Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2008-11-19
SIGMA 5 (2009), 065, 22 pages
Physics
High Energy Physics
High Energy Physics - Theory
Contribution to the Proceedings of the Workshop "Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Huncti
Scientific paper
10.3842/SIGMA.2009.065
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.
Levin Andrey M.
Olshanetsky Mikhail A.
Zotov Andrei V.
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