Monopoles and Modifications of Bundles over Elliptic Curves
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 mo...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2009 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2009
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149154 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Monopoles and Modifications of Bundles over Elliptic Curves / A.M. Levin, M.A. Olshanetsky, A.V. Zotov // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 21 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862538107588444160 |
|---|---|
| author | Levin, A.M. Olshanetsky, M.A. Zotov, A.V. |
| author_facet | Levin, A.M. Olshanetsky, M.A. Zotov, A.V. |
| citation_txt | Monopoles and Modifications of Bundles over Elliptic Curves / A.M. Levin, M.A. Olshanetsky, A.V. Zotov // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 21 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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 theta-functions with characteristic.
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| first_indexed | 2025-11-24T11:44:40Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-149154 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-24T11:44:40Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Levin, A.M. Olshanetsky, M.A. Zotov, A.V. 2019-02-19T17:47:50Z 2019-02-19T17:47:50Z 2009 Monopoles and Modifications of Bundles over Elliptic Curves / A.M. Levin, M.A. Olshanetsky, A.V. Zotov // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 21 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 14H70; 14F05; 33E05; 37K20; 81R12 https://nasplib.isofts.kiev.ua/handle/123456789/149154 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 theta-functions with characteristic. This paper is a contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions” (July 21–25, 2008, MPIM, Bonn, Germany). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Monopoles and Modifications of Bundles over Elliptic Curves Article published earlier |
| spellingShingle | Monopoles and Modifications of Bundles over Elliptic Curves Levin, A.M. Olshanetsky, M.A. Zotov, A.V. |
| title | Monopoles and Modifications of Bundles over Elliptic Curves |
| title_full | Monopoles and Modifications of Bundles over Elliptic Curves |
| title_fullStr | Monopoles and Modifications of Bundles over Elliptic Curves |
| title_full_unstemmed | Monopoles and Modifications of Bundles over Elliptic Curves |
| title_short | Monopoles and Modifications of Bundles over Elliptic Curves |
| title_sort | monopoles and modifications of bundles over elliptic curves |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149154 |
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