Monopoles and Modifications of Bundles over Elliptic Curves

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.

arXiv commentContribution to the Proceedings of the Workshop "Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Hunctions" (Bonn, July, 2008)

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Cited by in corpus (8)

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• Characteristic Classes of SL(N)-Bundles and Quantum Dynamical Elliptic R-Matrices
• Instantons and Yang-Mills Flows on Coset Spaces
• Classification of Isomonodromy Problems on Elliptic Curves
• Characteristic Classes and Integrable Systems. General Construction
• Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles
• Monopole solutions to the Bogomolny equation as three-dimensional generalizations of the Kronecker series
• Gauge theories, Simple Groups and Integrable Systems