complete 3-manifolds.

Findings

Fredholmness of monopole Dirac operators established

Explicit $L^2$-index formulas derived

Enhanced understanding of monopole operator analysis

Abstract

We prove the Fredholmness of Dirac operators of monopoles with Dirac-type singularities on oriented complete Riemannian $3$-folds, and we also calculate the $L^2$-indices of them.