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Symmetry, Integrability and Geometry: Methods and Applications Monopoles and Modifications of Bundles over Elliptic Curves ⋆

By Andrey M. Levin, Mikhail A. Olshanetsky and Andrei V. Zotov

Abstract

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 × (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 thetafunctions with characteristic. Key words: integrable systems; field theory; characteristic classes 2000 Mathematics Subject Classification: 14H70; 14F05; 33E05; 37K20; 81R12

Year: 2010
OAI identifier: oai:CiteSeerX.psu:10.1.1.170.330
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