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

Michael Finkelberg; Alexander Kuznetsov; Nikita Markarian; Ivan Mirković

doi:10.1007/s002200050560

paper

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

arXiv:math/9803124 · doi:10.1007/s002200050560

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 $C = P^{1} ∋ \infty$ be the projective line. Let $α \in H_{2} (X, Z)$ . The moduli space of $G$ -monopoles of topological charge $α$ (see e.g. [Jarvis]) is naturally identified with the space $M_{b} (X, α)$ of based maps from $(C, \infty)$ to $(X, B)$ of degree $α$ . The moduli space of $G$ -monopoles carries a natural hyperkähler structure, and hence a holomorphic symplectic structure. We propose a simple explicit formula for the symplectic structure on $M_{b} (X, α)$ . It generalizes the well known formula for $G = S L_{2}$ (see e.g. [Atiyah-Hitchin]). Let $P \supset B$ be a parabolic subgroup. The construction of the Poisson structure on $M_{b} (X, α)$ generalizes verbatim to the space of based maps $M = M_{b} (G / P, β)$ . In most cases the corresponding map $T^{*} M \to T M$ is not an isomorphism, i.e. $M$ splits into nontrivial symplectic leaves. These leaves are explicilty described.

v2: List of authors updated; v3: The formula for the symplectic form corrected; v4: Notations changed; v5: A few more corrections: final version

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

Abstract

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