Monopoles and Modifications of Bundles over Elliptic Curves

TL;DR

This paper explores modifications of bundles over elliptic curves, linking topological changes to characteristic classes, and introduces monopole solutions to establish connections with integrable systems and theta-functions.

Contribution

It provides a new description of bundle modifications via monopole solutions, including explicit Dirac monopole solutions and their functional equations, extending to non-Abelian cases.

Findings

Derived Dirac monopole solution on R × elliptic curve

Established functional equations generalizing Kronecker series

Defined Abelian and non-Abelian modifications using theta-functions

Abstract

Modifications of bundles over complex curves is an operation that allows one to construct a new bundle from a given one. Modifications can change a topological type of bundle. We describe the topological type in terms of the characteristic classes of the bundle. Being applied to the Higgs bundles modifications establish an equivalence between different classical integrable systems. Following Kapustin and Witten we define the modifications in terms of monopole solutions of the Bogomolny equation. We find the Dirac monopole solution in the case $R $\times$ (elliptic curve). This solution is a three-dimensional generalization of the Kronecker series. We give two representations for this solution and derive a functional equation for it generalizing the Kronecker results. We use it to define Abelian modifications for bundles of arbitrary rank. We also describe non-Abelian modifications in terms of theta-functions with characteristic.