Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 065, 22 pages      arXiv:0811.3056
Contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions”

Monopoles and Modifications of Bundles over Elliptic Curves

Andrey M. Levin a, b, Mikhail A. Olshanetsky a, c and Andrei V. Zotov a, c
a) Max Planck Institute of Mathematics, Bonn, Germany
b) Institute of Oceanology, Moscow, Russia
c) Institute of Theoretical and Experimental Physics, Moscow, Russia

Received November 20, 2008, in final form June 10, 2009; Published online June 25, 2009

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.

Key words: integrable systems; field theory; characteristic classes.

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