A note on a symplectic structure on the space of G-monopoles

Michael Finkelberg, Alexander Kuznetsov, Nikita Markarian, Ivan Mirković

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)


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 = ℙ1 ∋ ∞ be the projective line. Let α ∈ H2(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 = SLn. We propose a simple explicit formula for another natural symplectic structure on Mb(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 Mb(X, α) generalizes verbatim to the space of based maps script M sign = script M signb(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 languageEnglish
Pages (from-to)411-421
Number of pages11
JournalCommunications in Mathematical Physics
Issue number2
Publication statusPublished - 1999
Externally publishedYes


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