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      Monopoles and Modifications of Bundles over Elliptic Curves

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          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.

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          Most cited references10

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          Stable bundles and integrable systems

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            The τ-function of the universal whitham hierarchy, matrix models and topological field theories

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              Monodromy- and spectrum-preserving deformations I

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                Author and article information

                Journal
                19 November 2008
                2009-06-25
                Article
                10.3842/SIGMA.2009.065
                0811.3056
                eed3882d-eba2-4130-8f6d-38b535694c03

                http://creativecommons.org/licenses/by-nc-sa/3.0/

                History
                Custom metadata
                ITEP-TH-54/08
                SIGMA 5 (2009), 065, 22 pages
                Contribution to the Proceedings of the Workshop "Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Hunctions" (Bonn, July, 2008)
                hep-th math.AG

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