## Abstract

Let G be a semisimple complex Lie group with a Borel subgroup B. Let X = G/B be the flag manifold of G. Let C = ℙ^{1} ∋ ∞ be the projective line. Let α ∈ H_{2}(X, ℤ). The moduli space of G-monopoles of topological charge α is naturally identified with the space script M sign _{b}(X, α) of based maps from (C, ∞) to (X, B) of degree α. The moduli space of G-monopoles carries a natural hyperkähler structure, and hence a holomorphic symplectic structure. It was explicitly computed by R. Bielawski in case G = SL_{n}. We propose a simple explicit formula for another natural symplectic structure on M_{b}(X, α). It is tantalizingly similar to R. Bielawski's formula, but in general (rank > 1) the two structures do not coincide. Let P ⊃ B be a parabolic subgroup. The construction of the Poisson structure on M_{b}(X, α) generalizes verbatim to the space of based maps script M sign = script M sign_{b}(G/P, β). In most cases the corresponding map script T sign* script M sign → script T sign script M sign is not an isomorphism, i.e. script M sign splits into nontrivial symplectic leaves. These leaves are explicilty described.

Original language | English |
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Pages (from-to) | 411-421 |

Number of pages | 11 |

Journal | Communications in Mathematical Physics |

Volume | 201 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1999 |

Externally published | Yes |